Differential Equations, Dynamical Systems, and an Introduction to Chaos

Hirsch FM9780123820105 2012131 1235 Page i 1 DIFFERENTIAL EQUATIONS DYNAMICAL SYSTEMS AND AN INTRODUCTION TO CHAOS Morris W Hirsch University of California Berkeley Stephen Smale University of California Berkeley Robert L Devaney Boston University AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO Academic Press is an imprint of Elsevier Hirsch FM9780123820105 2012211 1200 Page ii 2 Academic Press is an imprint of Elsevier 225 Wyman Street Waltham MA 02451 USA The Boulevard Langford Lane Kidlington Oxford OX5 1GB UK c 2013 Elsevier Inc All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying recording or any information storage and retrieval system without permission in writing from the publisher Details on how to seek permission further information about the Publishers permissions policies and our arrangements with organizations such 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as a matter of products liability negligence or otherwise or from any use or operation of any methods products instructions or ideas contained in the material herein Library of Congress CataloginginPublication Data Hirsch Morris W 1933 Differential equations dynamical systems and an introduction to chaos 3rd ed Morris W Hirsch Stephen Smale Robert L Devaney p cm ISBN 9780123820105 hardback 1 Differential equations 2 Algebras Linear 3 Chaotic behavior in systems I Smale Stephen 1930 II Devaney Robert L 1948 III Title QA372H67 2013 51535dc23 2012002951 British Library CataloguinginPublication Data A catalogue record for this book is available from the British Library For information on all Academic Press publications visit our Website at wwwelsevierdirectcom Printed in the United States 12 13 14 15 16 10 9 8 7 6 5 4 3 2 1 Hirsch PrefaceThirdEdition9780123820105 2012130 958 Page xi 1 Preface to Third Edition The main new features in this edition consist of a number of additional explo rations together with numerous proof simplifications and revisions The new explorations include a sojourn into numerical methods that highlights how these methods sometimes fail which in turn provides an early glimpse of chaotic behavior Another new exploration involves the previously treated SIR model of infectious diseases only now considered with zombies as the infected population A third new exploration involves explaining the motion of a glider This edition has benefited from numerous helpful comments from a variety of readers Special thanks are due to Jamil Gomes de Abreu Eric Adams Adam Leighton Tiennyu Ma Lluis Fernand Mello Bogdan Przeradzki Charles Pugh Hal Smith and Richard Venti for their valuable insights and corrections ix Hirsch 04prexixiv9780123820105 2012217 2301 Page xi 1 Preface In the thirty years since the publication of the first edition of this book much has changed in the field of mathematics known as dynamical systems In the early 1970s we had very little access to highspeed computers and computer graphics The word chaos had never been used in a mathematical setting Most of the interest in the theory of differential equations and dynamical systems was confined to a relatively small group of mathematicians Things have changed dramatically in the ensuing three decades Comput ers are everywhere and software packages that can be used to approximate solutions of differential equations and view the results graphically are widely available As a consequence the analysis of nonlinear systems of differential equations is much more accessible than it once was The discovery of com plicated dynamical systems such as the horseshoe map homoclinic tangles the Lorenz system and their mathematical analysis convinced scientists that simple stable motions such as equilibria or periodic solutions were not always the most important behavior of solutions of differential equations The beauty and relative accessibility of these chaotic phenomena motivated scientists and engineers in many disciplines to look more carefully at the important differen tial equations in their own fields In many cases they found chaotic behavior in these systems as well Now dynamical systems phenomena appear in virtually every area of sci ence from the oscillating BelousovZhabotinsky reaction in chemistry to the chaotic Chua circuit in electrical engineering from complicated motions in celestial mechanics to the bifurcations arising in ecological systems xi Hirsch 04prexixiv9780123820105 2012217 2301 Page xii 2 xii Preface As a consequence the audience for a text on differential equations and dynamical systems is considerably larger and more diverse than it was in the 1970s We have accordingly made several major structural changes to this book including 1 The treatment of linear algebra has been scaled back We have dispensed with the generalities involved with abstract vector spaces and normed lin ear spaces We no longer include a complete proof of the reduction of all n n matrices to canonical form Rather we deal primarily with matrices no larger than 4 4 2 We have included a detailed discussion of the chaotic behavior in the Lorenz attractor the Shilnikov system and the doublescroll attractor 3 Many new applications are included previous applications have been updated 4 There are now several chapters dealing with discrete dynamical systems 5 We deal primarily with systems that are C thereby simplifying many of the hypotheses of theorems This book consists of three main parts The first deals with linear systems of differential equations together with some firstorder nonlinear equations The second is the main part of the text here we concentrate on nonlinear systems primarily twodimensional as well as applications of these systems in a wide variety of fields Part three deals with higher dimensional systems Here we emphasize the types of chaotic behavior that do not occur in planar systems as well as the principal means of studying such behaviorthe reduction to a discrete dynamical system Writing a book for a diverse audience whose backgrounds vary greatly poses a significant challenge We view this one as a text for a second course in differ ential equations that is aimed not only at mathematicians but also at scientists and engineers who are seeking to develop sufficient mathematical skills to analyze the types of differential equations that arise in their disciplines Many who come to this book will have strong backgrounds in linear algebra and real analysis but others will have less exposure to these fields To make this text accessible to both groups we begin with a fairly gentle introduction to lowdimensional systems of differential equations Much of this will be a review for readers with a more thorough background in differential equations so we intersperse some new topics throughout the early part of the book for those readers For example the first chapter deals with firstorder equations We begin it with a discussion of linear differential equations and the logistic popula tion model topics that should be familiar to anyone who has a rudimentary acquaintance with differential equations Beyond this review we discuss the logistic model with harvesting both constant and periodic This allows us to introduce bifurcations at an early stage as well as to describe Poincare maps Hirsch 04prexixiv9780123820105 2012217 2301 Page xiii 3 Preface xiii and periodic solutions These are topics that are not usually found in elemen tary differential equations courses yet they are accessible to anyone with a background in multivariable calculus Of course readers with a limited back ground may wish to skip these specialized topics at first and concentrate on the more elementary material Chapters 2 through 6 deal with linear systems of differential equations Again we begin slowly with Chapters 2 and 3 dealing only with planar sys tems of differential equations and twodimensional linear algebra Chapters 5 and 6 introduce higher dimensional linear systems however our emphasis remains on three and fourdimensional systems rather than completely gen eral ndimensional systems even though many of the techniques we describe extend easily to higher dimensions The core of the book lies in the second part Here we turn our atten tion to nonlinear systems Unlike linear systems nonlinear systems present some serious theoretical difficulties such as existence and uniqueness of solu tions dependence of solutions on initial conditions and parameters and the like Rather than plunge immediately into these difficult theoretical questions which require a solid background in real analysis we simply state the impor tant results in Chapter 7 and present a collection of examples that illustrate what these theorems say and do not say Proofs of all of the results are included in the final chapter of the book In the first few chapters in the nonlinear part of the book we introduce important techniques such as linearization near equilibria nullcline analysis stability properties limit sets and bifurcation theory In the latter half of this part we apply these ideas to a variety of systems that arise in biology electrical engineering mechanics and other fields Many of the chapters conclude with a section called Exploration These sections consist of a series of questions and numerical investigations dealing with a particular topic or application relevant to the preceding material In each Exploration we give a brief introduction to the topic at hand and provide references for further reading about this subject But we leave it to the reader to tackle the behavior of the resulting system using the material presented ear lier We often provide a series of introductory problems as well as hints as to how to proceed but in many cases a full analysis of the system could become a major research project You will not find answers in the back of the book for the questions in many cases nobody knows the complete answer Except of course you The final part of the book is devoted to the complicated nonlinear behav ior of higher dimensional systems known as chaotic behavior We introduce these ideas via the famous Lorenz system of differential equations As is often the case in dimensions three and higher we reduce the problem of com prehending the complicated behavior of this differential equation to that of understanding the dynamics of a discrete dynamical system or iterated Hirsch 04prexixiv9780123820105 2012217 2301 Page xiv 4 xiv Preface function So we then take a detour into the world of discrete systems dis cussing along the way how symbolic dynamics can be used to describe certain chaotic systems completely We then return to nonlinear differential equations to apply these techniques to other chaotic systems including those that arise when homoclinic orbits are present We maintain a website at mathbueduhsd devoted to issues regarding this text Look here for errata suggestions and other topics of interest to teachers and students of differential equations We welcome any contributions from readers at this site Hirsch Ch019780123820105 201222 1044 Page 1 1 1 FirstOrder Equations The purpose of this chapter is to develop some elementary yet important examples of firstorder differential equations The examples here illustrate some of the basic ideas in the theory of ordinary differential equations in the simplest possible setting We anticipate that the first few examples will be familiar to readers who have taken an introductory course in differential equations Later examples such as the logistic model with harvesting are included to give the reader a taste of certain topics eg bifurcations periodic solutions and Poincare maps that we will return to often throughout this book In later chapters our treatment of these topics will be much more systematic 11 The Simplest Example The differential equation familiar to all calculus students dx dt ax is the simplest It is also one of the most important First what does it mean Here x xt is an unknown realvalued function of a real variable t and dxdt is its derivative we will also use x or xt for the derivative In addi tion a is a parameter for each value of a we have a different differential Differential Equations Dynamical Systems and an Introduction to Chaos DOI 101016B9780123820105000014 c 2013 Elsevier Inc All rights reserved 1 2 Chapter 1 FirstOrder Equations equation The equation tells us that for every value of t the relationship xt axt is true The solutions of this equation are obtained from calculus if k is any real number then the function xt ke is a solution since xt ake axt Moreover there are no other solutions To see this let ut be any solution and compute the derivative of ute d at 174 pat at a uthe uthe utae aute aute 0 Therefore ute is a constant k so ut ke This proves our assertion Thus we have found all possible solutions of this differential equation We call the collection of all solutions of a differential equation the general solution of the equation The constant k appearing in this solution is completely determined if the value uo of a solution at a single point f is specified Suppose that a function xt satisfying the differential equation is also required to satisfy xt uo Then we must have ke up so that k upe Thus we have determined k and this equation therefore has a unique solution satisfying the specified initial condition xt up For simplicity we often take f 0 then k up There is no loss of generality in taking 0 for if ut is a solution with u0 uo then the function vt ut 9 is a solution with vt up It is common to restate this in the form of an initial value problem x ax x0 U A solution xt of an initial value problem must not only solve the differential equation but must also take on the prescribed initial value up at t 0 Note that there is a special solution of this differential equation when k 0 This is the constant solution xt 0 A constant solution like this is called an equilibrium solution or equilibrium point for the equation Equilibria are often among the most important solutions of differential equations The constant a in the equation x ax can be considered as a parameter If a changes the equation changes and so do the solutions Can we describe qualitatively the way the solutions change The sign of a is crucial here 1 Ifa0 lim ke equals co when k 0 and equals oo when k 0 Hirsch Ch019780123820105 201222 1044 Page 3 3 11 The Simplest Example 3 2 If a 0 keat constant 3 If a 0 limtkeat 0 The qualitative behavior of solutions is vividly illustrated by sketching the graphs of solutions as in Figure 11 Note that the behavior of solutions is quite different when a is positive and negative When a0 all nonzero solutions tend away from the equilibrium point at 0 as t increases whereas when a0 solutions tend toward the equi librium point We say that the equilibrium point is a source when nearby solutions tend away from it The equilibrium point is a sink when nearby solutions tend toward it We also describe solutions by drawing them on the phase line As the solu tion xt is a function of time we may view xt as a particle moving along the real line At the equilibrium point the particle remains at rest indicated by a solid dot while any other solution moves up or down the xaxis as indicated by the arrows in Figure 12 The equation x ax is stable in a certain sense if a 0 More precisely if a is replaced by another constant b with a sign that is the same as a then x t Figure 11 The solution graphs and phase line for x ax for a 0 Each graph represents a particular solution t x Figure 12 The solution graphs and phase line for x ax for a 0 4 Chapter 1 FirstOrder Equations the qualitative behavior of the solutions does not change But if a 0 the slightest change in a leads to a radical change in the behavior of solutions We therefore say that we have a bifurcation at a 0 in the oneparameter fam ily of equations x ax The concept of a bifurcation is one that will arise over and over in subsequent chapters of this book 12 The Logistic Population Model The differential equation x ax can be considered as a simplistic model of population growth when a 0 The quantity xt measures the population of some species at time t The assumption that leads to the differential equa tion is that the rate of growth of the population namely dxdt is directly proportional to the size of the population Of course this naive assumption omits many circumstances that govern actual population growth including for example the fact that actual populations cannot increase without bound To take this restriction into account we can make the following further assumptions about the population model 1 If the population is small the growth rate remains directly proportional to the size of the population 2 If the population grows too large however the growth rate becomes negative One differential equation that satisfies these assumptions is the logistic popu lation growth model This differential equation is x ax 1 N Here a and N are positive parameters a gives the rate of population growth when x is small while N represents a sort of ideal population or carrying capacity Note that if x is small the differential equation is essentially x ax since the term 1 xN 1 but if x N then x 0 Thus this simple equation satisfies the preceding assumptions We should add here that there are many other differential equations that correspond to these assumptions our choice is perhaps the simplest Without loss of generality we will assume that N 1 That is we will choose units so that the carrying capacity is exactly 1 unit of population and xt therefore represents the fraction of the ideal population present at time ft Therefore the logistic equation reduces to x fyx ax1 x 12 The Logistic Population Model 5 This is an example of a firstorder autonomous nonlinear differential equation It is first order since only the first derivative of x appears in the equation It is autonomous since the right side of the equation depends on x alone not on time f Plus it is nonlinear since fx is a nonlinear func tion of x The previous example x ax is a firstorder autonomous linear differential equation The solution of the logistic differential equation is easily found by the tried andtrue calculus method of separation and integration dx fy fon x1 x The method of partial fractions allows us to rewrite the left integral as 1 1 3 dx x I1x Integrating both sides and then solving for x yields Ket xt 2 1 Ke where K is the arbitrary constant that arises from integration Evaluating this expression at t 0 and solving for K gives x0 Ka 20 1x0 Using this we may rewrite this solution as x0e 1x0 x0e So this solution is valid for any initial population x0 When x0 1 we have an equilibrium solution since xt reduces to xt 1 Similarly xt 0 is an equilibrium solution Thus we have existence of solutions for the logistic differential equation We have no guarantee that these are all of the solutions of this equation at this stage we will return to this issue when we discuss the existence and uniqueness problem for differential equations in Chapter 7 To get a qualitative feeling for the behavior of solutions we sketch the slope field for this equation The right side of the differential equation determines the slope of the graph of any solution at each time t Thus we may plot little slope lines in the txplane as in Figure 13 with the slope of the line at tf x Hirsch Ch019780123820105 201222 1044 Page 6 6 6 Chapter 1 FirstOrder Equations x t x 1 x 0 Figure 13 Slope field solution graphs and phase line for x ax1 x 08 05 1 Figure 14 The graph of the function fx ax1 x with a 32 given by the quantity ax1 x Our solutions must therefore have graphs that are tangent to this slope field everywhere From these graphs we see immediately that in agreement with our assumptions all solutions for which x0 0 tend to the ideal population xt 1 For x0 0 solutions tend to although these solutions are irrelevant in the context of a population model Note that we can also read this behavior from the graph of the function fax ax1 x This graph displayed in Figure 14 crosses the xaxis at the two points x 0 and x 1 so these represent our equilibrium points When 0 x 1 we have f x 0 Therefore slopes are positive at any tx with 0 x 1 so solutions must increase in this region When x 0 or x 1 we have f x 0 so solutions must decrease as we see in both the solution graphs and the phase lines in Figure 13 We may read off the fact that x 0 is a source and x 1 is a sink from the graph of f in similar fashion Near 0 we have f x 0 if x 0 so slopes are positive and solutions increase but if x 0 then f x 0 so slopes are negative and solutions decrease Thus nearby solutions move away from 0 so 0 is a source Similarly 1 is a sink Hirsch Ch019780123820105 201222 1044 Page 7 7 13 Constant Harvesting and Bifurcations 7 x x 1 x 1 x 0 t Figure 15 Slope field solution graphs and phase line for x x x3 We may also determine this information analytically We have f ax a 2ax so that f a0 a 0 and f a1 a 0 Since f a0 0 slopes must increase through the value 0 as x passes through 0 That is slopes are negative below x 0 and positive above x 0 Thus solutions must tend away from x 0 In similar fashion f a1 0 forces solutions to tend toward x 1 making this equilibrium point a sink We will encounter many such derivative tests like this that predict the qualitative behavior near equilibria in subsequent chapters Example As a further illustration of these qualitative ideas consider the differential equation x gx x x3 There are three equilibrium points at x 0 1 Since gx 1 3x2 we have g0 1 so the equilibrium point 0 is a source Also g1 2 so the equilibrium points at 1 are both sinks Between these equilibria the sign of the slope field of this equation is nonzero From this information we can immediately display the phase line which is shown in Figure 15 13 Constant Harvesting and Bifurcations Now lets modify the logistic model to take into account harvesting of the pop ulation Suppose that the population obeys the logistic assumptions with the parameter a 1 but it is also harvested at the constant rate h The differential equation becomes x x1 x h where h 0 is a new parameter Hirsch Ch019780123820105 201222 1044 Page 8 8 8 Chapter 1 FirstOrder Equations x 05 h 14 h 14 h 14 fhx Figure 16 The graphs of the function fhx x1 x h Rather than solving this equation explicitly which can be donesee Exer cise 6 of this chapter we use the graph of the function fhx x1 x h to read off the qualitative behavior of solutions In Figure 16 we display the graph of fh in three different cases 0 h 14 h 14 and h 14 It is straightforward to check that fh has two roots when 0 h 14 one root when h 14 and no roots if h 14 as illustrated in the graphs As a consequence the differential equation has two equilibrium points xℓ and xr with 0 xℓ xr when 0 h 14 It is also easy to check that f hxℓ 0 so that xℓ is a source and f hxr 0 so that xr is a sink As h passes through h 14 we encounter another example of a bifurca tion The two equilibria xℓ and xr coalesce as h increases through 14 and then disappear when h 14 Moreover when h 14 we have fhx 0 for all x Mathematically this means that all solutions of the differential equation decrease to as time goes on We record this visually in the bifurcation diagram In Figure 17 we plot the parameter h horizontally Over each hvalue we plot the corresponding phase line The curve in this picture represents the equilibrium points for each value of h This gives another view of the sink and source merging into a single equilibrium point and then disappearing as h passes through 14 Ecologically this bifurcation corresponds to a disaster for the species under study For rates of harvesting 14 or lower the population persists provided the initial population is sufficiently large x0 xℓ But a very small change in the rate of harvesting when h 14 leads to a major change in the fate of the population at any rate of harvesting h 14 the species becomes extinct This phenomenon highlights the importance of detecting bifurcations in families of differential equationsa procedure that we will encounter many times in later chapters We should also mention that despite the simplicity of Hirsch Ch019780123820105 201222 1044 Page 9 9 13 Constant Harvesting and Bifurcations 9 x 14 h Figure 17 The bifurcation diagram for fhx x1 x h x x a a Figure 18 The bifurcation diagram for x x2 ax this population model the prediction that small changes in harvesting rates can lead to disastrous changes in population has been observed many times in real situations on earth Example As another example of a bifurcation consider the family of differ ential equations x gax x2 ax xx a which depends on a parameter a The equilibrium points are given by x 0 and x a We compute that g a0 a so 0 is a sink if a 0 and a source if a 0 Similarly g aa a so x a is a sink if a 0 and a source if a 0 We have a bifurcation at a 0 since there is only one equilibrium point when a 0 Moreover the equilibrium point at 0 changes from a source to a sink as a increases through 0 Similarly the equilibrium at x a changes from a sink to a source as a passes through 0 The bifurcation diagram for this family is shown in Figure 18 Hirsch Ch019780123820105 201222 1044 Page 10 10 10 Chapter 1 FirstOrder Equations 14 Periodic Harvesting and Periodic Solutions Now lets change our assumptions on the logistic model to reflect the fact that harvesting does not always occur at a constant rate For example populations of many species of fish are harvested at a higher rate in warmer months than in colder months So we assume that the population is harvested at a periodic rate One such model is then x f tx ax1 x h1 sin2πt where again a and h are positive parameters Thus the harvesting reaches a maximum rate 2h at time t 1 4 n where n is an integer representing the year and the harvesting reaches its minimum value 0 when t 3 4 n exactly one half year later Note that this differential equation now depends explicitly on time this is an example of a nonautonomous differential equation As in the autonomous case a solution xt of this equation must satisfy xt f txt for all t Also this differential equation is no longer separable so we cannot generate an analytic formula for its solution using the usual methods from calculus Thus we are forced to take a more qualitative approach see Figure 19 To describe the fate of the population in this case we first note that the right side of the differential equation is periodic with period 1 in the time variable that is f t 1x f tx This fact simplifies the problem of find ing solutions somewhat Suppose that we know the solution of all initial value problems not for all times but only for 0 t 1 Then in fact we know the solutions for all time For example suppose x1t is the solution that is defined for 0 t 1 and satisfies x10 x0 Suppose that x2t is the solution that satisfies x20 Figure 19 The slope field for fx x1 x h1 sin2πt Hirsch Ch019780123820105 201222 1044 Page 11 11 15 Computing the Poincare Map 11 x11 Then we can extend the solution x1 by defining x1t 1 x2t for 0 t 1 The extended function is a solution since we have x 1t 1 x 2t f tx2t f t 1x1t 1 Thus if we know the behavior of all solutions in the interval 0 t 1 then we can extrapolate in similar fashion to all time intervals and thereby know the behavior of solutions for all time Second suppose that we know the value at time t 1 of the solution satisfy ing any initial condition x0 x0 Then to each such initial condition x0 we can associate the value x1 of the solution xt that satisfies x0 x0 This gives us a function px0 x1 If we compose this function with itself we derive the value of the solution through x0 at time 2 that is ppx0 x2 If we compose this function with itself n times then we can compute the value of the solution curve at time n and hence we know the fate of the solution curve The function p is called a Poincare map for this differential equation Having such a function allows us to move from the realm of continuous dynami cal systems differential equations to the often easiertounderstand realm of discrete dynamical systems iterated functions For example suppose that we know that px0 x0 for some initial condition x0 that is x0 is a fixed point for the function p Then from our previous observations we know that xn x0 for each integer n Moreover for each time t with 0 t 1 we also have xt xt 1 and thus xt n xt for each integer n That is the solution satisfying the initial condition x0 x0 is a periodic function of t with period 1 Such solutions are called periodic solutions of the differential equation In Figure 110 we have displayed several solutions of the logistic equation with periodic harvesting Note that the solution satisfying the initial condi tion x0 x0 is a periodic solution and we have x0 px0 ppx0 Similarly the solution satisfying the initial condition x0 ˆx0 also appears to be a periodic solution so we should have pˆx0 ˆx0 Unfortunately it is usually the case that computing a Poincare map for a differential equation is impossible but for the logistic equation with periodic harvesting we get lucky 15 Computing the Poincare Map Before computing the Poincare map for this equation we need to introduce some important terminology To emphasize the dependence of a solution on Hirsch Ch019780123820105 201222 1044 Page 12 12 12 Chapter 1 FirstOrder Equations t 0 x0 x1 px0 x2 ppx0 x0 t 1 t 2 Figure 110 The Poincare map for x 5x1 x 081 sin2πt the initial value x0 we will denote the corresponding solution by φtx0 This function φ R R R is called the flow associated with the differential equation If we hold the variable x0 fixed then the function t φtx0 is just an alternative expression for the solution of the differential equation satisfying the initial condition x0 Sometimes we write this function as φtx0 Example For our first example x ax the flow is given by φtx0 x0eat For the logistic equation without harvesting the flow is φtx0 x0eat 1 x0 x0eat Now we return to the logistic differential equation with periodic harvesting x f tx ax1 x h1 sin2πt The solution that satisfies the initial condition x0 x0 is given by t φtx0 Although we do not have a formula for this expression we do 15 Computing the Poincaré Map 13 know that by the Fundamental Theorem of Calculus this solution satisfies t pt x0 0 fs ds 0 since 0 ta ft660 and 0x9 xo If we differentiate this solution with respect to xo using the Chain Rule we obtain 3 pa ag som 1t f Zisoomy 2 smds ax0 OX0 0x0 0 Now let 0 at 2 tx OX0 Note that 0 20 22 09 1 dX0 Differentiating z with respect to t we find 0 0 Zt oct 22 20 0X0 0X0 0 F 4 tx9 0 dX0 Again we do not know t xo explicitly but this equation does tell us that zt solves the differential equation a Zth F 46 tx9 20 OX0 14 Chapter 1 FirstOrder Equations with z0 1 Consequently via separation of variables we may compute that the solution of this equation is t af 2t exp A Sb s x0 ds Oxo 0 and so we find ag of a d xo exp s s xo ds 0X0 0x0 0 Since pxo 1x0 we have determined the derivative px9 of the Poin caré map note that pxo 0 Therefore p is an increasing function Differentiating once more we find ai P af Pp xo p 20 SG 50 exp a ubuxo du ds IX99Xo dXx0 0 0 which looks pretty intimidating However since ftx0 axo1 x h sin2z7t we have a2 of 2a Ox90X0 Thus we know in addition that px9 0 Consequently the graph of the Poincaré map is concave down This implies that the graph of p can cross the diagonal line y x at most two times that is there can be at most two values of x for which px x Therefore the Poincaré map has at most two fixed points These fixed points yield periodic solutions of the original differential equation These are solutions that satisfy xt 1 xt for all t Another way to say this is that the flow t x9 is a periodic function in t with period 1 when the initial condition xo is one of the fixed points We saw these two solutions in the particular case when h 08 in Figure 110 In Figure 111 we again see two solutions that appear to be periodic Note that one of these appears to attract all nearby solutions while the other appears to repel them Well return to these concepts often and make them more precise later in the book Hirsch Ch019780123820105 201222 1044 Page 15 15 16 Exploration A TwoParameter Family 15 1 1 2 3 4 5 Figure 111 Several solutions of x 5x1 x 081 sin2πt Recall that the differential equation also depends on the harvesting param eter h For small values of h there will be two fixed points such as shown in Figure 111 Differentiating f with respect to h we find f htx0 1 sin2πt Thus f h 0 except when t 34 This implies that the slopes of the slope field lines at each point tx0 decrease as h increases As a consequence the values of the Poincare map also decrease as h increases There is a unique value h therefore for which the Poincare map has exactly one fixed point For h h there are no fixed points for p so px0 x0 for all initial values It then follows that the population again dies out 16 Exploration A TwoParameter Family Consider the family of differential equations x fabx ax x3 b which depends on two parameters a and b The goal of this exploration is to combine all of the ideas in this chapter to put together a complete picture of the twodimensional parameter plane the abplane for this differential equation Feel free to use a computer to experiment with this differential Hirsch Ch019780123820105 201222 1044 Page 16 16 16 Chapter 1 FirstOrder Equations equation at first but then try the following to verify your observations rigorously 1 First fix a 1 Use the graph of f1b to construct the bifurcation diagram for this family of differential equations depending on b 2 Repeat the previous question for a 0 and then for a 1 3 What does the bifurcation diagram look like for other values of a 4 Now fix b and use the graph to construct the bifurcation diagram for this family which this time depends on a 5 In the abplane sketch the regions where the corresponding differential equation has different numbers of equilibrium points including a sketch of the boundary between these regions 6 Describe using phase lines and the graph of fabx the bifurcations that occur as the parameters pass through this boundary 7 Describe in detail the bifurcations that occur at a b 0 as a andor b vary 8 Consider the differential equation x x x3 bsin2πt where b is small What can you say about solutions of this equation Are there any periodic solutions 9 Experimentally what happens as b increases Do you observe any bifurcations Explain what you observe E X E R C I S E S 1 Find the general solution of the differential equation x ax 3 where a is a parameter What are the equilibrium points for this equation For which values of a are the equilibria sinks For which are they sources 2 For each of the following differential equations find all equilibrium solu tions and determine whether they are sinks sources or neither Also sketch the phase line a x x3 3x b x x4 x2 c x cosx d x sin2 x e x 1 x2 3 Each of the following families of differential equations depends on a parameter a Sketch the corresponding bifurcation diagrams a x x2 ax b x x3 ax c x x3 x a Exercises 17 fx x b Figure 112 Graph of the function f 4 Consider the function fx with a graph that is displayed in Figure 112 a Sketch the phase line corresponding to the differential equation x fx b Let gqx fx a Sketch the bifurcation diagram corresponding to the family of differential equations x gax c Describe the different bifurcations that occur in this family 5 Consider the family of differential equations x axsinx where a is a parameter a Sketch the phase line when a 0 b Use the graphs of ax and sin x to determine the qualitative behavior of all of the bifurcations that occur as a increases from 1 to 1 c Sketch the bifurcation diagram for this family of differential equations 6 Find the general solution of the logistic differential equation with con stant harvesting x xl1xh for all values of the parameter h 0 7 Consider the nonautonomous differential equation v x4 ift5 2x ift5 a Find a solution of this equation satisfying x0 4 Describe the qualitative behavior of this solution Hirsch Ch019780123820105 201222 1044 Page 18 18 18 Chapter 1 FirstOrder Equations b Find a solution of this equation satisfying x0 3 Describe the qualitative behavior of this solution c Describe the qualitative behavior of any solution of this system as t 8 Consider a firstorder linear equation of the form x ax f t where a R Let yt be any solution of this equation Prove that the general solution is yt c expat where c R is arbitrary 9 Consider a firstorder linear nonautonomous equation of the form xt atx a Find a formula involving integrals for the solution of this system b Prove that your formula gives the general solution of this system 10 Consider the differential equation x x cost a Find the general solution of this equation b Prove that there is a unique periodic solution for this equation c Compute the Poincare map p t 0 t 2π for this equation and use this to verify again that there is a unique periodic solution 11 Firstorder differential equations need not have solutions that are defined for all time a Find the general solution of the equation x x2 b Discuss the domains over which each solution is defined c Give an example of a differential equation for which the solution satisfying x0 0 is defined only for 1 t 1 12 Firstorder differential equations need not have unique solutions satisfy ing a given initial condition a Prove that there are infinitely many different solutions of the differ ential equations x x13 satisfying x0 0 b Discuss the corresponding situation that occurs for x xt x0x0 c Discuss the situation that occurs for x xt2 x0 0 13 Let x f x be an autonomous firstorder differential equation with an equilibrium point at x0 a Suppose f x0 0 What can you say about the behavior of solu tions near x0 Give examples b Suppose f x0 0 and f x0 0 What can you say now c Suppose f x0 f x0 0 but f x0 0 What can you say now Exercises 19 14 Consider the firstorder nonautonomous equation x ptx where pt is differentiable and periodic with period T Prove that all solutions of this equation are periodic with period T if and only if T ps ds 0 0 15 Consider the differential equation x f tx where ft x is continu ously differentiable in t and x Suppose that ftTx ftx for all t Suppose there are constants p q such that ftp0 ftq 0 for all t Prove that there is a periodic solution xt for this equation with p x0 q 16 Consider the differential equation x x 1 cost What can be said about the existence of periodic solutions for this equation Hirsch Ch029780123820105 201222 1059 Page 21 1 2 Planar Linear Systems In this chapter we begin the study of systems of differential equations A system of differential equations is a collection of n interrelated differential equations of the form x 1 f1tx1x2xn x 2 f2tx1x2xn x n fntx1x2xn Here the functions fj are realvalued functions of the n 1 variables x1x2 xn and t Unless otherwise specified we will always assume that the fj are C functions This means that the partial derivatives of all orders of the fj exist and are continuous To simplify notation we will use vector notation X x1 xn We often write the vector X as x1xn to save space Differential Equations Dynamical Systems and an Introduction to Chaos DOI 101016B9780123820105000026 c 2013 Elsevier Inc All rights reserved 21 Hirsch Ch029780123820105 201222 1059 Page 22 2 22 Chapter 2 Planar Linear Systems Our system may then be written more concisely as X FtX where FtX f1tx1xn fntx1xn A solution of this system is therefore a function of the form Xt x1t xnt that satisfies the equation so that Xt FtXt where Xt x 1tx nt Of course at this stage we have no guarantee that there is such a solution but we will begin to discuss this complicated question in Section 27 The system of equations is called autonomous if none of the fj depends on t so the system becomes X FX For most of the rest of this book we will be concerned with autonomous systems In analogy with firstorder differential equations a vector X0 for which FX0 0 is called an equilibrium point for the system An equilibrium point corresponds to a constant solution Xt X0 of the system as before Just to set some notation once and for all we will always denote real variables by lowercase letters such as xyx1x2t and so forth Realvalued functions will also be written in lowercase such as f xy or f1x1xnt We will reserve capital letters for vectors such as X x1xn or for vectorvalued functions such as Fxy f xygxy or Hx1xn h1x1xn hnx1xn We will denote ndimensional Euclidean space by Rn so that Rn consists of all vectors of the form X x1xn Hirsch Ch029780123820105 201222 1059 Page 23 3 21 SecondOrder Differential Equations 23 21 SecondOrder Differential Equations Many of the most important differential equations encountered in science and engineering are secondorder differential equations These are differential equations of the form x f txx Important examples of secondorder equations include Newtons equation mx f x the equation for an RLC circuit in electrical engineering LCx RCx x vt and the mainstay of most elementary differential equations courses the forced harmonic oscillator mx bx kx f t We discuss these and more complicated relatives of these equations at length as we go along First however we note that these equations are a special sub class of twodimensional systems of differential equations that are defined by simply introducing a second variable y x For example consider a secondorder constant coefficient equation of the form x ax bx 0 If we let y x then we may rewrite this equation as a system of firstorder equations x y y bx ay Any secondorder equation may be handled similarly Thus for the remainder of this book we will deal primarily with systems of equations Hirsch Ch029780123820105 201222 1059 Page 24 4 24 Chapter 2 Planar Linear Systems 22 Planar Systems In this chapter we will deal with autonomous systems in R2 which we will write in the form x f xy y gxy thus eliminating the annoying subscripts on the functions and variables As before we often use the abbreviated notation X FX where X xy and FX Fxy f xygxy In analogy with the slope fields of Chapter 1 we regard the right side of this equation as defining a vector field on R2 That is we think of Fxy as representing a vector with x and ycomponents that are f xy and gxy respectively We visualize this vector as being based at the point xy For example the vector field associated with the system x y y x is displayed in Figure 21 Note that in this case many of the vectors overlap making the pattern difficult to visualize For this reason we always draw a direction field instead which consists of scaled versions of the vectors A solution of this system should now be thought of as a parametrized curve in the plane of the form xtyt such that for each t the tangent vector at the point xtyt is Fxtyt That is the solution curve xtyt winds its way through the plane always tangent to the given vector Fxtyt based at xtyt Figure 21 Vector field direction field and several solutions for the system x yy x 22 Planar Systems 25 Example The curve xt asint yt a cost for any a R is a solution of the system x y yu7x since xt acost yt yt asint xt as required by the differential equation These curves define circles of radius a in the plane which are traversed in the clockwise direction as tf increases When a 0 the solutions are the constant functions xt 0 yt a Note that this example is equivalent to the secondorder differential equa tion x x by simply introducing the second variable y x This is an example of a linear secondorder differential equation which in more general form may be written atx btx ctx ft An important special case of this is the linear constant coefficient equation ax bx cx ft which we write as a system as x y c b f yosxyt a ea a An even more special case is the homogeneous equation in which ft 0 Example One of the simplest yet most important secondorder linear constantcoefficient differential equations is the equation for a harmonic oscil lator This equation models the motion of a mass attached to a spring The spring is attached to a vertical wall and the mass is allowed to slide along a Hirsch Ch029780123820105 201222 1059 Page 26 6 26 Chapter 2 Planar Linear Systems horizontal track We let x denote the displacement of the mass from its natu ral resting place with x 0 if the spring is stretched and x 0 if the spring is compressed Therefore the velocity of the moving mass is xt and the acceleration is xt The spring exerts a restorative force proportional to xt In addition there is a frictional force proportional to xt in the direction opposite to that of the motion There are three parameters for this system m denotes the mass of the oscillator b 0 is the damping constant and k 0 is the spring constant New tons law states that the force acting on the oscillator is equal to mass times acceleration Therefore the differential equation for the damped harmonic oscillator is mx bx kx 0 If b 0 the oscillator is said to be undamped otherwise we have a damped harmonic oscillator This is an example of a secondorder linear constant coefficient homogeneous differential equation As a system the harmonic oscillator equation becomes x y y k mx b my More generally the motion of the massspring system can be subjected to an external force such as moving the vertical wall back and forth periodically Such an external force usually depends only on time not position so we have a more general forced harmonic oscillator system mx bx kx f t where f t represents the external force This is now a nonautonomous secondorder linear equation 23 Preliminaries from Algebra Before proceeding further with systems of differential equations we need to recall some elementary facts regarding systems of algebraic equations We will often encounter simultaneous equations of the form ax by α cx dy β 23 Preliminaries from Algebra 27 where the values of a b c and d as well as w and are given In matrix form we may write this equation as a bx a c dy By We denote by A the 2 x 2 coefficient matrix a b s 4 This system of equations is easy to solve assuming that there is a solution There is a unique solution of these equations if and only if the determinant of A is nonzero Recall that this determinant is the quantity given by det A ad be If det A 0 we may or may not have solutions but if there is a solution then in fact there must be infinitely many solutions In the special case where a 6 0 we always have infinitely many solutions of 0 0 y 0 when detA 0 Indeed if the coefficient a of A is nonzero we have x bay and so b 94 dy 0 a Thus ad bcy 0 Since det A 0 the solutions of the equation assume the form bay y where y is arbitrary This says that every solution lies on a straight line through the origin in the plane A similar line of solutions occurs as long as at least one of the entries of A is nonzero We will not worry too much about the case where all entries of A are 0 in fact we will completely ignore it Let V and W be vectors in the plane We say that V and W are linearly independent if V and W do not lie along the same straight line through the origin The vectors V and W are linearly dependent if either V or W is the zero vector or both lie on the same line through the origin A geometric criterion for two vectors in the plane to be linearly indepen dent is that they do not point in the same or opposite directions That is two 28 Chapter 2 Planar Linear Systems nonzero vectors V and W are linearly independent if and only if V 4 4 W for any real number A An equivalent algebraic criterion for linear independence is given in the following proposition Proposition Suppose V 1 v2 and W Ww w2 Then V and W are linearly independent if and only if vy Wi t 0 u 2 For a proof see Exercise 11 of this chapter C Whenever we have a pair of linearly independent vectors V and W we may always write any vector Z IR in a unique way as a linear combination of V and W That is we may always find a pair of real numbers and such that ZaV pw Moreover w and 6 are unique To see this suppose Z 2 2 Then we must solve the equations z av pw 22 av Pwo where vw and zj are known But this system has a unique solution a 6 since yy Wy det 0 The linearly independent vectors V and W are said to define a basis for R Any vector Z has unique coordinates relative to V and W These coordinates are the pair a 8 for which Z aV BW Example The unit vectors FE 10 and F 01 obviously form a basis called the standard basis of R The coordinates of Z in this basis are just the usual Cartesian coordinates x y of Z a Example The vectors V 11 and V2 1 1 also form a basis of R Relative to this basis the coordinates of E are 1212 and those of 24 Planar Linear Systems 29 EF are 12 12 because 1 11 1l oJ 21 21 0 1fil 4 1l J 21 24K 1 These changes of coordinates will become important later a Example The vectors V 11 and V2 11 do not form a basis of R since these vectors are collinear Any linear combination of these vectors is of the form aB aV BV2 75 which yields only vectors on the straight line through the origin that is V and Vo a 24 Planar Linear Systems We now further restrict our attention to the most important class of planar systems of differential equations namely linear systems In the autonomous case these systems assume the simple form x axt by y cxdy where a bc and d are constants We may abbreviate this system by using the coefficient matrix A where a b A Then the linear system may be written as X AX Hirsch Ch029780123820105 201222 1059 Page 30 10 30 Chapter 2 Planar Linear Systems Note that the origin is always an equilibrium point for a linear system To find other equilibria we must solve the linear system of algebraic equations ax by 0 cx dy 0 This system has a nonzero solution if and only if det A 0 As we saw in the preceding if det A 0 then there is a straight line through the origin on which each point is an equilibrium Thus we have Proposition The planar linear system X AX has 1 A unique equilibrium point 00 if det A 0 2 A straight line of equilibrium points if det A 0 and A is not the 0matrix 25 Eigenvalues and Eigenvectors Now we turn to the question of finding nonequilibrium solutions of the linear system X AX The key observation here is this suppose V0 is a nonzero vector for which we have AV0 λV0 where λ R Then the function Xt eλtV0 is a solution of the system To see this we compute Xt λeλtV0 eλtλV0 eλtAV0 AeλtV0 AXt so Xt does indeed solve the system of equations Such a vector V0 and its associated scalar have names as follows Definition A nonzero vector V0 is called an eigenvector of A if AV0 λV0 for some λ The constant λ is called an eigenvalue of A 25 Eigenvalues and Eigenvectors 31 As we observed there is an important relationship between eigenvalues eigenvectors and solutions of systems of differential equations Theorem Suppose that Vo is an eigenvector for the matrix A with associ ated eigenvalue X Then the function Xt e Vo is a solution of the system X AX Note that if Vo is an eigenvector for A with eigenvalue 4 then any nonzero scalar multiple of Vo is also an eigenvector for A with eigenvalue i Indeed if AVo A Vo then A Vo a AVo a Vo for any nonzero constant a Example Consider 1 3 a 3 Then A has an eigenvector Vp 31 with associated eigenvalue A 2 since 1 3 3 6 3 1 1l 2 Ny Similarly V 11 is an eigenvector with associated eigenvalue 4 2 a Thus for the system 1 3 x 1 1 x we now know three solutions the equilibrium solution at the origin together with 3 1 Xt e 7 and Xte We will see that we can use these solutions to generate all solutions of this sys tem in a moment but first we address the question of how to find eigenvectors and eigenvalues 32 Chapter 2 Planar Linear Systems To produce an eigenvector V xy we must find a nonzero solution xy of the equation x x AG G y y Note that there are three unknowns in this system of equations the two components of V as well as A Let I denote the 2 x 2 identity matrix 1 0 i5 3 Then we may rewrite the equation in the form AADV 0 where 0 denotes the vector 00 Now A AJ is just a 2 x 2 matrix having entries involving the variable 4 so this linear system of equations has nonzero solutions if and only if det A AI 0 as we saw previously But this equation is just a quadratic equation in A and so its roots are easy to find This equation will appear over and over in the sequel it is called the characteristic equation As a function of A we call detA AJ the characteristic polynomial Thus the strategy to generate eigenvectors is first to find the roots of the characteristic equation This yields the eigenvalues Then we use each of these eigenvalues to generate in turn an associated eigenvector Example We return to the matrix 1 3 a 3 We have 1A 3 sare 9 So the characteristic equation is detA AD 1A1A 30 Simplifying we find 7 40 26 Solving Linear Systems 33 which yields the two eigenvalues A 2 Then for 4 2 we next solve the equation A21 0 y oy In component form this reduces to the system of equations 1 2x3y0 x12y0 or x 3y 0 as these equations are redundant Thus any vector of the form 3yy with y 4 0 is an eigenvector associated with A 2 In similar fashion any vector of the form yy with y 4 0 is an eigenvector associated with A 2 a Of course the astute reader will notice that there is more to the story of eigenvalues eigenvectors and solutions of differential equations than what we have described previously For example the roots of the characteristic equa tion may be complex or they may be repeated real numbers We will handle all of these cases shortly but first we return to the problem of solving linear systems 26 Solving Linear Systems As we saw in the example in the previous section if we find two real roots 41 and A2 with A 4 A2 of the characteristic equation then we may generate a pair of solutions of the system of differential equations of the form Xt eit V where V is the eigenvector associated with Note that each of these solutions is a straightline solution Indeed we have X0 Vj which is a nonzero point in the plane For each tf eV is a scalar multiple of V and so lies on the straight ray emanating from the origin and passing through Vj Note that if A 0 then lim Xt 00 too and lim Xt 00 too The magnitude of the solution Xf increases monotonically to oo along the ray through Vj as t increases and Xt tends to the origin along this ray in Hirsch Ch029780123820105 201222 1059 Page 34 14 34 Chapter 2 Planar Linear Systems backward time The exact opposite situation occurs if λi 0 whereas if λi 0 the solution Xit is the constant solution Xit Vi for all t So how do we find all solutions of the system given this pair of special solu tions The answer is now easy and important Suppose we have two distinct real eigenvalues λ1 and λ2 with eigenvectors V1 and V2 Then V1 and V2 are linearly independent as is easily checked see Exercise 14 of this chapter Thus V1 and V2 form a basis of R2 so given any point Z0 R2 we may find a unique pair of real numbers α and β for which αV1 βV2 Z0 Now consider the function Zt αX1t βX2t where the Xit are the preceding straightline solutions We claim that Zt is a solution of X AX To see this we compute Zt αX 1t βX 2t αAX1t βAX2t AαX1t βX2t This last step follows from the linearity of matrix multiplication see Exercise 13 of this chapter Thus we have shown that Zt AZt so Zt is a solu tion Moreover Zt is a solution that satisfies Z0 Z0 Finally we claim that Zt is the unique solution of X AX that satisfies Z0 Z0 Just as in Chapter 1 we suppose that Yt is another such solution with Y0 Z0 Then we may write Yt ζtV1 µtV2 with ζ0 α µ0 β Thus AYt Y t ζ tV1 µtV2 But AYt ζtAV1 µtAV2 λ1ζtV1 λ2µtV2 Therefore we have ζ t λ1ζt µt λ2µt 26 Solving Linear Systems 35 with 0 a 0 B As we saw in Chapter 1 it follows that ct sae wt Be so that Yt is indeed equal to Zt As a consequence we have now found the unique solution to the system X AX that satisfies X0 Zp for any Z R The collection of all such solutions is called the general solution of X AX That is the general solution is the collection of solutions of X AX that features a unique solution of the initial value problem X0 Z for each Z R We therefore have shown the theorem that follows Theorem Suppose A has a pair of real eigenvalues 4 4 d2 and associated eigenvectors V and V2 Then the general solution of the linear system X AX is given by Xt aeV Be Vp Example Consider the secondorder differential equation x 3x2x0 This is a specific case of the damped harmonic oscillator discussed earlier where the mass is 1 the spring constant is 2 and the damping constant is 3 As a system this equation may be rewritten 9 1 x 2 1 xax The characteristic equation is M3BAF2AFDAFD O so the system has eigenvalues 1 and 2 The eigenvector corresponding to the eigenvalue 1 is given by solving the equation x 0 ap In component form this equation becomes xty0 2x2y0 36 Chapter 2 Planar Linear Systems Thus one eigenvector associated with the eigenvalue 1 is 11 In similar fashion we compute that an eigenvector associated with the eigenvalue 2 is 1 2 Note that these two eigenvectors are linearly independent Therefore by the previous theorem the general solution of this system is tf 1 2 1 Xt ae 1 Be 1 That is the position of the mass is given by the first component of the solution xt ae Be and the velocity is given by the second component yt xt ae 2fpe 27 The Linearity Principle The theorem discussed in the previous section is a very special case of the fundamental theorem for ndimensional linear systems which we shall prove in Chapter 6 Section 61 Distinct Eigenvalues For the twodimensional version of this result note that if X AX is a planar linear system for which Yt and Yt are both solutions then just as before the function aYit BY2t is also a solution of this system We do not need real and distinct eigenvalues to prove this This fact is known as the Linearity Principle More important if the initial conditions Y0 and Y20 are linearly inde pendent vectors then these vectors form a basis of R Thus given any vector Xp R we may determine constants w and f such that Xp w Y 0 BY20 Then the Linearity Principle tells us that the solution Xt satisfying the initial condition X0 Xp is given by Xt a Yt BY2t We have therefore produced a solution of the system that solves any given initial value problem The Existence and Uniqueness Theorem for linear systems in Chap ter 6 will show that this solution is also unique This important result may then be summarized Theorem Let X AX be a planar system Suppose that Yt and Yt are solutions of this system and that the vectors Y0 and Y0 are linearly independent Then Xt aYit BY2t is the unique solution of this system that satisfies X0 a Y0 B Y20 Exercises 37 EXERCISES 1 Find the eigenvalues and eigenvectors of each of the following 2 x 2 matrices a 3 1 b 2 1 a 1 3 1 1 0 a b a 1 3 0 c V2 32 2 Find the general solution of each of the following linear systems 1 2 1 2 7 7 a X X b X x a 3X ox 1 2 1 2 7 7 cx xX dxX x 3 In Figure 22 you see four direction fields Match each of these direction fields with one of the systems in the previous exercise 4 Find the general solution of the system a b x X ca where be 0 1 NR RR 2 PAPA PPAR PP PAPA RRR EE eae PAPPP PPAR PP EPP VN NNR a PAPPP PPAR PP PAPE YN NNN TT PAPPPPPPP PP PAPE PPVNNSSOE TOIT 77 PAPA PPAR PA PAPE PEP VNNSTEIIIII ESF yy PPR A area ee FELAANNEDISI S72 rete AANA aa aad LAOH EAN a peer wednN TLS SS Tou fe atdd peeeeeeanS EET reads daag PTGS eewwwwnn TS OO Cee EEE AEE EEE eererrnn SY PUELEAEAAAEA EGS eer ree SYK Un a eS Ul ee eee NNN Ue a SSN NS ee SY VV VNNANKANNS SD 4 EPR PAULL AN NNR NSS SEES ee ee PYELV VV ANNAN SSS PPP PPPS SS DS PEPVAVNANR SNS SECIS PPP PPPS AS SA PEEP RVNNNSNS ODI 77 ttt Ps sss pss srrre Nat 7 44 LEEPER te HEH et VANNN Saas NNR ee ree HEP Pd PVN NS yy gg gy BANNAN ED ee Pededeegdy better eee POPPESESAA EEE TE eee ERR INN SELEIISNA NNN See G OATH ALAR SS pera NNN LLLELA LEAD EGEEAG SESS eSATA T LLLLLLLEAGEEDAEAG 2 hhh NNN ceredeerdeeredidt HRN TITINA NN ett A EE Figure 22 Match these direction fields with the systems in Exercise 2 38 Chapter 2 Planar Linear Systems 5 Find the general solution of the system 0 0 x 9 6 For the harmonic oscillator system x bx kx 0 find all values of b and k for which this system has real distinct eigenvalues Find the general solution of this system in these cases Find the solution of the system that satisfies the initial condition 01 Describe the motion of the mass in this particular case 7 Consider the 2 x 2 matrix a l a i Find the value ap of the parameter a for which A has repeated real eigenvalues What happens to the eigenvectors of this matrix as a approaches ao 8 Describe all possible 2 x 2 matrices with eigenvalues of 0 and 1 9 Give an example of a linear system for which eq is a solution for every constant a Sketch the direction field for this system What is the general solution of this system 10 Give an example of a system of differential equations for which tf 1 is a solution Sketch the direction field for this system What is the general solution of this system 11 Prove that two vectors V vj 7 and W w w are linearly inde pendent if and only if yy Wy det m1 0 12 Prove that if i jz are real eigenvalues of a 2 x 2 matrix then any nonzero column of the matrix A AJ is an eigenvector for ju 13 Let A be a 2 x 2 matrix and let V and V vectors in R Prove that AaV BV2 a AV BAV 14 Prove that the eigenvectors of a 2 x 2 matrix corresponding to distinct real eigenvalues are always linearly independent Hirsch Ch039780123820105 2012124 2216 Page 39 1 3 Phase Portraits for Planar Systems Given the Linearity Principle from the previous chapter we may now com pute the general solution of any planar system There is a seemingly endless number of distinct cases but we will see that these represent in the simplest possible form nearly all of the types of solutions we will encounter in the higherdimensional case 31 Real Distinct Eigenvalues Consider X AX and suppose that A has two real eigenvalues λ1 λ2 Assuming for the moment that λi 0 there are three cases to consider 1 λ1 0 λ2 2 λ1 λ2 0 3 0 λ1 λ2 We give a specific example of each case any system that falls into any one of these three categories may be handled similarly as we show later Differential Equations Dynamical Systems and an Introduction to Chaos DOI 101016B9780123820105000038 c 2013 Elsevier Inc All rights reserved 39 40 Chapter 3 Phase Portraits for Planar Systems Example Saddle First consider the simple system X AX where A 60 A 2 with A 0 A This can be solved immediately since the system decouples into two unrelated firstorder equations x Ax y hay We already know how to solve these equations but having in mind what comes later lets find the eigenvalues and eigenvectors The characteristic equation is A A1A 42 0 so A and Az are the eigenvalues An eigenvector corresponding to A is 10 and to Az is 01 Thus we find the general solution 1 0 Xt ae Ber 0 1 Since A 0 the straightline solutions of the form ae10 lie on the xaxis and tend to 00 as t oo This axis is called the stable line Since hz 0 the solutions Be20 1 lie on the yaxis and tend away from 00 as t oo this axis is the unstable line All other solutions with a 6B 4 0 tend to oo in the direction of the unstable line as t oo since Xt comes closer and closer to 0 Be2 as t increases In backward time these solutions tend to co in the direction of the stable line a In Figure 31 we have plotted the phase portrait of this system The phase portrait is a picture of a collection of representative solution curves of the system in R which we call the phase plane The equilibrium point of a system of this type eigenvalues satisfying 41 0 2 is called a saddle For a slightly more complicated example of this type consider X AX where 1 3 A As we saw in Chapter 2 the eigenvalues of A are 2 The eigenvector associ ated with A 2 is the vector 3 1 the eigenvector associated with 4 2 is 31 Real Distinct Eigenvalues 41 Figure 31 Saddle phase portrait for x xX y y 11 Thus we have an unstable line that contains straightline solutions of the form 3 Xt ae 7 each of which tends away from the origin as t oo The stable line contains the straightline solutions a 1 X2t Be i which tend toward the origin as t oo By the Linearity Principle any other solution assumes the form 3 1 xy ae pe 1 1 for some 8 Note that if a 0 as t oo we have ar 3 Xt ae 1 Xt whereas if 8 4 0 as t oo a 1 Xpe Xa0 Thus as time increases the typical solution approaches Xt while as time decreases this solution tends toward X2f just as in the previous case Figure 32 displays this phase portrait 42 Chapter 3 Phase Portraits for Planar Systems XS EAA CE Figure 32 Saddle phase portrait for xX x3y Yxy In the general case where A has a positive and negative eigenvalue we always find a similar stable and unstable line on which solutions tend toward or away from the origin All other solutions approach the unstable line as t oo and tend toward the stable line as tf oo Example Sink Now consider the case X AX where am 0 a0 2 but A Az 0 As before we find two straightline solutions and then the general solution 1 0 Xt ae per 0 1 Unlike the saddle case now all solutions tend to 00 as t oo The question is this How do they approach the origin To answer this we compute the slope dydx of a solution with 6B 4 0 We write xt ae yt pe and compute dy dyfdt d2Be 2B oayt dx dxdt dAyae Ala Since A2 A 0 it follows that these slopes approach 00 provided 6 0 Thus these solutions tend to the origin tangentially to the yaxis a 31 Real Distinct Eigenvalues 43 a b Figure 33 Phase portraits for a sink and a source Since 4 Az 0 we call 4 the stronger eigenvalue and 2 the weaker eigenvalue The reason for this in this particular case is that the xcoordinates of solutions tend to 0 much more quickly than the ycoordinates This accounts for why solutions except those on the line corresponding to A eigenvector tend to hug the straightline solution corresponding to the weaker eigenvalue as they approach the origin The phase portrait for this system is displayed in Figure 33a In this case the equilibrium point is called a sink More generally if the system has eigenvalues 41 Az 0 with eigenvectors u1 U2 and v1 v2 respectively then the general solution is get Bet 7 V2 The slope of this solution is given by dy Ayore uy ArBe2 vy dx hyaettyy ArBer2y Aye ArBe2 vy e772 Ayaettuy ArBer2vy J eA2t AyaeAiA2dty Arb v2 AyaeAiAvtyy A2BY which tends to the slope v21 of the A2eigenvector unless we have 6 0 If 0 our solution is the straightline solution corresponding to the eigen value A Thus in this case as well all solutions except those on the straight line corresponding to the stronger eigenvalue tend to the origin tangentially to the straightline solution corresponding to the weaker eigenvalue 44 Chapter 3 Phase Portraits for Planar Systems Example Source When the matrix a1 0 a0 2 satisfies 0 42 Aj our vector field may be regarded as the negative of the previous example The general solution and phase portrait remain the same except that all solutions now tend away from 00 along the same paths See Figure 33b a One may argue that we are presenting examples here that are much too simple Although this is true we will soon see that any system of differential equations with a matrix that has real distinct eigenvalues can be manipulated into the preceding special forms by changing coordinates Finally a special case occurs if one of the eigenvalues is equal to 0 As we have seen there is a straight line of equilibrium points in this case If the other eigenvalue A is nonzero then the sign of A determines whether the other solu tions tend toward or away from these equilibria see Exercises 10 and 11 of this chapter 32 Complex Eigenvalues It may happen that the roots of the characteristic polynomial are complex numbers In analogy with the real case we call these roots complex eigenvalues When the matrix A has complex eigenvalues we no longer have straightline solutions However we can still derive the general solution as before by using a few tricks involving complex numbers and functions The following examples indicate the general procedure Example Center Consider X AX with 0 Bp a5 4 and B 0 The characteristic polynomial is A 8 0 so the eigenvalues are now the imaginary numbers i8 Without worrying about the resulting complex vectors we react just as before to find the eigenvector corresponding to A iB We therefore solve 5 5o B iBy 0 32 Complex Eigenvalues 45 or iBx By since the second equation is redundant Thus we find a complex eigenvector 17 and so the function a 1 Xt el i is a complex solution of X AX Now in general it is not polite to hand someone a complex solution to a real system of differential equations but we can remedy this with the help of Eulers formula elBt cos Bt isin Bt Using this fact we rewrite the solution as Xt cosBtisinBt cosBtisinBt icosBtisinBt sinBticospt Better yet by breaking Xt into its real and imaginary parts we have Xt Xret 1Ximt where cos Bt sin Bt Xret Ximt sinBt cos Bt But now we see that both Xt and Xjf are real solutions of the original system To see this we simply check Xjet iXjqt X1 AXt AXet 1Ximt AXye iAXint Equating the real and imaginary parts of this equation yields X AXe and Xin AXjim which shows that both are indeed solutions Moreover since 1 0 Xre0 0 Xim 1 the linear combination of these solutions Xt 4 Xret OXimD 46 Chapter 3 Phase Portraits for Planar Systems Figure 34 Phase portrait for a center where c and c are arbitrary constants provides a solution to any initial value problem We claim that this is the general solution of this equation To prove this we need to show that these are the only solutions of this equation So suppose that this is not the case Let t Yt ut vt be another solution Consider the complex function ft ut ivteP Differentiating this expression and using the fact that Yt is a solution of the equation yields ft 0 Thus ut ivt is a complex constant times ePt From this it follows directly that Yt is a linear combination of Xt and Xjt Note that each of these solutions is a periodic function with period 27f Indeed the phase portrait shows that all solutions lie on circles centered at the origin These circles are traversed in the clockwise direction if B 0 counterclockwise if 6 0 See Figure 34 This type of system is called a center a Example Spiral Sink Spiral Source More generally consider X AX where a 8B a4 and af 0 The characteristic equation is now 47 2aAa B so the eigenvalues are Aaif An eigenvector associated with w if is determined by the equation a a iBx By 0 33 Repeated Eigenvalues 47 MEN MMEN WO WI ey L ies Figure 35 Phase portraits for a spiral sink and a spiral source Thus 17 is again an eigenvector and so we have complex solutions of the form 1 Xt tit i pt cos Bt iett sin Bt sin Bt cos Bt Xjet 1Ximt As before both Xt and Xjt yield real solutions of the system with initial conditions that are linearly independent Thus we find the general solution t sin Bt Xt qe cos ot H ae Or ae cos Bt Without the term e these solutions would wind periodically around circles centered at the origin The e term converts solutions into spirals that either spiral into the origin when 0 or away from the origin a 0 In these cases the equilibrium point is called a spiral sink or spiral source respectively See Figure 35 33 Repeated Eigenvalues The only remaining cases occur when A has repeated real eigenvalues One simple case occurs when A is a diagonal matrix of the form rX 0 s2 48 Chapter 3 Phase Portraits for Planar Systems The eigenvalues of A are both equal to A In this case every nonzero vector is an eigenvector since AVAV for any V R Thus solutions are of the form Xt acy Each such solution lies on a straight line through 00 and either tends to 00 if A 0 or away from 00 if A 0 So this is an easy case A more interesting case occurs when rn 1 a 2 Again both eigenvalues are equal to A but now there is only one linearly inde pendent eigenvector that is given by 10 Thus we have one straightline solution Xt ae 0 To find other solutions note that the system may be written x Axy yl yy Thus if y 4 0 we must have yt Be Therefore the differential equation for xt reads x Axt Be This is anonautonomous firstorder differential equation for xt One might first expect solutions of the form e but the nonautonomous term is also in this form As you perhaps saw in calculus the best option is to guess a solution of the form xt ae pte for some constants a and jt This technique is often called the method of undetermined coefficients Inserting this guess into the differential equation 34 Changing Coordinates 49 SQ Figure 36 Phase portrait for a system with repeated negative eigenvalues shows that 6 while w is arbitrary Thus the solution of the system may be written 1 t At At ae pe 0 7 This is in fact the general solution see Exercise 12 of this chapter Note that if A 0 each term in this solution tends to 0 as t oo This is clear for the we and Be terms For the term f te this is an immediate con sequence of lH6pitals rule Thus all solutions tend to 00 as t oo When A 0 all solutions tend away from 00 See Figure 36 In fact solutions tend toward or away from the origin in a direction tangent to the eigenvector 10 see Exercise 7 at the end of this chapter 34 Changing Coordinates Despite differences in the associated phase portraits we really have dealt with only three type of matrices in these past four sections a 0 a BrA l 0 wlPB aPO Ay Any 2 x 2 matrix that is in one of these three forms is said to be in canonical form Systems in this form may seem rather special but they are not Given any linear system X AX we can always change coordinates so that the new systems coefficient matrix is in canonical form and so is easily solved Here is how to do this 50 Chapter 3 Phase Portraits for Planar Systems A linear map or linear transformation on R is a function T R R of the form b r arty y cx dy That is T simply multiplies any vector by the 2 x 2 matrix a b c dj We will thus think of the linear map and its matrix as being interchangeable so that we also write a b T i Hopefully no confusion will result from this slight imprecision Now suppose that T is invertible This means that the matrix T has an inverse matrix S that satisfies TS ST I where I is the 2 x 2 identity matrix It is traditional to denote the inverse of a matrix T by T Asis easily checked the matrix 1 Sa d b detT c a serves as T if det T 4 0 If det T 0 we know from Chapter 2 that there are infinitely many vectors x y for which x 0 T 3 Thus there is no inverse matrix in this case for we would need GrrGr y y 0 for each such vector We have shown this Proposition T he 2 x 2 matrix T is invertible if and only ifdetT 40 O 34 Changing Coordinates 51 Now instead of considering a linear system X AX suppose we consider a different system Y TIATY for some invertible linear map T Note that if Yt is a solution of this new system then Xt TYt solves X AX Indeed we have TYt TYt TTATYt ATY1 as required That is the linear map T converts solutions of Y TATY to solutions of X AX Alternatively T takes solutions of X AX to solutions of Y TATY We therefore think of T as a change of coordinates that converts a given linear system into one with a different coefficient matrix What we hope to be able to do is find a linear map T that converts the given system into a sys tem of the form Y TATY that is easily solved And as you may have guessed we can always do this by finding a linear map that converts a given linear system to one in canonical form Example Real Eigenvalues Suppose the matrix A has two real distinct eigenvalues 4 and Az with associated eigenvectors V and V2 Let T be the matrix with columns Vj and V2 Thus TE V for j 12 where the E form the standard basis of R2 Also T Vj Therefore we have TAT Ej T AV T1AjVj 1 AjT V AjFj Thus the matrix TAT assumes the canonical form A 0 lap Al T AT 0 and the corresponding system is easy to solve a Example Asa further specific example suppose 1 0 a7 5 52 Chapter 3 Phase Portraits for Planar Systems The characteristic equation is 4 34 2 which yields eigenvalues 4 1 and A 2 An eigenvector corresponding to 4 1 is given by solving x 0 0 x 0 aro at SG y i Y 0 which yields an eigenvector 11 Similarly an eigenvector associated with A 2 is given by 01 We therefore have a pair of straightline solutions each tending to the origin as t oo The straightline solution corresponding to the weaker eigen value lies along the line y x the straightline solution corresponding to the stronger eigenvalue lies on the yaxis All other solutions tend to the origin tangentially to the line y x To put this sytem in canonical form we choose T to be the matrix with columns that are these eigenvectors 1 0 r1 3 so that 1 0 1 rie 9 Finally we compute 141 0 T AT 0 2 so TAT is in canonical form The general solution of the system Y TATY is 1 0 Y t 2t t ae 86 so the general solution of X AX is 1i 0 1 21 9 r j e 66 1 1 0 aet 2r oe0e7 34 Changing Coordinates 53 NY Tt WE Figure 37 Change of variables Tin the case of a real sink Thus the linear map T converts the phase portrait for the system 1 0 re My to that of X AX as shown in Figure 37 a Note that we really do not have to go through the step of converting a specific system to one in canonical form once we have the eigenvalues and eigenvectors we can simply write down the general solution We take this extra step because when we attempt to classify all possible linear systems the canonical form of the system will greatly simplify this process Example Complex Eigenvalues Now suppose that the matrix A has complex eigenvalues w if with 6 4 0 Then we may find a complex eigen vector V iV2 corresponding to a i where both V and V2 are real vectors We claim that V and V2 are linearly independent vectors in R If this were not the case then we would have V cV2 for some c R But then we have ACV iV2 iBV1 1V2 1Bc 1 V9 But we also have AV 1V2 c 1 AV3 So we conclude that AV2 a if V2 This is a contradiction since the left side is a real vector while the right is complex Since Vj iV2 is an eigenvector associated with a i8 we have AV 1V2 i8V 1V2 54 Chapter 3 Phase Portraits for Planar Systems Equating the real and imaginary components of this vector equation we find AV avy B V2 AV BV aVp Let T be the matrix with columns V and V2 Thus TE V for j 12 Now consider TAT We have T7ATE T7aVi BV2 aF BE and similarly TAT Ey BE takp Thus the matrix TAT is in the canonical form lyp B T AT b We saw that the system Y TATY has phase portrait corresponding to a spiral sink center or spiral source depending on whether a 0 a 0 or a 0 Therefore the phase portrait of X AX is equivalent to one of these after changing coordinates using T a Example Another Harmonic Oscillator Consider the secondorder equa tion x 4x0 This corresponds to an undamped harmonic oscillator with mass 1 and spring constant 4 As a system we have yo 0 1 XxX 1 5 X AX The characteristic equation is 740 34 Changing Coordinates 55 so that the eigenvalues are 27 A complex eigenvector associated with A 2 is a solution of the system 2ixty0 4x 2iy0 One such solution is the vector 127 So we have a complex solution of the form it 1 2it 22 Breaking this solution into its real and imaginary parts we find the general solution cos 2t sin2t XD a1 5 an e wen Thus the position of this oscillator is given by xt c cos2t c2 sin2t which is a periodic function of period z Now let T be the matrix with columns that are the real and imaginary parts of the eigenvector 121 that is 1 0 rj 3 Then we compute easily that 1470 2 T AT é 5 5 which is in canonical form The phase portraits of these systems are shown in Figure 38 Note that T maps the circular solutions of the system Y TATY to elliptic solutions of X AX a Example Repeated Eigenvalues Suppose A has a single real eigenvalue A If there exists a pair of linearly independent eigenvectors then in fact A must be in the form rX 0 a0 4 so the system X AX is easily solved see Exercise 15 of this chapter 56 Chapter 3 Phase Portraits for Planar Systems T Ha Figure 38 Change of variables Tin the case of a center For the more complicated case lets assume that V is an eigenvector and that every other eigenvector is a multiple of V Let W be any vector for which V and W are linearly independent Then we have AW puVvW for some constants v R Note that 4 4 0 for otherwise we would have a second linearly independent eigenvector W with eigenvalue v We claim that v i If v 40 a computation shows that aw 4 v w 4 v vA vA This says that v is a second eigenvalue different from 4 Thus we must have vA Finally let U 12 W Then Xr AUVWVAU bb Thus if we define TE V TE U we get typ 1 T AT 0 a as required X AX is therefore again in canonical form after this change of coordinates a Exercises 57 EXERCISES 1 In Figure 39 you see six phase portraits Match each of these phase portraits with one of the following linear systems 3 5 3 2 3 2 2 3 2 Z 3 3 5 3 5 3 5 33 4 2 0G 3 2 For each of the following systems of the form X AX a Find the eigenvalues and eigenvectors of A b Find the matrix T that puts A in canonical form c Find the general solution of both X AX and Y TATY d Sketch the phase portraits of both systems 1 Gi A s ii A s Awa SS FA WSS EZ é Figure 39 Match these phase portraits with the systems in Exercise 1 58 Chapter 3 Phase Portraits for Planar Systems ees 1 1 1 1 dul A 5 iv A s 1 1 1 1 v A vi A 1 3 Find the general solution of the following harmonic oscillator equations a x2xx0 b x2xx0 4 Consider the harmonic oscillator system 9 1 v 48 where b 0k 0 and the mass m 1 a For which values of k b does this system have complex eigenvalues Repeated eigenvalues Real and distinct eigenvalues b Find the general solution of this system in each case c Describe the motion of the mass when the mass is released from the initial position x 1 with zero velocity in each of the cases in part a 5 Sketch the phase portrait of X AX where a 1 A For which values of a do you find a bifurcation Describe the phase portrait for avalues above and below the bifurcation point 6 Consider the system 2a b va 8x Sketch the regions in the abplane where this system has different types of canonical forms 7 Consider the system rk 1 e 2 with A 40 Show that all solutions tend to respectively away from the origin tangentially to the eigenvector 10 when A 0 respectively A0 Exercises 59 8 Find all 2 x 2 matrices that have pure imaginary eigenvalues That is determine conditions on the entries of a matrix that guarantee the matrix has pure imaginary eigenvalues 9 Determine a computable condition that guarantees that if a matrix A has complex eigenvalues with nonzero imaginary parts then solutions of X AX travel around the origin in the counterclockwise direction 10 Consider the system a b x x where a d 4 0 but ad bc 0 Find the general solution of this system and sketch the phase portrait 11 Find the general solution and describe completely the phase portrait for 0 1 x 0 Xx 12 Prove that 1 t At At ve 2 is the general solution of aA 1 X x 13 Prove that a 2 x 2 matrix A always satisfies its own characteristic equa tion That is if 4 aA B 0 is the characteristic equation associated with A then the matrix A aA BI is the 0matrix 14 Suppose the 2 x 2 matrix A has repeated eigenvalues 2 Let V R Using the previous problem show that either V is an eigenvector for A or else AAJV is an eigenvector for A 15 Suppose the matrix A has repeated real eigenvalues A and there exist a pair of linearly independent eigenvectors associated with A Prove that A 0 A 16 Consider the nonlinear system x y y x Use the methods of this chapter to describe the phase portrait IPN Hh Pelt Uf Classification of Planar Systems In this chapter we summarize what we have accomplished so far using a dynamical systems point of view Among other things this means that we would like to have a complete dictionary of all possible behaviors of 2 x 2 linear systems One of the dictionaries we present here is geometric the trace determinant plane The other dictionary is more dynamic This involves the notion of conjugate systems 41 The TraceDeterminant Plane For a matrix a b a0 a we know that the eigenvalues are the roots of the characteristic equation which may be written a da ad bc 0 Differential Equations Dynamical Systems and an Introduction to Chaos DOI 101016B978012382010500004X 2013 Elsevier Inc All rights reserved 61 62 Chapter 4 Classification of Planar Systems The constant term in this equation is detA The coefficient of 4 also has a name The quantity a d is called the trace of A and is denoted by tr A Thus the eigenvalues satisfy 7 trAA detA 0 and are given by 1 ha 5 trAt ytray 4detA Note that Ay A trA and A4A det A so the trace is the sum of the eigenvalues of A while the determinant is the product of the eigenvalues of A We will also write T tr A and D det A Knowing T and D tells us the eigenvalues of A and therefore virtually everything about the geometry of solutions of X AX For example the values of T and D tell us whether solutions spiral into or away from the origin whether we have a center and so forth We may display this classification visually by painting a picture in the trace determinant plane In this picture a matrix with trace T and determinant D corresponds to the point with coordinates TD The location of this point in the TDplane then determines the geometry of the phase portrait as before For example the sign of T 4D tells us that the eigenvalues are 1 Complex with nonzero imaginary part if T 4D 0 2 Real and distinct if T 4D 0 3 Real and repeated if T 4D 0 Thus the location of TD relative to the parabola T 4D 0 in the TD plane tells us all we need to know about the eigenvalues of A from an algebraic point of view In terms of phase portraits however we can say more If T 4D 0 then the real part of the eigenvalues is T2 and so we have a 1 Spiral sink if T 0 2 Spiral source if T 0 3 Center if T 0 If T 4D 0 we have a similar breakdown into cases In this region both eigenvalues are real If D 0 then we have a saddle This follows since D is the product of the eigenvalues one of which must be positive the other negative Equivalently if D 0 we compute T T4D 41 The TraceDeterminant Plane 63 so that T VT 4D Thus we have TVT4D0 TvVT4D 0 so the eigenvalues are real and have different signs If D 0 and T 0 then both TVT4D 0 so we have a real sink Similarly T 0 and D 0 lead to a real source When D Oand T F 0 we have one zero eigenvalue while both eigenvalues vanish if D T 0 Plotting all of this verbal information in the TDplane gives us a visual sum mary of all of the different types of linear systems The preceding equations partition the TDplane into various regions in which systems of a particular type reside See Figure 41 This yields a geometric classification of 2 x 2 linear systems A couple of remarks are in order First the tracedeterminant plane is a two dimensional representation of what really is a fourdimensional space since 2 x 2 matrices are determined by four parameters the entries of the matrix Thus there are infinitely many different matrices corresponding to each point in the TDplane Although all of these matrices share the same eigenvalue con figuration there may be subtle differences in the phase portraits such as the direction of rotation for centers and spiral sinks and sources or the possibility of one or two independent eigenvectors in the repeated eigenvalue case We also think of the tracedeterminant plane as the analogue of the bifur cation diagram for planar linear systems A oneparameter family of linear systems corresponds to a curve in the TDplane When this curve crosses the Taxis the positive Daxis or the parabola T 4D 0 the phase portrait of the linear system undergoes a bifurcation there is a major change in the geometry of the phase portrait Finally note that we may obtain quite a bit of information about the system from D and T without ever computing the eigenvalues For example if D 0 we know that we have a saddle at the origin Similarly if both D and T are positive then we have a source at the origin 64 Chapter 4 Classification of Planar Systems ONS NN fii Det grrr a GES WE AT QS WEY ips cots Spe T4D AY QZ ML INS Tr Figure 41 The tracedeterminant plane Any resemblance to any of the authors faces is purely coincidental 42 Dynamical Classification In this section we give a different more dynamical classification of planar lin ear systems From a dynamical systems point of view we are usually interested primarily in the longterm behavior of solutions of differential equations Thus two systems are equivalent if their solutions share the same fate To make this precise we recall some terminology introduced in Chapter 1 Section 15 To emphasize the dependence of solutions on both time and the initial con ditions Xo we let 6Xo denote the solution that satisfies the initial condition Xo That is 69Xo Xo The function t Xo Xo is called the flow of the differential equation while is called the time t map of the flow For example let 20 X 0 3 Then the time t map is given by 1X0s Yo xe yor 42 Dynamical Classification 65 Thus the flow is a function that depends on both time and initial values We will consider two systems to be dynamically equivalent if there is a func tion h that takes one flow to the other We require that this function be a homeomorphism that is h is a onetoone onto and continuous function with an inverse that is also continuous Definition Suppose X AX and X BX have flows 4 and 8 These two systems are topologically conjugate if there exists a homeomorphism h R R that satisfies thXo WAt Xo The homeomorphism h is called a conjugacy Thus a conjugacy takes the solution curves of X AX to those of X BX Example For the onedimensional linear differential equations x Ax and x Aox we have the flows tx xoe for j 12 Suppose that A and Az are nonzero and have the same sign Then let xh2a1 ifx0 hx an x ae if x 0 where we recall that Xr hola exp 2 tos Al Note that h isa homeomorphism of the real line We claim that h is a conjugacy between x 4 x and x dx To see this we check that when xq 0 doA h tx0 xpe2 xl et th as required A similar computation works when x9 0 a Hirsch Ch049780123820105 2012125 140 Page 66 6 66 Chapter 4 Classification of Planar Systems There are several things to note here First λ1 and λ2 must have the same sign for otherwise we have h0 in which case h is not a homeo morphism This agrees with our notion of dynamical equivalence If λ1 and λ2 have the same sign then their solutions behave similarly as either both tend to the origin or both tend away from the origin Also note that if λ2 λ1 then h is not differentiable at the origin whereas if λ2 λ1 then h1x xλ1λ2 is not differentiable at the origin This is the reason we require h to be only a homeomorphism and not a diffeomor phism a differentiable homeomorphism with a differentiable inverse If we assume differentiability then we must have λ1 λ2 which does not yield a very interesting notion of equivalence This gives a classification of autonomous linear a firstorder differen tial equations which agrees with our qualitative observations in Chapter 1 There are three conjugacy classes the sinks the sources and the special inbetween case x 0 where all solutions are constants Now we move to the planar version of this scenario We first note that we only need to decide on conjugacies among systems with matrices in canonical form For as we saw in Chapter 3 if the linear map T R2 R2 puts A in canonical form then T takes the time t map of the flow of Y T1ATY to the time t map for X AX Our classification of planar linear systems now proceeds just as in the one dimensional case We will stay away from the case where the system has eigenvalues with real part equal to 0 but you will tackle this case in the Exercises at the end of this chapter Definition A matrix A is hyperbolic if none of its eigenvalues has real part 0 We also say that the system X AX is hyperbolic Theorem Suppose that the 22 matrices A1 and A2 are hyperbolic Then the linear systems X AiX are conjugate if and only if each matrix has the same number of eigenvalues with negative real part Thus two matrices yield conjugate linear systems if both sets of eigenvalues fall into the same category 1 One eigenvalue is positive and the other is negative 2 Both eigenvalues have negative real parts 3 Both eigenvalues have positive real parts Before proving this note that this theorem implies that a system with a spiral sink is conjugate to a system with a real sink Of course Even though their phase portraits look very different it is nevertheless the case that all solutions of both systems share the same fate They tend to the origin as t 42 Dynamical Classification 67 Proof Recall from before that we may assume that all of the systems are in canonical form Then the proof divides into three distinct cases Case 1 Suppose we have two linear systems X AX for i 12 such that each A has eigenvalues 4 0 jz Thus each system has a saddle at the origin This is the easy case As we saw earlier the real differential equations x Ajx have conjugate flows via the homeomorphism A2Ay ifx0 x ifx In tar ifx0 Similarly the equations y jy have conjugate flows via an analogous function h Now define Ax y hy x hpy Then one checks immediately that H provides a conjugacy between these two systems Case 2 Consider the system X AX where A is in canonical form with eigenvalues that have negative real parts We further assume that the matrix A is not in the form XW 1 0 A with 2 0 Thus in canonical form A assumes one of the two forms a 6 dX 0 0G 9 with A 2 0 We will show that in either a or b the system is conjugate to X BX where l1 0 p4 It then follows that any two systems of this form are conjugate Consider the unit circle in the plane parametrized by the curve X0 cossin 0 6 27 We denote this circle by S We first claim that the 68 Chapter 4 Classification of Planar Systems vector field determined by a matrix in the preceding form must point inside S In case 2a we have that the vector field on S is given by acosé Bsin AXG cos ano The outwardpointing normal vector to S at X0 is cos N sind The dot product of these two vectors satisfies AX0N acos6 sin6 0 since a 0 This shows that AX6 does indeed point inside S Case 2b is even easier As a consequence each nonzero solution of X AX crosses S exactly once Let denote the time t map for this system and let t Tx y denote the time at which x y meets S Thus bay xy 1 Let 8 denote the time t map for the system X BX Clearly F my exe y We now define a conjugacy H between these two systems If x y 4 00 let B A Ax y Pr xy Prixy xy and set H00 00 Geometrically the value of Hxy is given by fol lowing the solution curve of X AX exactly t xy time units forward or backward until the solution reaches S and then following the solution of X BX starting at that point on S and proceeding in the opposite time direction exactly t time units See Figure 42 To see that H gives a conjugacy note first that A T xy Tx y s 42 Dynamical Classification 69 0 O7 Y Hx y 0 Y x Y Figure 42 The definition of txy since Gee GY PECs SI Therefore we have H8 Gy 6804 64 56202 GY 3 Hxy So H is a conjugacy Now we show that H is a homeomorphism We can construct an inverse for H by simply reversing the process defining H That is let A B Gx y 1 xy Pry xy xy and set G00 00 Here tx y is the time for the solution of X BX through xy to reach S An easy computation shows that txy logr where r x y Clearly G H so H is onetoone and onto Also G is continuous at x y 4 00 since G may be written pA xy Gx y Pogr which is a composition of continuous functions For continuity of G at the origin suppose that x y is close to the origin so that r is small Observe that as r 0 logr oo Now xryr is a point on S and for r sufficiently small o4 logr Maps the unit circle very close to 00 This shows that G is continuous at 00 70 Chapter 4 Classification of Planar Systems We thus need only show continuity of H For this we need to show that Txy is continuous But t is determined by the equation oA 1 We write bd x y x0 yt Taking the partial derivative of pA x y with respect to t we find Ol 4 0 S yl t2 t2 5 een eM OW tt xtxt ytyt Vv xt vt arent Gen Goo lotion WwO yYoO But the latter dot product is nonzero when t Tx y since the vector field given by xt yt points inside S So ohay 0 x ar lt y at txyxy Thus we may apply the Implicit Function Theorem to show that t is differentiable at xy and thus continuous Continuity of H at the origin follows as in the case of G H Thus H is a homeomorphism and we have a conjugacy between X AX and X BX Note that this proof works equally well if the eigenvalues have positive real parts Case 3 Finally suppose that rn 1 a0 3 with 4 0 The associated vector field need not point inside the unit circle in this case However if we let 1 0 T then the vector field given by YTATY Exercises 71 now does have this property provided 0 is sufficiently small Indeed rn T AT 0 so 6 0 T AT cos cos isin0cosé sind sind Thus if we choose A this dot product is negative Therefore the change of variables T puts us into the situation where the same proof as shown in Case 2 applies This completes the proof in one direction The only if part of the proof follows immediately a 43 Exploration A 3D Parameter Space Consider the threeparameter family of linear systems given by a b X x where a b and c are parameters 1 First fix a 0 Describe the analogue of the tracedeterminant plane in the bcplane That is identify the bcvalues in this plane where the corre sponding system has saddles centers spiral sinks and so on Sketch these regions in the bcplane 2 Repeat the previous task when a 0 and when a 0 3 Describe the bifurcations that occur as a changes from positive to negative 4 Now put all of the previous pieces of information together and give a description of the full threedimensional parameter space for this system You could build a 3D model of this space create a flipbook animation of the changes as say a varies or use a computer model to visualize this image In any event your model should accurately capture all of the distinct regions in this space EXERCISES 1 Consider the oneparameter family of linear systems given by 2 a2 xX v2 x a2 0 72 Chapter 4 Classification of Planar Systems a Sketch the path traced out by this family of linear systems in the tracedeterminant plane as a varies b Discuss any bifurcations that occur along this path and compute the corresponding values of a 2 Sketch the analogue of the tracedeterminant plane for the two parameter family of systems ab x 2x in the abplane That is identify the regions in the abplane where this system has similar phase portraits 3 Consider the harmonic oscillator equation with m 1 x bx kx 0 where b 0 and k 0 Identify the regions in the relevant portion of the bkplane where the corresponding system has similar phase portraits 4 Prove that Hx y xy provides a conjugacy between ye 1 1 1 1 x ix and vj i 5 For each of the following systems find an explicit conjugacy between their flows 1 1 1 0 x 74 and Y 12 Y 0 1 0 2 b xf 0 X and Y 5 0 Y 6 Prove that any two linear systems with the same eigenvalues i6 B 4 0 are conjugate What happens if the systems have eigenvalues 76 and tiy with B 4 y What if y f 7 Consider all linear systems with exactly one eigenvalue equal to 0 Which of these systems are conjugate Prove this 8 Consider all linear systems with two zero eigenvalues Which of these systems are conjugate Prove this 9 Provide a complete description of the conjugacy classes for 2 x 2 systems in the nonhyperbolic case Hirsch Ch059780123820105 201222 1117 Page 73 1 5 HigherDimensional Linear Algebra As in Chapter 2 we need to make another detour into the world of linear algebra before proceeding to the solution of higherdimensional linear sys tems of differential equations There are many different canonical forms for matrices in higher dimensions but most of the algebraic ideas involved in changing coordinates to put matrices into these forms are already present in the 2 2 case In particular the case of matrices with distinct real or complex eigenvalues can be handled with minimal additional algebraic com plications so we deal with this case first This is the generic case as we show in Section 56 Matrices with repeated eigenvalues demand more sophisticated concepts from linear algebra we provide this background in Section 54 We assume throughout this chapter that the reader is familiar with solving systems of linear algebraic equations by putting the associated matrix in reduced row echelon form 51 Preliminaries from Linear Algebra In this section we generalize many of the algebraic notions of Section 23 to higher dimensions We denote a vector X Rn in coordinate form as X x1 xn Differential Equations Dynamical Systems and an Introduction to Chaos DOI 101016B9780123820105000051 c 2013 Elsevier Inc All rights reserved 73 Hirsch Ch059780123820105 201222 1117 Page 74 2 74 Chapter 5 HigherDimensional Linear Algebra In the plane we called a pair of vectors V and W linearly independent if they were not collinear Equivalently V and W were linearly independent if there were no nonzero real numbers α and β such that αV βW is the zero vector More generally in Rn a collection of vectors V1Vk in Rn is said to be linearly independent if whenever α1V1 αkVk 0 with αj R it follows that each αj 0 If we can find such α1αk not all of which are 0 then the vectors are linearly dependent Note that if V1Vk are linearly independent and W is the linear combination W β1V1 βkVk then the βj are unique This follows since if we could also write W γ1V1 γkVk then we would have 0 W W β1 γ1V1 βk γkVk which forces βj γj for each j by linear independence of the Vj Example The vectors 100 010 and 001 are clearly linearly independent in R3 More generally let Ej be the vector in Rn where the jth component is 1 and all other components are 0 Then the vectors E1En are linearly independent in Rn The collection of vectors E1En is called the standard basis of Rn We will discuss the concept of a basis in Section 54 Example The vectors 100 110 and 111 in R3 are also linearly independent for if we have α1 1 0 0 α2 1 1 0 α3 1 1 1 α1 α2 α3 α2 α3 α3 0 0 0 then the third component says that α3 0 The fact that α3 0 in the second component then says that α2 0 and finally the first component similarly Hirsch Ch059780123820105 201222 1117 Page 75 3 51 Preliminaries from Linear Algebra 75 tells us that α1 0 On the other hand the vectors 111 123 and 234 are linearly dependent for we have 1 1 1 1 1 1 2 3 1 2 3 4 0 0 0 When solving linear systems of differential equations we often encounter special subsets of Rn called subspaces A subspace of Rn is a collection of all possible linear combinations of a given nonempty set of vectors More precisely given V1Vk Rn the set S α1V1 αkVk αj R is a subspace of Rn In this case we say that S is spanned by V1Vk Equiv alently it can be shown see Exercise 12 at the end of this chapter that a subspace S is a nonempty subset of Rn having the following two properties 1 If XY S then X Y S 2 If X S and α R then αX S Note that the zero vector lies in every subspace of Rn and that any linear combination of vectors in a subspace S also lies in S Example Any straight line through the origin in Rn is a subspace of Rn since this line may be written as tV t R for some nonzero V Rn The single vector V spans this subspace The plane P defined by x y z 0 in R3 is a subspace of R3 Indeed any vector V in P may be written in the form xyx y or V x 1 0 1 y 0 1 1 which shows that the vectors 101 and 011 span P In linear algebra one often encounters rectangular n m matrices but in differential equations most often these matrices are square n n Conse quently we will assume that all matrices in this chapter are n n We write 76 Chapter 5 HigherDimensional Linear Algebra such a matrix M1 42 Ain M1 2 An A Gni 4n2 Ann more compactly as A ajj For X x1X IR we define the product AX to be the vector Vel Ajj AX Viel AnjXj so that the ith entry in this vector is the dot product of the ith row of A with the vector X Matrix sums are defined in the obvious way If A aj and B bj are n X n matrices then we define A B C where C aj bj Matrix arithmetric has some obvious linearity properties 1 Aky X ky Xy k AX ky AX where k ER Xj R 2 A BBA 3 ABCABC The product of the n x n matrices A and B is defined to be the n x n matrix AB cj where n Ci S Aik dkjs k1 so that cj is the dot product of the ith row of A with the jth column of B One checks easily that if A B and C are n x n matrices then 1 ABC ABC 2 AB C AB AC 3 A BC AC BC 4 kAB kKAB AkB for any ke R All of the preceding properties of matrix arithmetic are easily checked by writing out the jjentries of the corresponding matrices It is important to remember that matrix multiplication is not commutative so that AB 4 BA in general For example 1 0f1l 1f1 i 1 10 1 1 2 51 Preliminaries from Linear Algebra 77 whereas 1 11 0 2 1 0 1l 1 1 Also matrix cancellation is usually forbidden if AB AC then we do not necessarily have B C as in 11f1 0f1 0f4 1 faz 172 1 iJlo oJa oJ tt aN 12 In particular if AB is the zero matrix it does not follow that one of A or B is also the zero matrix The n x n matrix A is invertible if there exists an n x n matrix C for which AC CAI where I is the n x n identity matrix that has 1s along the diag onal and 0s elsewhere The matrix C is called the inverse of A Note that if A has an inverse then this inverse is unique For if AB BA I as well then C CI CAB CAB IB B The inverse of A is denoted by A7 If A is invertible then the vector equation AX V has a unique solution for any V R Indeed AV is one solution Moreover it is the only one for if Y is another solution then we have YA AYA NAY A lV For the converse of this statement recall that the equation AX V has unique solutions if and only if the reduced row echelon form of the matrix A is the identity matrix The reduced row echelon form of A is obtained by applying to Aa sequence of elementary row operations of the form 1 Add k times row i of A to row j 2 Interchange row i and j 3 Multiply row iby k 40 Note that these elementary row operations correspond exactly to the opera tions that are used to solve linear systems of algebraic equations 1 Add k times equation i to equation j 2 Interchange equations i and j 3 Multiply equation i by k 0 78 Chapter 5 HigherDimensional Linear Algebra Each of these elementary row operations may be represented by multiplying A by an elementary matrix For example if L j is the matrix that has 1s along the diagonal k for some choice of i and j i j and all other entries are 0 then LA is the matrix obtained by performing row operation 1 on A Similarly if L has 1s along the diagonal with the exception that 0 but 1 and all other entries are 0 then LA is the matrix that results after performing row operation 2 on A Finally if L is the identity matrix with a k instead of 1 in the ii position then LA is the matrix obtained by performing row operation 3 A matrix L in one of these three forms is called an elementary matrix Each elementary matrix is invertible since its inverse is given by the matrix that simply undoes the corresponding row operation As a con sequence any product of elementary matrices is invertible Therefore if IjL are the elementary matrices that correspond to the row operations that put A into the reduced row echelon form that is the identity matrix then Ly L A7 That is if the vector equation AX V has unique solu tions for any V R then A is invertible Thus we have our first important result Proposition Let A be ann xn matrix Then the system of algebraic equa tions AX V has a unique solution for any V R if and only if A is invertible O Thus the natural question now is How do we tell if A is invertible One answer is provided by the following result Proposition The matrix A is invertible if and only if the columns of A form a linearly independent set of vectors Proof Suppose first that A is invertible and has columns Vj V We have AE V where the FE form the standard basis of IR If the V are not lin early independent we may find real numbers a0 not all zero such that 0 Vj 0 But then n n 0 YS ajAkj A S Fj jl jl Thus the equation AX 0 has two solutions the nonzero vector y and the 0 vector This contradicts the previous proposition Conversely suppose that the Vj are linearly independent If A is not invertible then we may find a pair of vectors X and Xz with X 4 X and AX AX Therefore the nonzero vector Z X X satisfies AZ 0 Let 51 Preliminaries from Linear Algebra 79 Z aQ Then we have n 0AZ ajVj jl so that the Vj are not linearly independent This contradiction establishes the result O A more computable criterion for determining whether or not a matrix is invertible as in the 2 x 2 case is given by the determinant of A Given the n x nmatrix A we will denote by Ajj the 1 1 x n 1 matrix obtained by deleting the ith row and jth column of A Definition The determinant of A aj is defined inductively by n detA 01aiz det Aix k1 Note that we know the determinant of a 2 x 2 matrix so this induction makes sense for k 2 Example From the definition we compute 12 3 det 4 5 6 lI1det 6 2det 46 3 det 4 8 9 7 9 7 8 7 8 9 31290 We remark that the definition of det A just given involves expanding along the first row of A One can equally well expand along the jth row so that n det A SoH aj det Aj k1 We will not prove this fact the proof is an entirely straightforward though tedious calculation Similarly det A can be calculated by expanding along a given column see Exercise 1 of this chapter a 80 Chapter 5 HigherDimensional Linear Algebra Example Expanding the matrix in the previous example along the second and third rows yields the same result 1 2 3 det4 5 6 4det 23 5det Is 6det a 8 9 7 9 7 8 7 8 9 24 60 360 2 3 1 3 1 2 7 det 8det 9det 5 21448270 Incidentally note that this matrix is not invertible since 1 2 3 1 0 4 5 6 20 7 8 9 1 0 a The determinant of certain types of matrices is easy to compute A matrix aj is called upper triangular if all entries below the main diagonal are 0 That is aij 0 if i j Lower triangular matrices are defined similarly We have the following proposition Proposition fA is an upper or lower triangular n x n matrix then det A is the product of the entries along the diagonal That is detajj 41 nn UU The proof is a straightforward application of induction The following proposition describes the effects that elementary row operations have on the determinant of a matrix Proposition Let A and B ben x n matrices 1 Suppose matrix B is obtained by adding a multiple of one row of A to another row of A Then det B det A 2 Suppose B is obtained by interchanging two rows of A Then det B det A 3 Suppose B is obtained by multiplying each element of a row of A by k Then det B kdet A Proof The proof of the proposition is straightforward when A is a 2 x 2 matrix so we use induction Suppose A is k x k with k 2 To compute det B we expand along a row that is left untouched by the row operation By induc tion on k we see that det B is a sum of determinants of size k 1 x k1 Hirsch Ch059780123820105 201222 1117 Page 81 9 51 Preliminaries from Linear Algebra 81 Each of these subdeterminants has precisely the same row operation per formed on them as in the case of the full matrix By induction it follows that each of these subdeterminants is multiplied by 1 1 or k in 1 to 3 respectively Thus detB has the same property In particular we note that if L is an elementary matrix then detLA detLdetA Indeed detL 1 1 or k in cases 1 through 3 see Exercise 7 at the end of this chapter The preceding proposition now yields a criterion for A to be invertible Corollary Invertibility Criterion The matrix A is invertible if and only if detA 0 Proof By elementary row operations we can manipulate any matrix A into an upper triangular matrix Then A is invertible if and only if all diagonal entries of this rowreduced matrix are nonzero In particular the determinant of this matrix is nonzero Now by the preceding observation row operations multiply detA by nonzero numbers so we see that all of the diagonal entries are nonzero if and only if detA is also nonzero This concludes the proof This section concludes with a further important property of determinants Proposition detAB detAdetB Proof If either A or B is noninvertible then AB is also noninvertible see Exercise 11 of this chapter Thus the proposition is true since both sides of the equation are zero If A is invertible then we can write A L1 LnI where each Lj is an elementary matrix Thus detAB detL1 LnB detL1detL2 LnB detL1detL2detLndetB detL1 LndetB detAdetB Hirsch Ch059780123820105 201222 1117 Page 82 10 82 Chapter 5 HigherDimensional Linear Algebra 52 Eigenvalues and Eigenvectors As we saw in Chapter 3 eigenvalues and eigenvectors play a central role in the process of solving linear systems of differential equations Definition A vector V is an eigenvector of an n n matrix A if V is a nonzero solution to the system of linear equations A λIV 0 The quan tity λ is called an eigenvalue of A and V is an eigenvector associated with λ Just as in Chapter 2 the eigenvalues of a matrix A may be real or complex and the associated eigenvectors may have complex entries By the Invertibility Criterion of the previous section it follows that λ is an eigenvalue of A if and only if λ is a root of the characteristic equation detA λI 0 Since A is n n this is a polynomial equation of degree n which therefore has exactly n roots counted with multiplicity As we saw in R2 there are many different types of solutions of systems of differential equations and these types depend on the configuration of the eigenvalues of A and the resulting canonical forms There are many many more types of canonical forms in higher dimensions We will describe these types in this and the following sections but we will relegate some of the more specialized proofs of these facts to the exercises at the end of this chapter Suppose first that λ1λℓ are real and distinct eigenvalues of A with associated eigenvectors V1Vℓ Here distinct means that no two of the eigenvalues are equal Thus AVk λkVk for each k We claim that the Vk are linearly independent If not we may choose a maximal subset of the Vi that are linearly independent say V1Vj Then any other eigenvector may be written in a unique way as a linear combination of V1Vj Say Vj1 is one such eigenvector Then we may find αi not all 0 such that Vj1 α1V1 αjVj Multiplying both sides of this equation by A we find λj1Vj1 α1AV1 αjAVj α1λ1V1 αjλjVj Hirsch Ch059780123820105 201222 1117 Page 83 11 52 Eigenvalues and Eigenvectors 83 Now λj1 0 for otherwise we would have α1λ1V1 αjλjVj 0 with each λi 0 This contradicts the fact that V1Vj are linearly indepen dent Thus we have Vj1 α1 λ1 λj1 V1 αj λj λj1 Vj Since the λi are distinct we have now written Vj1 in two different ways as a linear combination of V1Vj This contradicts the fact that this set of vectors is linearly independent We have proved the following proposition Proposition Suppose λ1λℓ are real and distinct eigenvalues for A with associated eigenvectors V1Vℓ Then the Vj are linearly independent Of primary importance when we return to differential equations is the following corollary Corollary Suppose A is an n n matrix with real distinct eigenvalues Then there is a matrix T such that T1AT λ1 λn where all of the entries off the diagonal are 0 Proof Let Vj be an eigenvector associated to λj Consider the linear map T for which TEj Vj where the Ej form the standard basis of Rn That is T is the matrix with columns that are V1Vn Since the Vj are linearly independent T is invertible and we have T1ATEj T1AVj λjT1Vj λjEj That is the jth column of T1AT is just the vector λjEj as required 84 Chapter 5 HigherDimensional Linear Algebra Example Let 12 I1 A0 3 2 0 2 2 Expanding det A AI along the first column we find that the characteristic equation of A is 32 2 det AAD 2aer a 1 A3A2 A 4 1AA2A so the eigenvalues are 21 and 1 The eigenvector corresponding to A 2 is given by solving the equations A 2IX 0 which yields x2yz0 y2z0 2y4z0 These equations reduce to x3z0 y2z0 Thus V 321 is an eigenvector associated to A 2 In similar fashion we find that 100 is an eigenvector associated to A 1 while 0 12 is an eigenvector associated to A 1 Then we set 3 1 0 T 2 0 1 1 0 2 A simple calculation shows that 2 0 O ATT0O 1 O J 0 0 l Hirsch Ch059780123820105 201222 1117 Page 85 13 53 Complex Eigenvalues 85 Since detT 3 T is invertible and we have T1AT 2 0 0 0 1 0 0 0 1 53 Complex Eigenvalues Now we treat the case where A has nonreal complex eigenvalues Suppose α iβ is an eigenvalue of A with β 0 Since the characteristic equation for A has real coefficients it follows that if α iβ is an eigenvalue then so is its complex conjugate α iβ α iβ Another way to see this is the following Let V be an eigenvector associated to α iβ Then the equation AV α iβV shows that V is a vector with complex entries We write V x1 iy1 xn iyn Let V denote the complex conjugate of V V x1 iy1 xn iyn Then we have AV AV α iβV α iβV which shows that V is an eigenvector associated to the eigenvalue α iβ Notice that we have temporarily stepped out of the real world of Rn and into the world Cn of complex vectors This is not really a problem since all of the previous linear algebraic results hold equally well for complex vectors Now suppose that A is a 2n 2n matrix with distinct nonreal eigenval ues αj iβj for j 1n Let Vj and Vj denote the associated eigenvectors 86 Chapter 5 HigherDimensional Linear Algebra Then just as in the previous proposition this collection of eigenvectors is linearly independent That is if we have n YiGVj4jVj 0 jl where the cj and d are now complex numbers then we must have cj dj 0 for each j Now we change coordinates to put A into canonical form Let 1 Wy 449 j Waj Vi Vi Note that W2 and W are both real vectors Indeed W is just the real part of V while W is its imaginary part So working with the W brings us back home to R Proposition The vectors W W2n are linearly independent Proof Suppose not Then we can find real numbers Gj and dj for j1n such that n Do GWoj1 4jWaj 0 jl but not all of the c and dj are zero So we have l n 5 DL GVi Vj idiVjVj 0 jl from which we find n Do G id Vj G id Vj 0 jl Since the V and the V are linearly independent we must have c id 0 from which we conclude c dj 0 for all j This contradiction establishes the result L 53 Complex Eigenvalues 87 Note that we have 1 AWj1 5 AV AV 1 5 ip Vj a iB V a ip Vj Vj Vj Vj 2 2 aWj1 BW Similarly we compute AW BW j1 W Now consider the linear map T for which TE W for j 12n That is the matrix associated to T has columns Wj W2 Note that this matrix has real entries Since the Wj are linearly independent it follows from Section 51 that T is invertible Now consider the matrix T AT We have T7AT Ej T7AWpj1 TWoj1 B Wj aFpj1 BEaj and similarly T7TAT Ej BExj1 Ej Therefore the matrix associated to T AT is D TAT 7 Dn where each Dj is a 2 x 2 matrix of the form Dj Bi By a This is our canonical form for matrices with distinct nonreal eigenvalues Combining the results of this and the previous section we have the follow ing theorem 88 Chapter 5 HigherDimensional Linear Algebra Theorem Suppose that then x n matrix A has distinct eigenvalues Then we may choose a linear map T so that A Xr 1 k T AT D De where the Dj are 2 x 2 matrices in the form Dj i Py 1 Bi i 54 Bases and Subspaces To deal with the case of a matrix with repeated eigenvalues we need some further algebraic concepts Recall that the collection of all linear combinations of a given finite set of vectors is called a subspace of R More precisely given Vi5 Ve R the set S a Vi aV aj R is a subspace of IR In this case we say that S is spanned by Vj Vx Definition Let S be a subspace of R A collection of vectors Vj Vz is a basis of S if the V are linearly independent and span S Note that a subspace always has a basis for if S is spanned by Vj Vi we can always throw away certain of the V to reach a linearly independent subset of these vectors that spans S More precisely if the V are not linearly independent then we may find one of these vectors say Vz for which Ve BiVi B1Ve1 54 Bases and Subspaces 89 Thus we can write any vector in S as a linear combination of the Vi Vi1 alone the vector V is extraneous Continuing in this fashion we eventually reach a linearly independent subset of the V that spans S More important for our purposes is the following proposition Proposition Every basis of a subspace S CR has the same number of elements Proof We first observe that the system of k linear equations in k unknowns given by AX A RExKe 0 Ay X1 agKpexkere 0 always has a nonzero solution Indeed using row reduction we may first solve for one unknown in terms of the others and then we may eliminate this unknown to obtain a system of k 1 equations in k 1 unknowns Thus we are finished by induction the first case k 1 being obvious Now suppose that Vj Vg is a basis for the subspace S Suppose that WiWke is also a basis of S with 0 Then each Wj is a linear combination of the Vj so we have constants aj such that k Wj SaiVi forj1k2 i1 By the preceding observation the system of k equations ke S aijxj 0 fori1k jl has a nonzero solution c4 Then ké k k k ke Vami Lo Levi LLaiavio jl jl i1 i1 jl so that the Wj are linearly dependent This contradiction completes the proof O Hirsch Ch059780123820105 201222 1117 Page 90 18 90 Chapter 5 HigherDimensional Linear Algebra As a consequence of this result we may define the dimension of a subspace S as the number of vectors that form any basis for S In particular Rn is a subspace of itself and its dimension is clearly n Furthermore any other subspace of Rn must have dimension less than n for otherwise we would have a collection of more than n vectors in Rn that are linearly independent This cannot happen by the previous proposition The set consisting of only the 0 vector is also a subspace and we define its dimension to be zero We write dimS for the dimension of the subspace S Example A straight line through the origin in Rn forms a onedimensional subspace of Rn since any vector on this line may be written uniquely as tV where V Rn is a fixed nonzero vector lying on the line and t R is arbitrary Clearly the single vector V forms a basis for this subspace Example The plane P in R3 defined by x y z 0 is a twodimensional subspace of R3 The vectors 101 and 011 both lie in P and are linearly independent If W P we may write W x y y x x 1 0 1 y 0 1 1 so these vectors also span P As in the planar case we say that a function T Rn Rn is linear if TX AX for some n n matrix A T is called a linear map or linear transformation Using the properties of matrices discussed in Section 51 we have TαX βY αTX βTY for any αβ R and XY Rn We say that the linear map T is invertible if the matrix A associated to T has an inverse For the study of linear systems of differential equations the most important types of subspaces are the kernels and ranges of linear maps We define the kernel of T denoted KerT to be the set of vectors mapped to 0 by T The range of T consists of all vectors W for which there exists a vector V for which TV W This of course is a familiar concept from calculus The difference here is that the range of T is always a subspace of Rn Hirsch Ch059780123820105 201222 1117 Page 91 19 54 Bases and Subspaces 91 Example Consider the linear map TX 0 1 0 0 0 1 0 0 0 X If X xyz then TX y z 0 Thus KerT consists of all vectors of the form α00 while RangeT is the set of vectors of the form βγ 0 where αβγ R Both sets are clearly subspaces Example Let TX AX 1 2 3 4 5 6 7 8 9 X For KerT we seek vectors X that satisfy AX 0 Using row reduction we find that the reduced row echelon form of A is the matrix 1 0 1 0 1 2 0 0 0 Thus the solutions X xyz of AX 0 satisfy x z y 2z Therefore any vector in KerT is of the form z2zz so KerT has dimension 1 For RangeT note that the columns of A are vectors in RangeT since they are the images of 100 010 and 001 respectively These vectors are not linearly independent since 1 1 4 7 2 2 5 8 3 6 9 However 147 and 258 are linearly independent so these two vectors give a basis of RangeT 92 Chapter 5 HigherDimensional Linear Algebra Proposition Let T R R be a linear map Then Ker T and Range T are both subspaces of IR Moreover dim Ker T dim Range T n Proof First suppose that Ker T 0 Let EE be the standard basis of IR Then we claim that TE TE are linearly independent If this is not the case then we may find not all 0 such that n 7 TE 0 jl But then we have n T S aE 0 jl which implies that jE Ker T so that aE 0 Thus each a 0 which is a contradiction so the vectors TE are linearly independent But then given V R we may write n V Bj TE jl for some f8n Thus n VT Bie jl which shows that Range T R Thus both Ker T and Range T are subspaces of IR and we have dim Ker T 0 and dim Range T n If Ker T 0 we may find a nonzero vector V Ker T Clearly Ta V 0 for any a ER so all vectors of the form a V lie in Ker T If Ker T contains additional vectors choose one and call it V2 Then Ker T contains all linear combinations of Vj and V3 since Ta1 V 02 V2 a TV a2TV2 0 Continuing in this fashion we obtain a set of linearly independent vectors that span Ker T thus showing that Ker T is a subspace Note that this process must Hirsch Ch059780123820105 201222 1117 Page 93 21 55 Repeated Eigenvalues 93 end since every collection of more than n vectors in Rn is linearly dependent A similar argument works to show that RangeT is a subspace Now suppose that V1Vk form a basis of KerT where 0 k n the case where k n being obvious Choose vectors Wk1Wn so that V1VkWk1Wn form a basis of Rn Let Zj TWj for each j Then the vectors Zj are linearly independent for if we had αk1Zk1 αnZn 0 then we would also have Tαk1Wk1 αnWn 0 This implies that αk1Wk1 αnWn KerT But this is impossible since we cannot write any Wj and thus any linear com bination of the Wj as a linear combination of the Vi This proves that the sum of the dimensions of KerT and RangeT is n We remark that it is easy to find a set of vectors that spans RangeT simply take the set of vectors that comprise the columns of the matrix associated to T This works since the ith column vector of this matrix is the image of the stan dard basis vector Ei under T In particular if these column vectors are linearly independent then KerT 0 and there is a unique solution to the equation TX V for every V Rn Thus we have this corollary Corollary 1 If T Rn Rn is a linear map with dim KerT 0 then T is invertible 55 Repeated Eigenvalues In this section we describe the canonical forms that arise when a matrix has repeated eigenvalues Rather than spending an inordinate amount of time developing the general theory in this case we will give the details only for 3 3 and 4 4 matrices with repeated eigenvalues More general cases are relegated to the exercises of this chapter We justify this omission in the next section where we show that the typical matrix has distinct eigenvalues and thus can be handled as in the previous sec tion If you happen to meet a random matrix while walking down the street 94 Chapter 5 HigherDimensional Linear Algebra the chances are very good that this matrix will have distinct eigenvalues The most general result regarding matrices with repeated eigenvalues is given by the following proposition Proposition Let A be ann x n matrix Then there is a change of coordinates T for which By T AT Bx where each of the Bjs is a square matrix and all other entries are zero of one of the following forms A 1 Q h A 1 Q h i nr ii Mets oy oE r Q where a B 1 0 a n and where a BA IR with B 0 The special cases where B A or a 6B 5 Sp are of course allowed We first consider the case of R If A has repeated eigenvalues in R then all eigenvalues must be real There are then two cases Either there are two distinct eigenvalues one of which is repeated or else all eigenvlaues are the same The former case can be handled by a process similar to that described in Chapter 3 so we restrict our attention here to the case where A has a single eigenvalue A of multiplicity 3 Hirsch Ch059780123820105 201222 1117 Page 95 23 55 Repeated Eigenvalues 95 Proposition Suppose A is a 3 3 matrix for which λ is the only eigenvalue Then we may find a change of coordinates T such that T1AT assumes one of the following three forms i λ 0 0 0 λ 0 0 0 λ ii λ 1 0 0 λ 0 0 0 λ iii λ 1 0 0 λ 1 0 0 λ Proof Let K be the kernel of A λI Any vector in K is an eigenvector of A There are then three subcases depending on whether the dimension of K is 1 2 or 3 If the dimension of K is 3 then A λIV 0 for any V R3 Thus A λI This yields matrix i Suppose the dimension of K is 2 Let R be the range of A λI Then R has dimension 1 since dimK dimR 3 as we saw in the previous section We claim that R K If this is not the case let V R be a nonzero vector Since A λIV R and R is onedimensional we must have A λIV µV for some µ 0 But then AV λ µV so we have found a new eigenvalue λ µ This contradicts our assumption so we must have R K Now let V1 R be nonzero Since V1 K V1 is an eigenvector and so A λIV1 0 Since V1 also lies in R we may find V2 R3 K with A λIV2 V1 Since K is twodimensional we may choose a second vec tor V3 K such that V1 and V3 are linearly independent Note that V3 is also an eigenvector If we now choose the change of coordinates TEj Vj for j 123 then it follows easily that T1AT assumes the form of case ii Finally suppose that K has dimension 1 Thus R has dimension 2 We claim that in this case K R If this is not the case then A λIR R and so A λI is invertible on R Thus if V R there is a unique W R for which A λIW V In particular we have AV AA λIW A2 λAW A λIAW This shows that if V R then so too is AV Thus A also preserves the subspace R It then follows immediately that A must have an eigenvector in R but this then says that K R and we have a contradiction Next we claim that A λIR K To see this note that A λIR is one dimensional since K R If A λIR K there is a nonzero vector V K for which A λIR tV where t R But then A λIV tV for some Hirsch Ch059780123820105 201222 1117 Page 96 24 96 Chapter 5 HigherDimensional Linear Algebra t Rt 0 and so AV t λV yields another new eigenvalue Thus we must in fact have A λIR K Now let V1 K be an eigenvector for A As before there exists V2 R such that A λIV2 V1 Since V2 R there exists V3 such that A λIV3 V2 Note that A λI2V3 V1 The Vj are easily seen to be linearly inde pendent Moreover the linear map defined by TEj Vj finally puts A into canonical form iii This completes the proof Example Suppose A 2 0 1 0 2 1 1 1 2 Expanding along the first row we find detA λI 2 λ2 λ2 1 2 λ 2 λ3 so the only eigenvalue is 2 Solving A 2IV 0 yields only one indepen dent eigenvector V1 110 so we are in case iii of the proposition We compute A 2I2 1 1 0 1 1 0 0 0 0 so that the vector V3 100 solves A 2I2V3 V1 We also have A 2IV3 V2 001 As in the preceding we let TEj Vj for j 123 so that T 1 0 1 1 0 0 0 1 0 Then T1AT assumes the canonical form T1AT 2 1 0 0 2 1 0 0 2 Hirsch Ch059780123820105 201222 1117 Page 97 25 55 Repeated Eigenvalues 97 Example Now suppose A 1 1 0 1 3 0 1 1 2 Again expanding along the first row we find detA λI 1 λ3 λ2 λ 2 λ 2 λ3 so again the only eigenvalue is 2 This time however we have A 2I 1 1 0 1 1 0 1 1 0 so that we have two linearly independent eigenvectors xyz for which we must have x y while z is arbitrary Note that A 2I2 is the zero matrix so we may choose any vector that is not an eigenvector as V2 say V2 100 Then A 2IV2 V1 111 is an eigenvector A second linearly independent eigenvector is then V3 001 for example Defining TEj Vj as usual then yields the canonical form T1AT 2 1 0 0 2 0 0 0 2 Now we turn to the 4 4 case The case of all real eigenvalues is similar to the 3 3 case though a little more complicated algebraically and is left as an exercise at the end of this chapter Thus we assume that A has repeated complex eigenvalues α iβ with β 0 There are just two cases either we can find a pair of linearly independent eigenvectors corresponding to α iβ or we can find only one such eigenvec tor In the former case let V1 and V2 be the independent eigenvectors The V1 and V2 are linearly independent eigenvectors for α iβ As before choose the real vectors W1 V1 V12 W2 iV1 V12 W3 V2 V22 W4 iV2 V22 98 Chapter 5 HigherDimensional Linear Algebra If we set TE Wj then changing coordinates via T puts A in canonical form a B 0 0 p a 0 0 T AT 0 0 aw Bl 0 0 6 a If we find only one eigenvector V for w if then we solve the system of equations A a iBIX Vj as in the case of repeated real eigenvalues The proof of the previous proposition shows that we can always find a nonzero solution V2 of these equations Then choose the W as before and set TE Wj Then T puts A into the canonical form a B 1 O p a 0 1 T AT 0 0 a BI 0 0 6 a For example we compute TATE TAW3 T7AV2 V22 T7Vi oe iB V22 Vi a iBV22 TVi Vi2 V2 Va2 iB V2 V22 FE a 3 BEg Example Let 1 1 0 1 2 1 1 O A1o 0 1 2 0 0O 1 1 The characteristic equation after a little computation is 7 1 0 Thus A has eigenvalues i each repeated twice Solving the system AiIX 0 yields one linearly independent com plex eigenvector V 11 i00 associated to i Then Vj is an eigenvector associated to the eigenvalue i 55 Repeated Eigenvalues 99 Next we solve the system A ux V to find V2 001 i1 Then V2 solves the system A iI X Vj Finally choose W Vit Vi2Re Vy W2 iVi Vi2ImV W3 V2 V22 Re V2 Wg iV2 V22 Im V2 and let TE W for j 14 We have 1 0 0 O 1 0O 0 O f1 1 0 0 Jl 1 0 0 T 0 0 1 1 rT 0 O 0 17 0 oO 1 0 0 oO 1 1 and we find the canonical form 0 1 1 O il1 0 0 1 reat 0 0 0 1f 0 0 1 0 7 Example Let 2 0 1 0 0 2 01 A1o 0 2 0f 0 1 0 2 The characteristic equation for A is 222AY 1 0 so the eigenvalues are 2 i and 2 with multiplicity 2 Solving the equations A2i1IX 0 yields an eigenvector V 0 10 1 for 2 i Let W 000 1 and W 0 100 be the real and imaginary parts of V Solving the equations A 2IX 0 yields only one eigenvector associated to 2 namely W3 1000 Then we solve A 21 X W3 to find W4 100 Chapter 5 HigherDimensional Linear Algebra 00 10 Setting TE W as usual puts A into the canonical form 2 1 0 0 1 2 0 0 peal 0 02 1 0 00 2 as is easily checked a 56 Genericity We have mentioned several times that most matrices have distinct eigenval ues Our goal in this section is to make this precise Recall that a set C R is open if whenever X U there is an open ball about X contained in U that is for some a 0 depending on X the open ball about X of radius a Y ER YX a is contained in U Using geometrical language we say that if X belongs to an open set U any point sufficiently near to X also belongs to U Another kind of subset of R is a dense set U C R is dense if there are points in arbitrarily close to each point in R More precisely if X R then for every 0 there exists some Y U with X Y Equivalently U is dense in R if VNU is nonempty for every nonempty open set V C R For example the rational numbers form a dense subset of R as do the irrational numbers Similarly x y Rboth x and y are rational is a dense subset of the plane An interesting kind of subset of R is a set that is both open and dense Such a set U is characterized by the following properties Every point in the com plement of 7 can be approximated arbitrarily closely by points of U since U is dense but no point in U can be approximated arbitrarily closely by points in the complement because U is open Here is a simple example of an open and dense subset of R Vxy R xy Al Hirsch Ch059780123820105 201222 1117 Page 101 29 56 Genericity 101 This of course is the complement in R2 of the hyperbola defined by xy 1 Suppose x0y0 V Then x0y0 1 and if x x0 y y0 are small enough then xy 1 this proves that V is open Given any x0y0 R2 we can find xy as close as we like to x0y0 with xy 1 this proves that V is dense An open and dense set is a very fat set as the following proposition shows Proposition Let V1Vm be open and dense subsets of Rn Then V V1 Vm is also open and dense Proof It can be easily shown that the intersection of a finite number of open sets is open so V is open To prove that V is dense let U Rn be a nonempty open set Then U V1 is nonempty since V1 is dense Because U and V1 are open U V1 is also open Since U V1 is open and nonempty U V1 V2 is nonempty because V2 is dense Since V2 is open U V1 V2 is open Thus U V1 V2 V3 is nonempty and so on So U V is nonempty which proves that V is dense in Rn We therefore think of a subset of Rn as being large if this set contains an open and dense subset To make precise what we mean by most matrices we need to transfer the notion of an open and dense set to the set of all matrices Let LRn denote the set of n n matrices or equivalently the set of linear maps of Rn In order to discuss open and dense sets in LRn we need to have a notion of how far apart two given matrices in LRn are But we can do this by simply writing all of the entries of a matrix as one long vector in a specified order and thereby thinking of LRn as Rn2 Theorem The set M of matrices in LRn that have n distinct eigenvalues is open and dense in LRn Proof We first prove that M is dense Let A LRn Suppose that A has some repeated eigenvalues The proposition from the previous section states that we can find a matrix T such that T1AT assumes one of two forms Either we have a canonical form with blocks along the diagonal of the form i λ 1 λ 1 1 λ or ii C2 I2 C2 I2 I2 C2 102 Chapter 5 HigherDimensional Linear Algebra where a 6A R with 6 4 0 and a Bp 1 0 a n i or else we have a pair of separate diagonal blocks A or C Either case can be handled as follows Choose distinct values 4 such that A A is as small as desired and replace the preceding block i with Ai 1 A2 1 oy Aj This new block now has distinct eigenvalues In block ii we may similarly replace each 2 x 2 block a 6 B a with distinct ajs The new matrix thus has distinct eigenvalues a In this fashion we find a new matrix B arbitrarily close to T AT with distinct eigen values Then the matrix TBT also has distinct eigenvalues and moreover this matrix is arbitrarily close to A Indeed the funtion F LR LCR given by FM TMT where T is a fixed invertible matrix is a continuous function on LR and thus takes matrices close to TAT to new matrices close to A This shows that M is dense To prove that M is open consider the characteristic polynomial of a matrix A LR If we vary the entries of A slightly then the characteristic polyno mials coefficients vary only slightly Therefore the roots of this polynomial in C move only slightly as well Thus if we begin with a matrix that has distinct eigenvalues nearby matrices have this property as well This proves that M is open A property P of matrices is a generic property if the set of matrices hav ing property P contains an open and dense set in LR Thus a property is generic if it is shared by some open and dense set of matrices and perhaps other matrices as well Intuitively speaking a generic property is one that almost all matrices have Thus having all distinct eigenvalues is a generic property of n x m matrices Hirsch Ch059780123820105 201222 1117 Page 103 31 Exercises 103 E X E R C I S E S 1 Prove that the determinant of a 3 3 matrix can be computed by expanding along any row or column 2 Find the eigenvalues and eigenvectors of the following matrices a 0 0 1 0 1 0 1 0 0 b 0 0 1 0 2 0 3 0 0 c 1 1 1 1 1 1 1 1 1 d 0 0 2 0 2 0 2 0 0 e 3 0 0 1 0 1 2 2 1 2 1 4 1 0 0 3 3 Describe the regions in abcspace where the matrix 0 0 a 0 b 0 c 0 0 has real complex and repeated eigenvalues 4 Describe the regions in abcspace where the matrix a 0 0 a 0 a b 0 0 c a 0 a 0 0 a has real complex and repeated eigenvalues 5 Put the following matrices in canonical form a 0 0 1 0 1 0 1 0 0 b 1 0 1 0 1 0 0 0 1 c 0 1 0 1 0 0 1 1 1 d 0 1 0 1 0 0 1 1 1 e 1 0 1 0 1 0 1 0 1 f 1 1 0 1 1 1 0 1 1 104 Chapter 5 HigherDimensional Linear Algebra LON fo ao g 1 1 l h 00 1 0 0 1 0 1 0 0 0 6 Suppose that a 5 x 5 matrix has eigenvalues 2 and 1 i List all possible canonical forms for a matrix of this type 7 Let L be the elementary matrix that interchanges the ith and jth rows of a given matrix That is L has 1s along the diagonal with the exception that 0 but 1 Prove that det L 1 8 Find a basis for both Ker T and Range T when T is the matrix 12 111 19 6 b 1 1 1 c 1 4 1 111 2 7 1 9 Suppose A is a 4 x 4 matrix that has a single real eigenvalue 4 and only one independent eigenvector Prove that A may be put in canonical form X 1 0 0 0 Al O 0 0A 17 000A 10 Suppose A is a 4 x 4 matrix with a single real eigenvalue and two linearly independent eigenvectors Describe the possible canonical forms for A and show that A may indeed be transformed into one of these canonical forms Describe explicitly the conditions under which A is transformed into a particular form 11 Show that if A andor B are noninvertible matrices then AB is also non invertible 12 Suppose that S is a subset of R having the following properties a If XY SthenXYeS b IfX S anda eRthenaxéeéS Prove that S may be written as the collection of all possible linear combinations of a finite set of vectors 13 Which of the following subsets of R are open andor dense Give a brief reason in each case a Uyly 0 b Uby e y 1 c U3 x y x is irrational Hirsch Ch059780123820105 201222 1117 Page 105 33 Exercises 105 d U4 xyx and y are not integers e U5 is the complement of a set C1 where C1 is closed and not dense f U6 is the complement of a set C2 that contains exactly 6 billion and 2 distinct points 14 Each of the following properties defines a subset of real n n matrices Which of these sets are open andor dense in the LRn Give a brief reason in each case a Det A 0 b Trace A is rational c Entries of A are not integers d 3 detA 4 e 1 λ 1 for every eigenvalue λ f A has no real eigenvalues g Each real eigenvalue of A has multiplicity 1 15 Which of the following properties of linear maps on Rn are generic a λ 1 for every eigenvalue λ b n 2 one eigenvalue is not real c n 3 one eigenvalue is not real d No solution of X AX is periodic except the zero solution e There are n distinct eigenvalues each with distinct imaginary parts f AX X and AX X for all X 0 Hirsch Ch069780123820105 201222 1139 Page 107 1 6 HigherDimensional Linear Systems After our little sojourn into the world of linear algebra its time to return to differential equations and in particular to the task of solving higher dimensional linear systems with constant coefficients As in the linear algebra chapter we have to deal with a number of different cases 61 Distinct Eigenvalues Consider first a linear system X AX where the n n matrix A has n dis tinct real eigenvalues λ1λn By the results in Chapter 5 there is a change of coordinates T so that the new system Y T1ATY assumes the particularly simple form y 1 λ1y1 y n λnyn Differential Equations Dynamical Systems and an Introduction to Chaos DOI 101016B9780123820105000063 c 2013 Elsevier Inc All rights reserved 107 108 Chapter 6 HigherDimensional Linear Systems The linear map T is the map that takes the standard basis vector EF to the eigenvector V associated with Aj Clearly a function of the form Cl eit Yt cyernt is a solution of Y TATY that satisfies the initial condition Y0 c15C As in Chapter 3 this is the only such solution for if wt Wt Wp Tt is another solution then differentiating each expression wjt expAjt we find d hjt hit a vive v w Ajwje J 0 Thus wt cjexpAt for each j Therefore the collection of solutions Yt yields the general solution of Y TATY It then follows that Xt TYt is the general solution of X AX so this general solution may be written in the form n Xt So gei Vj jl Now suppose that the eigenvalues 41A of A are negative while the eigenvalues Ax41A are positive Since there are no zero eigenvalues the system is hyperbolic Then any solution that starts in the subspace spanned by the vectors V Vz must first of all stay in that subspace for all time since Ckt1 Cy 0 Second each such solution tends to the origin as f oo In analogy with the terminology introduced for planar systems we call this subspace the stable subspace Similarly the subspace spanned by Vj1V contains solutions that move away from the origin This subspace is the unstable subspace All other solutions tend toward the stable subspace as time goes backward and toward the unstable subspace as time increases Therefore this system is a higherdimensional analogue of a saddle Hirsch Ch069780123820105 201222 1139 Page 109 3 61 Distinct Eigenvalues 109 Example Consider X 1 2 1 0 3 2 0 2 2 X In Chapter 5 Section 52 we showed that this matrix has eigenvalues 21 and 1 with associated eigenvectors 321 100 and 012 respectively Therefore the matrix T 3 1 0 2 0 1 1 0 2 converts X AX to Y T1ATY 2 0 0 0 1 0 0 0 1 Y which we can solve immediately Multiplying the solution by T then yields the general solution Xt c1e2t 3 2 1 c2et 1 0 0 c3et 0 1 2 of X AX The line through the origin and 012 is the stable line while the plane spanned by 321 and 100 is the unstable plane A collection of solutions of this system as well as the system Y T1ATY is displayed in Figure 61 Example If the 3 3 matrix A has three real distinct eigenvalues that are negative then we may find a change of coordinates so that the system assumes the form Y T1ATY λ1 0 0 0 λ2 0 0 0 λ3 Y where λ3 λ2 λ1 0 All solutions therefore tend to the origin and so we have a higherdimensional sink See Figure 62 For an initial condition x0y0z0 with all three coordinates nonzero the corresponding solution tends to the origin tangentially to the xaxis see Exercise 2 at the end of the chapter Hirsch Ch069780123820105 201222 1139 Page 110 4 110 Chapter 6 HigherDimensional Linear Systems z x y T 01 2 Figure 61 Stable and unstable subspaces of a saddle in dimension 3 On the left the system is in canonical form z y x Figure 62 A sink in three dimensions Now suppose that the n n matrix A has n distinct eigenvalues of which k1 are real and k2 are nonreal so that n k1 2k2 Then as in Chapter 5 we may change coordinates so that the system assumes the form x j λjxj u ℓ αℓuℓ βℓvℓ v ℓ βℓuℓ αℓvℓ for j 1k1 and ℓ 1k2 As in Chapter 3 we therefore have solutions of the form xjt cjeλjt uℓt pℓeαℓt cosβℓt qℓeαℓt sinβℓt vℓt pℓeαℓt sinβℓt qℓeαℓt cosβℓt As before it is straightforward to check that this is the general solution We have therefore shown the following theorem 61 Distinct Eigenvalues 111 Theorem Consider the system X AX where A has distinct eigenvalues Aly Ak R and a iB0 iB C Let T be the matrix that puts A in the canonical form Ay Xr lan ki T AT Bi Br where B a P i Bi a Then the general solution of X AX is TYt where cet Chy enh t aye cos Bit bye sin Bit Yt oly t gs ey t aye sin Bit bye cos Bit ag 2 cos By t by e2 sin By t ag ew sin By t bp e2 cos Biot As usual the columns of the matrix T in this theorem are the eigenvectors or the real and imaginary parts of the eigenvectors corresponding to each eigenvalue Also as before the subspace spanned by the eigenvectors corres ponding to eigenvalues with negative resp positive real parts is the stable resp unstable subspace Example Consider the system 0 1 0 X1 0 0 X 0 0 l Hirsch Ch069780123820105 201222 1139 Page 112 6 112 Chapter 6 HigherDimensional Linear Systems x2 y2 a2 Figure 63 Phase portrait for a spiral center with a matrix that is already in canonical form The eigenvalues are i1 The solution satisfying the initial condition x0y0z0 is given by Yt x0 cost sint 0 y0 sint cost 0 z0et 0 0 1 so this is the general solution The phase portrait for this system is displayed in Figure 63 The stable line lies along the zaxis whereas all solutions in the xyplane travel around circles centered at the origin In fact each solution that does not lie on the stable line actually lies on a cylinder in R3 given by x2 y2 constant These solutions spiral toward the periodic solution in the xyplane if z0 0 Example Now consider X AX where A 01 0 1 1 1 11 1 0 01 The characteristic equation is λ3 08λ2 081λ 101 0 Hirsch Ch069780123820105 201222 1139 Page 113 7 61 Distinct Eigenvalues 113 which we have kindly factored for you into 1 λλ2 02λ 101 0 Therefore the eigenvalues are the roots of this equation which are 1 and 01 i Solving A 01 iIX 0 yields the eigenvector i11 asso ciated with 01 i Let V1 Rei11 011 and V2 Imi11 100 Solving A IX 0 yields V3 010 as an eigenvector corre sponding to λ 1 Then the matrix with columns that are the Vi T 0 1 0 1 0 1 1 0 0 converts X AX into Y 01 1 0 1 01 0 0 0 1 Y This system has an unstable line along the zaxis while the xyplane is the stable plane Note that solutions spiral into 0 in the stable plane We call this system a spiral saddle See Figure 64 Typical solutions off the stable plane spi ral toward the zaxis while the zcoordinate meanwhile increases or decreases See Figure 65 Figure 64 A spiral saddle in canonical form Hirsch Ch069780123820105 201222 1139 Page 114 8 114 Chapter 6 HigherDimensional Linear Systems Figure 65 Typical spiral saddle solutions tend to spiral toward the unstable line 62 Harmonic Oscillators Consider a pair of undamped harmonic oscillators with equations x 1 ω2 1x1 x 2 ω2 2x2 We can almost solve these equations by inspection as visions of sinωt and cosωt pass through our minds But lets push on a bit first to illustrate the theorem in the previous section in the case of nonreal eigenvalues but more importantly to introduce some interesting geometry We first introduce the new variables yj x j for j 12 so that the equations may be written as a system x j yj y j ω2 j xj In matrix form this system is X AX where X x1y1x2y2 and A 0 1 ω2 1 0 0 1 ω2 2 0 This system has eigenvalues iω1 and iω2 An eigenvector corresponding to iω1 is V1 1iω100 while V2 001iω2 is associated with iω2 Let W1 Hirsch Ch069780123820105 201222 1139 Page 115 9 62 Harmonic Oscillators 115 and W2 be the real and imaginary parts of V1 and let W3 and W4 be the same for V2 Then as usual we let TEj Wj and the linear map T puts this system into canonical form with the matrix T1AT 0 ω1 ω1 0 0 ω2 ω2 0 We then see that the general solution of Y T1ATY is Yt x1t y1t x2t y2t a1 cosω1t b1 sinω1t a1 sinω1t b1 cosω1t a2 cosω2t b2 sinω2t a2 sinω2t b2 cosω2t just as we originally expected We could say that this is the end of the story and stop here since we have the formulas for the solution However lets push on a bit more Each pair of solutions xjtyjt for j 12 is clearly a periodic solu tion of the equation with period 2πωj but this does not mean that the full fourdimensional solution is a periodic function Indeed the full solution is a periodic function with period τ if and only if there exist integers m and n such that ω1τ m2π and ω2τ n2π Thus for periodicity we must have τ 2πm ω1 2πn ω2 or equivalently ω2 ω1 n m That is the ratio of the two frequencies of the oscillators must be a rational number In Figure 66 we have plotted x1tx2t for the particular solution of this system when the ratio of the frequencies is 52 When the ratio of the frequencies is irrational something very different happens To understand this we make another and much more familiar change of coordinates In canonical form our system currently is x j ωjyj y j ωjxj Hirsch Ch069780123820105 201222 1139 Page 116 10 116 Chapter 6 HigherDimensional Linear Systems x1 x2 Figure 66 A solution with frequency ratio 52 projected into the x1x2plane Note that x2t oscillates five times and x1t only twice before returning to the initial position Lets now introduce polar coordinates rjθj in place of the xj and yj variables Differentiating r2 j x2 j y2 j we find 2rjr j 2xjx j 2yjy j 2xjyjωj 2xjyjωj 0 Therefore r j 0 for each j Also differentiating the equation tanθj yj xj yields sec2θjθ j y jxj yjx j x2 j ωjr2 j r2 j cos2 θj from which we find θ j ωj Hirsch Ch069780123820105 201222 1139 Page 117 11 62 Harmonic Oscillators 117 So in polar coordinates these equations really are quite simple r j 0 θ j ωj The first equation tells us that both r1 and r2 remain constant along any solution Then no matter what we pick for our initial r1 and r2 values the θj equations remain the same Thus we may as well restrict our attention to r1 r2 1 The resulting set of points in R4 is a torusthe surface of a doughnutalthough this is a little difficult to visualize in fourdimensional space However we know that we have two independent variables on this set namely θ1 and θ2 and both are periodic with period 2π So this is akin to the two independent circular directions that parametrize the familiar torus in R3 Restricted to this torus the equations now read θ 1 ω1 θ 2 ω2 It is convenient to think of θ1 and θ2 as variables in a square of sidelength 2π where we glue together the opposite sides θj 0 and θj 2π to make the torus In this square our vector field now has constant slope θ 2 θ 1 ω2 ω1 Therefore solutions lie along straight lines with slope ω2ω1 in this square When a solution reaches the edge θ1 2π say at θ2 c it instantly reap pears on the edge θ1 0 with θ2 coordinate given by c and then continues onward with slope ω2ω1 A similar identification occurs when the solution meets θ2 2π So now we have a simplified geometric vision of what happens to these solu tions on these tori But what really happens The answer depends on the ratio ω2ω1 If this ratio is a rational number say nm then the solution starting at θ10θ20 will pass through the torus horizontally exactly m times and vertically n times before returning to its starting point This is the periodic solution we observed previously Incidentally the picture of the straightline solutions in the θ1θ2plane is not at all the same as our depiction of solutions in the x1x2plane as shown in Figure 66 In the irrational case something quite different occurs See Figure 67 To understand what is happening here we return to the notion of a Poincare map discussed in Chapter 1 Consider the circle θ1 0 the left edge of our square Hirsch Ch069780123820105 201222 1139 Page 118 12 118 Chapter 6 HigherDimensional Linear Systems Figure 67 A solution with frequency ratio 2 projected into the x1x2plane the left curve computed up to time 50π the right to time 100π representation of the torus Given an initial point on this circle say θ2 x0 we follow the solution starting at this point until it next hits θ1 2π By our identification this solution has now returned to the circle θ1 0 The solution may cross the boundary θ2 2π several times in making this transit but it does eventually return to θ1 0 So we may define the Poincare map on θ1 0 by assigning to x0 on this circle the corresponding coordinate of the point of first return Suppose that this first return occurs at the point θ2τ where τ is the time for which θ1τ 2π Since θ1t θ10 ω1t we have τ 2πω1 Thus θ2τ x0 ω22πω1 Therefore the Poincare map on the circle may be written as f x0 x0 2πω2ω1 mod2π where x0 θ20 is our initial θ2 coordinate on the circle See Figure 68 Thus the Poincare map on the circle is just the function that rotates points on the circle by angle 2πω2ω1 Since ω2ω1 is irrational this function is called an irrational rotation of the circle Definition The set of points x0x1 f x0x2 f f x0xn f xn1 is called the orbit of x0 under iteration of f The orbit of x0 tracks how our solution successively crosses θ1 2π as time increases Proposition Suppose ω2ω1 is irrational Then the orbit of any initial point x0 on the circle θ1 0 is dense in the circle Hirsch Ch069780123820105 201222 1139 Page 119 13 62 Harmonic Oscillators 119 x1 fx0 θ1 0 θ1 2π x0 x2 Figure 68 Poincare map on the circle θ1 0 in the θ1θ2torus Proof Recall from Chapter 5 Section 6 that a subset of the circle is dense if there are points in this subset that are arbitrarily close to any point whatsoever in the circle Therefore we must show that given any point z on the circle and any ϵ 0 there is a point xn on the orbit of x0 such that z xn ϵ where z and xn are measured mod 2π To see this observe first that there must be nm for which m n and xn xm ϵ Indeed we know that the orbit of x0 is not a finite set of points since ω2ω1 is irrational Thus there must be at least two of these points where the distance apart is less than ϵ since the circle has finite circumference These are the points xn and xm actually there must be infinitely many such points Now rotate these points in the reverse direction exactly n times The points xn and xm are rotated to x0 and xmn respectively We find after this rotation that x0 xmn ϵ Now xmn is given by rotating the circle through angle m n2πω2ω1 in which mod 2π is therefore a rotation of angle less than ϵ Thus performing this rotation again we find x2mn xmn ϵ as well and inductively xkmn xk1mn ϵ for each k Thus we have found a sequence of points obtained by repeated rotation through angle m n2πω2ω1 and each of these points is within ϵ of its predecessor Thus there must be a point of this form within ϵ of z Since the orbit of x0 is dense in the circle θ1 0 it follows that the straight line solutions connecting these points in the square are also dense and so the original solutions are dense in the torus on which they reside This accounts Hirsch Ch069780123820105 201222 1139 Page 120 14 120 Chapter 6 HigherDimensional Linear Systems for the densely packed solution shown projected into the x1x2plane shown in Figure 67 when ω2ω1 2 Returning to the actual motion of the oscillators we see that when ω2ω1 is irrational the masses do not move in periodic fashion However they do come back very close to their initial positions over and over again as time goes on due to the density of these solutions on the torus These types of motions are called quasiperiodic motions In Exercise 7 at the end of this chapter we investigate a related set of equations namely a pair of coupled oscillators 63 Repeated Eigenvalues As we saw in the previous chapter the solution of systems with repeated real eigenvalues reduces to solving systems with matrices that contain blocks of the form λ 1 λ 1 1 λ Example Let X λ 1 0 0 λ 1 0 0 λ X The only eigenvalue for this system is λ and its only eigenvector is 100 We may solve this system as we did in Chapter 3 by first noting that x 3 λx3 so we must have x3t c3eλt Now we must have x 2 λx2 c3eλt As in Chapter 3 we guess a solution of the form x2t c2eλt αteλt Substituting this guess into the differential equation for x 2 we determine that α c3 and find x2t c2eλt c3teλt Hirsch Ch069780123820105 201222 1139 Page 121 15 63 Repeated Eigenvalues 121 Finally the equation x 1 λx1 c2eλt c3teλt suggests the guess x1t c1eλt αteλt βt2eλt Solving as before we find x1t c1eλt c2teλt c3 t2 2 eλt Altogether we find Xt c1eλt 1 0 0 c2eλt t 1 0 c3eλt t22 t 1 which is the general solution Despite the presence of the polynomial terms in this solution when λ 0 the exponential term dominates and all solu tions do tend to zero Some representative solutions when λ 0 are shown in Figure 69 Note that there is only one straightline solution for this system this solution lies on the xaxis Also the xyplane is invariant and solutions there behave exactly as in the planar repeated eigenvalue case x z Figure 69 Phase portrait for repeated real eigenvalues Hirsch Ch069780123820105 201222 1139 Page 122 16 122 Chapter 6 HigherDimensional Linear Systems Example Consider the fourdimensional system x 1 x1 x2 x3 x 2 x2 x4 x 3 x3 x4 x 4 x4 We may write this system in matrix form as X AX 1 1 1 0 0 1 0 1 0 0 1 1 0 0 0 1 X Since A is upper triangular all of the eigenvalues are equal to 1 Solving A IX 0 we find two independent eigenvectors V1 1000 and W1 0110 This reduces the possible canonical forms for A to two pos sibilities Solving A IX V1 yields one solution V2 0100 and solving A IX W1 yields another solution W2 0001 Thus we know that the system X AX may be tranformed into Y T1ATY 1 1 0 0 0 1 0 0 0 0 1 1 0 0 0 1 Y where the matrix T is given by T 1 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 Solutions of Y T1ATY therefore are given by y1t c1et c2tet y2t c2et y3t c3et c4tet y4t c4et 64 The Exponential of a Matrix 123 Applying the change of coordinates T we find the general solution of the original system t t xt cje ote xt ce ce cyte x3t c3e cate x4t cge 2 64 The Exponential of a Matrix We turn now to an alternative and elegant approach to solving linear systems using the exponential of a matrix In a certain sense this is the more natural way to attack these systems Recall how we solved the 1 x 1 system of linear equations x ax where our matrix was now simply a We did not go through the process of finding eigenvalues and eigenvectors here well actually we did but the process was pretty simple Rather we just exponentiated the matrix a to find the general solution xt cexpat In fact this process works in the general case where Ais nx n All we need to know is how to exponentiate a matrix Heres how Recall from calculus that the exponential function can be expressed as the infinite series OO Lk x x c ki k0 We know that this series converges for every x IR Now we can add matrices we can raise them to the power k and we can multiply each entry by 1k So this suggests that we can use this series to exponentiate them as well Definition Let A be an n x n matrix We define the exponential of A to be the matrix given by OO Ak A expA S a k0 Of course we have to worry about what it means for this sum of matrices to converge but lets put that off and try to compute a few examples first 124 Chapter 6 HigherDimensional Linear Systems Example Let A 0 A ny Then we have k A 0 Po so that Co So atyk 0 k0 ee 0 expA x 1 k 0 e 0 a k0 as you may have guessed a Example For a slightly more complicated example let B A 5 We compute A Oe i 1 0 4 p47 45 psf 9 1 A pI A B ie so we find oo 2k 00 2k1 B xB Sept Sy pk ar 2k 10 2k1 expA 0 k1 oe 2k p cB yipk Loo 10 2k1 ar 2k cosB sinB Zs sinB cosB 64 The Exponential of a Matrix 125 Example Now let rn 1 s0 3 with 4 4 0 With an eye toward what comes later we compute not exp A but rather exptA We have tayk tka ko 0 tA Thus we find tA tA ae te k0 k0 el te exptA 0 oo k 0 ta LE k0 Note that in each of these three examples the matrix expA is a matrix with entries that are infinite series We therefore say that the infinite series of matrices expA converges absolutely if each of its individual terms does so In each of the preceding cases this convergence was clear Unfortunately in the case of a general matrix A this is not so clear To prove convergence here we need to work a little harder Let ajjk denote the ijentry of A Let a max a We have n aj2 Yana na k1 n aj3 y aaa a klé1 JaiK n hak Thus we have a bound for the ijentry of the n x m matrix expA oo oo OO klk k aijk aijk na na ear sS sa sk Gr seem k0 k0 k0 k0 so that this series converges absolutely by the comparison test Therefore the matrix exp A makes sense for any A LR 126 Chapter 6 HigherDimensional Linear Systems The following result shows that matrix exponentiation shares many of the familiar properties of the usual exponential function Proposition Let AB and T ben x n matrices Then 1 IfB TAT then expB T expAT 2 If AB BA then expA B expA expB 3 expA expA Proof The proof of 1 follows from the identities TA BT TAT TBT and TAT TAT Therefore n 4k n 1 k A TAT mperey eer k0 k0 and 1 follows by taking limits To prove 2 observe that because AB BA we have by the binomial theorem Ai BK nal fo A B n ah jtkn Therefore we must show that Al BK Al SS BE 2 y4 8 n0 jkn j0 k0 This is not as obvious as it may seem since we are dealing here with series of matrices not series of real numbers So we will prove this in the following lemma which then proves 2 Putting B A in 2 gives 3 L Lemma For any n x n matrices A and B we have Al BK SAI BE o 4yo8 n0 jtkn j0 k0 Proof We know that each of these infinite series of matrices converges We just have to check that they converge to each other To do this consider the partial sums 2m i pk Al B rmd 4 n0 jkn 64 The Exponential of a Matrix 127 and m i m Al Bk An 55 and By oz j0 k0 We need to show that the matrices y2 Q m6 tend to the zero matrix as m oo Toward that end for a matrix M mj we let M maxmjj We will show that 72 mBm 0 as m oo A computation shows that 1 Al BK 1 Al Bk Y2m AmBm S lk 0 jt where denotes the sum over terms with indices satisfying jtk2m 0jm m1k2m while denotes the sum corresponding to jtk2m m1j2m 0km Therefore Al Bk Al BK IIy2mOmBall lll to WE lll j k j k Now 1 Al BE a A a YF lhliglls COIFI YX gl J j0 J km1 This tends to 0 as m oo since as we saw previously SONG expallAll o0 o 7 Similarly n Al BK arti ll gl as m oo Therefore limmooY2m mBm 0 proving the lemma 128 Chapter 6 HigherDimensional Linear Systems Observe that statement 3 of the proposition implies that expA is invert ible for every matrix A This is analogous to the fact that e 0 for every real number a There is a very simple relationship between the eigenvectors of A and those of expA Proposition If V R is an eigenvector of A associated with the eigenvalue i then V is also an eigenvector of expA associated with e Proof From AV XV we obtain n k AXV expAV im TT k0 nok Xr Jim Ev k0 dUG k0 eV OU Now lets return to the setting of systems of differential equations Let A be an nm X n matrix and consider the system X AX Recall that LR denotes the set of all n x m matrices We have a function R LR which assigns the matrix exptA to t R Since LR is identified with R it makes sense to speak of the derivative of this function Proposition d Th exptA AexptA exptAA In other words the derivative of the matrixvalued function t exptA is another matrixvalued function AexptA Proof We have d expt hA exptA de OPA jit h expfA exphHA exptA io A h0 h exphA I tA exptA lim i exptAA 64 The Exponential of a Matrix 129 That the last limit equals A follows from the series definition of expHA Note that A commutes with each term of the series for exptA thus with exptA This proves the proposition O Now we return to solving systems of differential equations The following may be considered the fundamental theorem of linear differential equations with constant coefficients Theorem Let A be ann x n matrix Then the solution of the initial value problem X AX with X0 Xo is Xt exptA Xo Moreover this is the only such solution Proof The preceding proposition shows that d d exptA Xo exptA Xp AexptA Xp dt dt Moreover since exp0AX Xo it follows that this is a solution of the initial value problem To see that there are no other solutions let Yt be another solution satisfying Y0 Xo and set Zt exptA Yt Then d 1 ZtHh 5 exp Yt exptA Yt AexptA Yt exptAAYt expfAA A YH 0 Therefore Zt is a constant Setting t 0 shows Zt Xp so that Yt exptA Xo This completes the proof of the theorem Note that this proof is identical to that given in Chapter 1 Section 11 Only the meaning of the letter A has changed Example Consider the system aA 1 va x 130 Chapter 6 HigherDimensional Linear Systems By the theorem the general solution is tA Xt exptAXo exp 0 a Xo But this is precisely the exponential of the matrix we computed earlier We find that eth teth XH 0 eft Xo Note that this agrees with our computations in Chapter 3 a 65 Nonautonomous Linear Systems Up to this point virtually all of the linear systems of differential equations that we have encountered have been autonomous There are however certain types of nonautonomous systems that often arise in applications One such system is of the form X AtX where At ajjt is an 1 x n matrix that depends continuously on time We will investigate these types of systems further when we encounter the variational equation in subsequent chapters Here we restrict our attention to a different type of nonautonomous linear system given by XAXGt where A is a constant n x n matrix and GR R is a forcing term that depends explicitly on t This is an example of a firstorder linear nonho mogeneous system of equations Example The Forced Harmonic Oscillator If we apply an external force to the harmonic oscillator system the differential equation governing the motion becomes x bx kx ft where ft measures the external force An important special case occurs when this force is a periodic function of time which corresponds for example to moving the table on which the massspring apparatus resides back and forth periodically As a system the forced harmonic oscillator equation becomes 90 1 0 x 2 ix Gt where Gt 0 a 65 Nonautonomous Linear Systems 131 For a nonhomogeneous system the equation that results from dropping the timedependent term namely X AX is called the homogeneous equation We know how to find the general solution of this system Borrowing the nota tion from the previous section the solution satisfying the initial condition X0 Xo is Xt exptA Xo so this is the general solution of the homogeneous equation To find the general solution of the nonhomogeneous equation suppose that we have one particular solution Zt of this equation So Zt AZt Gt If Xf is any solution of the homogeneous equation then the function Yt Xt Zf is another solution of the nonhomogeneous equation This follows since we have YXZ AXAZGt AX Z Gt AY Gt Therefore since we know all solutions of the homogeneous equation we can now find the general solution to the nonhomogeneous equation provided that we can find just one particular solution of this equation Often one gets such a solution by simply guessing it in calculus this method is usually called the method of undetermined coefficients Unfortunately guessing a solution does not always work The following method called variation of parameters does work in all cases However there is no guarantee that we can actually evaluate the required integrals Theorem Variation of Parameters Consider the nonhomogeneous equa tion XxX AX4 Gt where A isan n x n matrix and Gt is a continuous function of t Then t Xt exptA f expsa co 0 is a solution of this equation satisfying X0 Xo 132 Chapter 6 HigherDimensional Linear Systems Proof Differentiating Xt we obtain t Xt AexptA f expisa ca 0 1 t exptA exes Gsds 0 t AexptA f exp oa Gt 0 AXt Gt We now give several applications of this result in the case of the periodically forced harmonic oscillator Assume first that we have a damped oscillator that is forced by cost so the period of the forcing term is 277 The system is XAX Gt where Gt 0cost and A is the matrix 0 1 a2 4 with bk 0 We claim that there is a unique periodic solution of this system which has period 27 To prove this we must first find a solution Xf satisfy ing X0 Xp X27 By variation of parameters we need to find Xo such that 20 Xo exp27 A Xo exp27 A expsA Gs ds 0 Now the term 20 exp2rA expsA Gs ds 0 is a constant vector that we denote by W Therefore we must solve the equation exp27 A I Xo W There is a unique solution to this equation since the matrix exp27 A I is invertible For if this matrix were not invertible there would be a nonzero vector V with exp2z A I V 0 or in other words the matrix exp27 A would have an eigenvalue 1 But from the previous section the eigenvalues of exp27 A are given by exp27j 65 Nonautonomous Linear Systems 133 where the A are the eigenvalues of A But each A has real part less than 0 so the magnitude of exp27r is smaller than 1 Thus the matrix exp27 A I is indeed invertible and the unique initial value leading to a 27periodic solution is Xo exp2 A 1 W So let Xt be this periodic solution with X0 Xo This solution is called the steadystate solution If Yo is any other initial condition then we may write Yo Yo Xo Xo so the solution through Yo is given by t Yt exptA Yo Xo exptA Xo exptA expsA Gsds 0 exptAYo Xo X0 The first term in this expression tends to 0 as t oo since it is a solution of the homogeneous equation Thus every solution of this system tends to the steady state solution as t oo Physically this is clear The motion of the damped and unforced oscillator tends to equilibrium leaving only the motion due to the periodic forcing We have proved the following theorem Theorem Consider the forced damped harmonic oscillator equation x bx kx cost with k b 0 Then all solutions of this equation tend to the steadystate solution which is periodic with period 27 Now consider a particular example of a forced undamped harmonic oscillator 9 1 0 x 5 x4 covir where the period of the forcing is now 277w with w 1 Let 0 1 s The solution of the homogeneous equation is Xt exptAXp cost sint x XP 0 sint cost 134 Chapter 6 HigherDimensional Linear Systems Variation of parameters provides a solution of the nonhomogeneous equation starting at the origin t 0 Yt exptA expsA oss ds 0 t mown f 85 2 9 sins coss cosws 0 t sins cosws 7 expra cos s cosws ds 0 t l sinw 1ssinw 1s 9 expia are 1scosw 1s ds 0 Recalling that cost sint expTA sint cos and using the fact that wA1 evaluation of this integral plus a long computation yields cosw 1t 1 cos 1t i oa1 o1 YO 2 exptA f sinw 1t 1 sin 1t a1 o1 2 1 1 exptA 0 1 coswt 17 1 oer expta 0 Thus the general solution of this equation is w 17 1 coswt Yt exptA x 0 321 wsinat The first term in this expression is periodic with period 27 while the second has period 27w Unlike the damped case this solution does not necessarily yield a periodic motion Indeed this solution is periodic if and only if w is a Hirsch Ch069780123820105 201222 1139 Page 135 29 Exercises 135 rational number If ω is irrational the motion is quasiperiodic just as we saw in Section 62 E X E R C I S E S 1 Find the general solution for X AX where A is given by a 0 0 1 0 1 0 1 0 0 b 1 0 1 0 1 0 1 0 1 c 0 1 0 1 0 0 1 1 1 d 0 1 0 1 0 0 1 1 1 e 1 0 1 0 1 0 0 0 1 f 1 1 0 1 1 1 0 1 1 g 1 0 1 1 1 1 0 0 1 h 1 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 2 Consider the linear system X λ1 0 0 0 λ2 0 0 0 λ3 X where λ3 λ2 λ1 0 Describe how the solution through an arbitrary initial value tends to the origin 3 Give an example of a 3 3 matrix A for which all nonequilibrium solutions of X AX are periodic with period 2π Sketch the phase portrait 4 Find the general solution of X 0 1 1 0 1 0 0 1 0 0 0 1 0 0 1 0 X 5 Consider the system X 0 0 a 0 b 0 a 0 0 X depending on the two parameters a and b Hirsch Ch069780123820105 201222 1139 Page 136 30 136 Chapter 6 HigherDimensional Linear Systems a Find the general solution of this system b Sketch the region in the abplane where this system has different types of phase portraits 6 Consider the system X a 0 b 0 b 0 b 0 a X depending on the two parameters a and b a Find the general solution of this system b Sketch the region in the abplane where this system has different types of phase portraits 7 Coupled Harmonic Oscillators In this series of exercises you are asked to generalize the material on harmonic oscillators in Section 62 to the case where the oscillators are coupled Suppose there are two masses m1 and m2 attached to springs and walls as shown in Figure 610 The springs connecting mj to the walls both have spring constants k1 while the spring connecting m1 and m2 has spring constant k2 This coupling means that the motion of either mass affects the behavior of the other Let xj denote the displacement of each mass from its rest position and assume that both masses are equal to 1 The differential equations for these coupled oscillators are then given by x 1 k1 k2x1 k2x2 x 2 k2x1 k1 k2x2 These equations are derived as follows If m1 is moved to the right x1 0 the left spring is stretched and exerts a restorative force on m1 given by k1x1 Meanwhile the central spring is compressed so it exerts a restorative force on m1 given by k2x1 If the right spring is stretched then the central spring is compressed and exerts a restorative force on m1 given by k2x2 since x2 0 The forces on m2 are similar a Write these equations as a firstorder linear system b Determine the eigenvalues and eigenvectors of the corresponding matrix k1 k2 k1 m1 m2 Figure 610 A coupled oscillator Exercises 137 c Find the general solution d Let w k and w kj 2k What can be said about the periodicity of solutions relative to the Prove this 8 Suppose X AX where A is a 4 x 4 matrix with eigenvalues that are i2 and i3 Describe this flow 9 Suppose X AX where A is a 4 x 4 matrix with eigenvalues that are i and 1 i Describe this flow 10 Suppose X AX where A is a 4 x 4 matrix with eigenvalues that are i and 1 Describe this flow 11 Consider the system X AX where X x56s 0 WM OW 0 0 W2 A W2 0 1 1 and q is irrational Describe qualitatively how a solution behaves when at time 0 each x is nonzero with the exception that a x5 0 b x5 0 c 3x4x50 d x3 x4 x5 x 0 12 Compute the exponentials of the following matrices 5 6 2 1 2 1 0 1 3 w 3 6 J 4 0 1 2 2 0 0 Xr 0 0 e0 0 3 0 3 OF yl Aa O 0 0 0 0 1 3 0 1d 1 0 0 0 i 0 f1li 0 1 0 0 0 y co 2 i O11 0 0 0 1 0 0 0 13 Find an example of two matrices A B such that expA B expA expB Hirsch Ch069780123820105 201222 1139 Page 138 32 138 Chapter 6 HigherDimensional Linear Systems 14 Show that if AB BA then a expAexpB expBexpA b expAB BexpA 15 Consider the triplet of harmonic oscillators x 1 x1 x 2 2x2 x 3 ω2x3 where ω is irrational What can you say about the qualitative behavior of solutions of this sixdimensional system Hirsch Ch079780123820105 201222 1214 Page 139 1 7 Nonlinear Systems In this chapter we begin the study of nonlinear differential equations In linear constant coefficient systems we can always find the explicit solution of any initial value problem however this is rarely the case for nonlinear systems In fact basic properties such as the existence and uniqueness of solutions which was so obvious in the linear case no longer hold for nonlinear systems As we shall see some nonlinear systems have no solutions whatsoever to a given initial value problem On the other hand there are other systems that have infinitely many differ ent such solutions Even if we do find a solution of such a system this solution need not be defined for all time for example the solution may tend to in finite time Other questions also arise For example what happens if we vary the initial condition of a system ever so slightly Does the corresponding solu tion vary continuously All of this is clear for linear systems but not at all clear in the nonlinear case This means that the underlying theory behind nonlinear systems of differential equations is quite a bit more complicated than that for linear systems In practice most nonlinear systems that arise are nice in the sense that we do have existence and uniqueness of solutions as well as continuity of solutions when initial conditions are varied and other natural properties Thus we have a choice Given a nonlinear system we could simply plunge ahead and either hope that or if possible verify that in each specific case the systems solutions behave nicely Alternatively we could take a long pause at Differential Equations Dynamical Systems and an Introduction to Chaos DOI 101016B9780123820105000075 c 2013 Elsevier Inc All rights reserved 139 Hirsch Ch079780123820105 201222 1214 Page 140 2 140 Chapter 7 Nonlinear Systems this stage to develop the necessary hypotheses that guarantee that solutions of a given nonlinear system behave nicely In this book we pursue a compromise route In this chapter we spell out in precise detail many of the theoretical results that govern the behavior of solu tions of differential equations We present examples of how and when these results fail but we will not prove these theorems here Rather we will postpone all of the technicalities until Chapter 17 primarily because understanding this material demands a firm and extensive background in the principles of real analysis In subsequent chapters we will make use of the results stated here but readers who are primarily interested in applications of differential equations or in understanding how specific nonlinear systems may be ana lyzed need not get bogged down in these details here Readers who want the technical details may take a detour to Chapter 17 now 71 Dynamical Systems As mentioned previously most nonlinear systems of differential equations are impossible to solve analytically One reason for this is the unfortunate fact that we simply do not have enough functions with specific names that we can use to write down explicit solutions of these systems Equally problematic is the fact that as we shall see higherdimensional systems may exhibit chaotic behav ior a property that makes knowing a particular explicit solution essentially worthless in the larger scheme of understanding the behavior of the system Thus to begin to understand these systems we are forced to resort to different means These are the techniques that arise in the field of dynamical systems We will use a combination of analytic geometric and topological techniques to derive rigorous results about the behavior of solutions of these equations We begin by collecting together some of the terminology regarding dynam ical systems that we have introduced at various points in the preceding chapters A dynamical system is a way of describing the passage in time of all points of a given space S The space S could be thought of for example as the space of states of some physical system Mathematically S might be a Euclidean space or an open subset of Euclidean space or some other space such as a surface in R3 When we consider dynamical systems that arise in mechanics the space S will be the set of possible positions and velocities of the system For the sake of simplicity we will assume throughout that the space S is Euclidean space Rn although in certain cases the important dynamical behavior will be confined to a particular subset of Rn Given an initial position X Rn a dynamical system on Rn tells us where X is located 1 unit of time later 2 units of time later and so on We denote these Hirsch Ch079780123820105 201222 1214 Page 141 3 71 Dynamical Systems 141 new positions of X by X1X2 and so forth At time zero X is located at posi tion X0 One unit before time zero X was at X1 In general the trajectory of X is given by Xt If we measure the positions Xt using only integer time values we have an example of a discrete dynamical system which we shall study in Chapter 15 If time is measured continuously with t R we have a continuous dynamical system If the system depends on time in a continuously differen tiable manner we have a smooth dynamical system These are the three princi pal types of dynamical systems that arise in the study of systems of differential equations and they will form the backbone of Chapters 8 through 14 The function that takes t to Xt yields either a sequence of points or a curve in Rn that represents the life history of X as time runs from to Dif ferent branches of dynamical systems make different assumptions about how the function Xt depends on t For example ergodic theory deals with such functions under the assumption that they preserve a measure on Rn Topolog ical dynamics deals with such functions under the assumption that Xt varies only continuously In the case of differential equations we will usually assume that the function Xt is continuously differentiable The map φt Rn Rn that takes X into Xt is defined for each t and from our interpretation of Xt as a state moving in time it is reasonable to expect φt to have φt as its inverse Also φ0 should be the identity function φ0X X and φtφsX φtsX is also a natural condition We formalize all of this in the following definition Definition A smooth dynamical system on Rn is a continuously differentiable function φ R Rn Rn where φtX φtX satisfies 1 φ0 Rn Rn is the identity function φ0X0 X0 2 The composition φt φs φts for each ts R Recall that a function is continuously differentiable if all of its partial deriva tives exist and are continuous throughout its domain It is traditional to call a continuously differentiable function a C1 function If the function is k times continuously differentiable it is called a Ck function Note that the preceding definition implies that the map φt Rn Rn is C1 for each t and has a C1 inverse φt take s t in part 2 Example For the firstorder differential equation x ax the function φtx0 x0 expat gives the solutions of this equation and also defines a smooth dynamical system on R 142 Chapter 7 Nonlinear Systems Example Let A bean x n matrix Then the function Xo exptA Xo defines a smooth dynamical system on R Clearly 9 exp0 I and as we saw in the previous chapter we have Pts expt sA exptaexpsA Gr 0 ds a Note that these examples are intimately related to the system of differential equations X AX In general a smooth dynamical system always yields a vector field on R via this rule Given let FX d 1X dt t0 Then is just the time t map associated with the flow of X FX Conversely the differential equation X FX generates a smooth dynam ical system provided the time t map of the flow is well defined and continu ously differentiable for all time Unfortunately this is not always the case 72 The Existence and Uniqueness Theorem We turn now to the fundamental theorem of differential equations the Exis tence and Uniqueness Theorem Consider the system of differential equations X FX where F R R Recall that a solution of this system is a function X J IR defined on some interval J C R such that for all t J Xt FXt Geometrically Xt is a curve in R with a tangent vector Xt that exists for all t J and equals FXt As in previous chapters we think of this vector as being based at Xt so that the map F R R defines a vector field on R An initial condition or initial value for a solution X J R is a specifi cation of the form Xt Xo where t J and Xp R For simplicity we usually take 0 The main problem in differential equations is to find the solution of any initial value problemthat is to determine the solution that of the system that satisfies the initial condition X0 Xo for each Xp R Unfortunately nonlinear differential equations may have no solutions that satisfy certain initial conditions 72 The Existence and Uniqueness Theorem 143 Example Consider the simple firstorder differential equation 1 ifx0 x 1 ifx0 This vector field on R points to the left when x 0 and to the right if x 0 Consequently there is no solution that satisfies the initial condition x0 0 Indeed such a solution must initially decrease since x0 1 but for all negative values of x solutions must increase This cannot happen Note fur ther that solutions are never defined for all time For example if xp 0 then the solution through x is given by xt xo t but this solution is only valid for oo t x for the same reason as before The problem in this example is that the vector field is not continuous at 0 whenever a vector field is discontinuous we face the possibility that nearby vectors may point in opposing directions thereby causing solutions to halt at these bad points a Beyond the problem of existence of solutions of nonlinear differential equa tions we also must confront the fact that certain equations may have many different solutions to the same initial value problem Example Consider the differential equation x 3x79 The identically zero function u R R given by ut 0 is clearly a solution with initial condition u0 0 But ut f is also a solution satisfying this initial condition Moreover for any t 0 the function given by ut 0 iftT mVtty iftt is also a solution satisfying the initial condition u 0 0 Although the dif ferential equation in this example is continuous at x 0 the problems arise because x3 is not differentiable at this point a From these two examples it is clear that to ensure existence and uniqueness of solutions certain conditions must be imposed on the function F In the first example F was not continuous at the problematic point 0 while in the sec ond example F failed to be differentiable at 0 It turns out that the assumption that F is continuously differentiable is sufficient to guarantee both existence and uniqueness of solutions as we shall see Fortunately differential equa tions that are not continuously differentiable rarely arise in applications so 144 Chapter 7 Nonlinear Systems the phenomenon of nonexistence or nonuniqueness of solutions with given initial conditions is quite exceptional The following is the fundamental local theorem of ordinary differential equations The important proof of this theorem is contained in Chapter 17 The Existence and Uniqueness Theorem Consider the initial value problem x FX Xt Xo where Xy R Suppose that F R R is C Then first there exists a solu tion of this initial value problem and second this is the only such solution More precisely there exists an a 0 and a unique solution Xt a to a R of this differential equation satisfying the initial condition Xt Xo Without dwelling on the details here the proof of this theorem depends on an important technique known as Picard iteration Before moving on we illustrate how the Picard iteration scheme used in the proof of the theorem works in several special examples The basic idea behind this iterative process is to construct a sequence of functions that converges to the solution of the differential equation The sequence of functions ut is defined inductively by uot xo where xp is the given initial condition and then t nail a0 f Fuisds 0 Example Consider the simple differential equation x x We will produce the solution of this equation satisfying x0 x We know of course that this solution is given by xt xoe We will construct a sequence of functions uxt one for each k that converges to the actual solution xt as k oo We start with ug t xX the given initial value Then we set t t m a f Flwsasa0 f xd 0 0 72 The Existence and Uniqueness Theorem 145 so that ut xp txp Given uw we define t t Un t xo f Fan ds xo oc sxo ds 0 0 so that ut x9 tx x0 You can probably see where this is heading Inductively we set t mnait a0 f Plus as 0 and so k1 ti Ukyit Xo S 7 i0 As k 00 ugt converges to CO ti 0 5 xe xt i0 which is the solution of our original equation a Example For an example of Picard iteration applied to a system of differential equations consider the linear system f90 1 xX FX 5 x with initial condition X0 10 As we have seen the solution of this initial value problem is cost X sin Using Picard iteration we have 1 Upt t t 1 1 1 0 1 vin 0 fa f 22 0 0 146 Chapter 7 Nonlinear Systems 2 l s 12 wxto a f 4 0 2 l s 182 Us f 2 2p ds in is 0 11724 t44 and we see the infinite series for the cosine and sine functions emerging from this iteration a Now suppose that we have two solutions Yt and Zt of the differential equation X FX and that Yt and Zt satisfy Yt Zt Suppose that both solutions are defined on an interval J The Existence and Unique ness Theorem guarantees that Yt Zt for all in an interval about f which may a priori be smaller than J However this is not the case To see this suppose that J is the largest interval on which Yt Zt Let t be an endpoint of J By continuity we have Yt Zt The theorem then guarantees that in fact Yt and Zt agree on an open interval containing t This contradicts the assertion that J is the largest interval on which the two solutions agree Thus we can always assume that we have a unique solution defined on a maximal time domain There is however no guarantee that a solution Xt can be defined for all time no matter how nice FX is Example Consider the differential equation in R given by x 14x This equation has as solutions the functions xt tanfc where c is a constant Such a function cannot be extended over an interval larger than Iv 1 Iv ctc 2 2 since xt 00 ast c72 a This example is typical for we have the following theorem Theorem Let U C R be an open set and let F U R be C Let Xt be a solution of X FX defined on a maximal open interval J a B C R with 73 Continuous Dependence of Solutions 147 B oo Then given any closed and bounded set K C U there is some t a B with Xt K The theorem says that if a solution Xt cannot be extended to a larger time interval then this solution leaves any closed and bounded set in U This implies that Xt must come arbitrarily close to the boundary of U as t B Similar results hold as t a 73 Continuous Dependence of Solutions For the Existence and Uniqueness Theorem to be at all interesting in any phys ical or even mathematical sense this result needs to be complemented by the property that the solution Xt depends continuously on the initial condition X0 The next theorem gives a precise statement of this property Theorem Consider the differential equation X FX where F R IR is C Suppose that Xt is a solution of this equation that is defined on the closed interval to t1 with Xto Xo Then there is a neighborhood U C R of Xo and a constant K such that if Yo U then there is a unique solution Yt also defined on to t with Yt Yo Moreover Yt satisfies Yt Xt Yo XolexpKt t forallt tt This result says that if the solutions Xt and Yt start out close together then they remain close together for tf close to Although these solutions may separate from each other they do so no faster than exponentially In particular since the right side of this inequality depends on Yo Xo which we may assume is small we have Corollary Continuous Dependence on Initial Conditions Let t X be the flow of the system X FX where F is C Then is a continuous function of X Example Let k 0 For the system 1 0 X 0 x Hirsch Ch079780123820105 201222 1214 Page 148 10 148 Chapter 7 Nonlinear Systems Xt 1 1η3 Yη3t Yη2 Yη1 η3ekτ η2ekτ η1ekτ eτ Figure 71 The solutions Yηt separate exponentially from Xt but nonetheless are continuous in their initial conditions we know that the solution Xt satisfying X0 10 is given by Xt et0 For any η 0 let Yηt be the solution satisfying Yη0 1η Then Yηt etηekt As in the theorem we have Yηt Xt ηekt 0 η 0ekt Yη0 X0ekt The solutions Yη do indeed separate from Xt as we see in Figure 71 but they do so at most exponentially Moreover for any fixed time t we have Yηt Xt as η 0 Differential equations often depend on parameters For example the har monic oscillator equations depend on the parameters b the damping con stant and k the spring constant Then the natural question is how do solutions of these equations depend on these parameters As in the previous case solutions depend continuously on these parameters provided that the system depends on the parameters in a continuously differentiable fashion We can see this easily by using a special little trick Suppose the system X FaX Hirsch Ch079780123820105 201222 1214 Page 149 11 74 The Variational Equation 149 depends on the parameter a in a C1 fashion Lets consider an artificially augmented system of differential equations given by x 1 f1x1xna x n fnx1xna a 0 This is now an autonomous system of n 1 differential equations Although this expansion of the system may seem trivial we may now invoke the previous result about continuous dependence of solutions on initial conditions to verify that solutions of the original system depend continuously on a as well Theorem Continuous Dependence on Parameters Let X FaX be a system of differential equations for which Fa is continuously differentiable in both X and a Then the flow of this system depends continuously on a as well 74 The Variational Equation Consider an autonomous system X FX where as usual F is assumed to be C1 The flow φtX of this system is a function of both t and X From the results of the previous section we know that φ is continuous in the variable X We also know that φ is differentiable in the variable t since t φtX is just the solution curve through X In fact φ is also differentiable in the variable X we will prove the following theorem in Chapter 17 Theorem Smoothness of Flows Consider the system X FX where F is C1 Then the flow φtX of this system is a C1 function that is φt and φX exist and are continuous in t and X Note that we can compute φt for any value of t as long as we know the solution passing through X0 for we have φ t tX0 FφtX0 We also have φ X tX0 DφtX0 150 Chapter 7 Nonlinear Systems where D is the Jacobian of the function X X To compute 00X however it appears that we need to know the solution through Xo as well as the solutions through all nearby initial positions since we need to com pute the partial derivatives of the various components of However we can get around this difficulty by introducing the variational equation along the solution through Xo To accomplish this we need to take another brief detour into the world of nonautonomous differential equations Let At be a family of n x m matrices that depends continuously on t The system X AtX is a linear nonautonomous system We have an existence and uniqueness theorem for these types of equations Theorem Let At be a continuous family of n x n matrices defined for t la 8 Then the initial value problem X AtX Xt Xo has a unique solution that is defined on the entire interval a B Note that there is some additional content to this theorem We do not assume that the right side is a C function in t Continuity of At suffices to guarantee existence and uniqueness of solutions Example Consider the firstorder linear nonautonomous differential equation x atx The unique solution of this equation satisfying x0 xp is given by t xt xo exp as 0 as is easily checked using the methods of Chapter 1 All we need is that at is continuous so that xt atxt we do not need differentiability of at for this to be true a Note that solutions of linear nonautonomous equations satisfy the Linear ity Principle That is if Yt and Zf are two solutions of such a system then so too is wYt BZt for any constants a and B Hirsch Ch079780123820105 201222 1214 Page 151 13 74 The Variational Equation 151 Now we return to the autonomous nonlinear system X FX Let Xt be a particular solution of this system defined for t in some interval J αβ Fix t0 J and set Xt0 X0 For each t J let At DFXt where DFXt denotes the Jacobian matrix of F at the point Xt Rn Since F is C1 At DFXt is a continuous family of n n matrices Consider the nonautonomous linear equation U AtU This equation is known as the variational equation along the solution Xt By the previous theorem we know that this variational equation has a solution defined on all of J for every initial condition Ut0 U0 The significance of this equation is that if Ut is the solution of the variational equation that satisfies Ut0 U0 then the function t Xt Ut is a good approximation to the solution Yt of the autonomous equation with initial value Yt0 X0 U0 provided U0 is sufficiently small This is the content of the following result Proposition Consider the system X FX where F is C1 Suppose 1 Xt is a solution of X FX which is defined for all t αβ and satisfies Xt0 X0 2 Ut is the solution to the variational equation along Xt that satisfies Ut0 U0 3 Yt is the solution of X FX that satisfies Yt0 X0 U0 Then lim U00 Yt Xt Ut U0 converges to 0 uniformly in t αβ Technically this means that for every ϵ 0 there exists δ 0 such that if U0 δ then Yt Xt Ut ϵU0 for all t αβ Thus as U0 0 the curve t Xt Ut is a better and better approximation to Yt In many applications the solution of the 152 Chapter 7 Nonlinear Systems variational equation Xt Uf is used in place of Yt this is convenient because Ut depends linearly on Up by the Linearity Principle Example Consider the nonlinear system of equations x x y yy We will discuss this system in more detail in the next chapter For now note that we know one solution of this system explicitily namely the equilib rium solution at the origin Xt 00 The variational equation along this solution is given by fi 0 U DFyU U which is an autonomous linear system We obtain the solutions of this equation immediately they are given by xoe Ut 2 The preceding result then guarantees that the solution of the nonlinear equa tion through xp yp and defined on the interval t T is as close as we wish to Ut provided xo yo is sufficiently close to the origin a Note that the arguments in the example are perfectly general Given any nonlinear system of differential equations X FX with an equilibrium point at X9 we may consider the variational equation along this solution But DFyx is a constant matrix A The variational equation is then U AU which is an autonomous linear system This system is called the linearized system at Xo We know that flow of the linearized system is exptA Up so the preced ing result says that near an equilibrium point of a nonlinear system the phase portrait resembles that of the corresponding linearized system We will make the term it resembles more precise in the next chapter Using the previous proposition we may now compute 00X assuming we know the solution Xt Theorem Let X FX be a system of differential equations where F is C Let Xt be a solution of this system satisfying the initial condition X0 Xo and defined for t aB and let Ut Up be the solution to the variational equation along Xt that satisfies U0 Up Up Then DXo Up UCt Up 75 Exploration Numerical Methods 153 That is 0b 0X applied to Up is given by solving the corresponding variational equation starting at Up Proof Using the proposition we have for all t a 8 Xo hUo bt X Ut hU DebulXq Up linn SONU r 4 UHV yey yy h0 h h0 h Example Asan illustration of these ideas consider the differential equation x x An easy integration shows that the solution xt satisfying the initial condition x0 xo is Xo t xt mio Thus we have ag 1 tx ax 4 xot 12 On the other hand the variational equation for xt is 2 u 2xthu ey xot1 The solution of this equation satisfying the initial condition u0 up is given by G 1 ut uy Xot1 as required a 75 Exploration Numerical Methods In this exploration we first describe three different methods for approximat ing the solutions of firstorder differential equations Your task will be to evaluate the effectiveness of each method Each of these methods involves an iterative process whereby we find a sequence of points t x that approximates selected points t xt along the graph of a solution of the firstorder differential equation x f tx In 154 Chapter 7 Nonlinear Systems each case we begin with an initial value x0 xo Thus f 0 and xp is our given initial value We need to produce t and x In each of the three methods we will generate the t recursively by choosing a step size At and simply incrementing t at each stage by At Thus in each case thoy te At Choosing At small will hopefully improve the accuracy of the method Therefore to describe each method we only need to determine the values of xx In each case x41 is the xcoordinate of the point that sits directly over t41 On a certain straight line through tx in the txplane Thus all we need to do is to provide you with the slope of this straight line and then x41 is determined Each of the three methods involves a different straight line Eulers Method Here x is generated by moving Aft time units along the straight line generated by the slope field at the point t x Since the slope at this point is f t xx taking this short step puts us at Xk1 Xk f tks x4 At Improved Eulers Method In this method we use the average of two slopes to move from tg xg to th1Xk41 The first slope is just that of the slope field at t xz namely my f ths Xk The second is the slope of the slope field at the point t1 yg where yx is the terminal point determined by Eulers method applied at t x That is Mk f tht Yk Where yp x fth xR At Then we have mn Xk Xk At FourthOrder RungeKutta Method This method is the one most often used to solve differential equations There are more sophisticated numer ical methods that are specifically designed for special situations but this method has served as a generalpurpose solver for decades In this method we will determine four slopes mx nz pg and qx The step from t xz to tht1Xk41 is given by moving along a straight line with a slope that is a weighted average of these four values M 2ng 2pKE hu qt Teer At 75 Exploration Numerical Methods 155 These slopes are determined as follows a mx is given as in Eulers method mi f thy Xk b mg is the slope at the point obtained by moving halfway along the slope field line at ty xz to the intermediate point t At2 yx so that At At nk f tk 9 Yk where Vk Xk Mk c px is the slope at the point obtained by moving halfway along a different straight line at tx where the slope is now mx rather than mx as before Thus At At Pef tk 92k where Zk Xk Mk d Finally qx is the slope at the point t1 wz where we use a line with slope px at ty x to determine this point Thus dk f ther We where wexp pp At Your goal in this exploration is to compare the effectiveness of these three methods by evaluating the errors made in carrying out this procedure in several examples We suggest that you use a spreadsheet to make these lengthy calculations 1 First just to make sure you comprehend the rather terse descriptions of the preceding three methods draw a picture in the txplane that illus trates the process of moving from ti xx to te41Xz41 in each of the three cases 2 Now lets investigate how the various methods work when applied to an especially simple differential equation x x a Find the explicit solution xt of this equation satisfying the ini tial condition x01 now theres a free gift from the math department b Use Eulers method to approximate the value of x1 e using the step size At 01 That is recursively determine t and x for k 110 using At 01 and starting with f 0 and x 1 c Repeat the previous step with At half the size namely 005 d Again use Eulers method this time reducing the step size by a factor of 5 so that At 001 to approximate x1 156 Chapter 7 Nonlinear Systems e Repeat the previous three steps using the Improved Eulers method with the same step sizes f Repeat using RungeKutta g You now have nine different approximations for the value of x1 e three for each method Calculate the error in each case For the record use the value e 271828182845235360287 in calculating the error h Calculate how the error changes as you change the step size from 01 to 005 and then from 005 to 001 That is if 9 denotes the error made using step size A compute both 01005 and 005 001 3 Repeat the previous exploration this time for the nonautonomous equation x 2t1 x Use the value tan 1 1557407724654 4 Discuss how the errors change as you shorten the step size by a factor of two or a factor of five Why in particular is the RungeKutta method called a fourthorder method 76 Exploration Numerical Methods and Chaos In this exploration we will see how numerical methods can sometimes fail dra matically This is also our first encounter with chaos a topic that will reappear numerous times later in the book 1 Consider the simple nonautonomous differential equation ay esiny dt Sketch the graph of the solution to this equation that satisfies the initial condition y0 03 2 Use Eulers method with a step size At 03 to approximate the value of the previous solution at y10 It is probably easiest to use a spreadsheet to carry out this method How does your numerical solution compare to the actual solution 3 Repeat the previous calculation with step sizes At 00010002 and 0003 What happens now This behavior is called sensitive dependence on initial conditions the hallmark of the phenomenon known as chaos 4 Repeat step 2 but now for initial conditions y0 0301 and y0 0302 Why is this behavior called senstitive dependence on initial con ditions Exercises 157 5 What causes Eulers method to behave in this manner 6 Repeat steps 2 and 3 now using the RungeKutta method Is there any change 7 Can you come up with other differential equations for which these num erical methods break down EXERCISES 1 Write out the first few terms of the Picard iteration scheme for each of the following initial value problems Where possible find explicit solutions and describe the domain of this solution a x x2 x0 2 b x x43 x0 0 c x x x0 1 d x cosx x0 0 e x 12x x1 1 2 Let A be an n x n matrix Show that the Picard method for solving X AX X0 Xo gives the solution exptA Xo 3 Derive the Taylor series for sin2t by applying the Picard method to the firstorder system corresponding to the secondorder initial value problem x 4x x00 x0 2 4 Verify the Linearity Principle for linear nonautonomous systems of differential equations 5 Consider the firstorder equation x xt What can you say about solutions that satisfy x0 0 x0 a 40 6 Discuss the existence and uniqueness of solutions of the equation x x where a 0 and x0 0 7 Let At be a continuous family of mx m matrices and let Pt be the matrix solution to the initial value problem P AtP P0 Po Show that t det Pt det Po exp Tr As 0 8 Construct an example ofa firstorder differential equation on R for which there are no solutions to any initial value problem 9 Construct an example ofa differential equation depending on a parameter a for which some solutions do not depend continuously on a Hirsch Ch089780123820105 2012125 1452 Page 159 1 8 Equilibria in Nonlinear Systems To avoid some of the technicalities we encountered in the previous chapter we will henceforth assume that our differential equations are C except when specifically noted This means that the right side of the differential equation is k times continuously differentiable for all k This will at the very least allow us to keep the number of hypotheses in our theorems to a minimum As we have seen it is often impossible to write down explicit solutions of nonlinear systems of differential equations The one exception to this occurs when we have equilibrium solutions Provided we can solve the algebraic equations we can write down the equilibria explicitly Often these are the most important solutions of a particular nonlinear system More important given our extended work on linear systems we can usually use the technique of linearization to determine the behavior of solutions near equilibrium points We describe this process in detail in this chapter 81 Some Illustrative Examples In this section we consider several planar nonlinear systems of differential equations Each will have an equilibrium point at the origin Our goal is to see that the solutions of the nonlinear system near the origin resemble those of the linearized system at least in certain cases Differential Equations Dynamical Systems and an Introduction to Chaos DOI 101016B9780123820105000087 c 2013 Elsevier Inc All rights reserved 159 160 Chapter 8 Equilibria in Nonlinear Systems As a first example consider the system x x 4 y yy There is a single equilibrium point at the origin To picture nearby solutions we note that when y is small y is much smaller Thus near the origin at least the differential equation x x y is very close to x x In Chapter 7 Section 74 we showed that the flow of this system near the origin is also close to that of the linearized system X DFX This suggests that we consider instead the linearized equation x x yuy derived by simply dropping the higherorder term We can of course solve this system immediately We have a saddle at the origin with a stable line along the yaxis and an unstable line along the xaxis Now lets go back to the original nonlinear system Luckily we can also solve this system explicitly For the second equation y y yields yt yoe Inserting this into the first equation we must solve x xtye 7 This is a firstorder nonautonomous equation with solutions that can be determined as in calculus by guessing a particular solution of the form ce Inserting this guess into the equation yields a particular solution 1 xt ype at 3 Thus any function of the form ot lia ar xt ce 3 yoe is a solution of this equation as is easily checked The general solution is then 1 1 xt x 5 ef sye 3 3 yt yor If yo 0 we find a straightline solution xt xoe yt 0 just as in the linear case However unlike the linear case the yaxis is no longer home to a Hirsch Ch089780123820105 2012125 1452 Page 161 3 81 Some Illustrative Examples 161 solution that tends to the origin Indeed the vector field along the yaxis is given by y2y which is not tangent to the axis rather all nonzero vectors point to the right along this axis On the other hand there is a curve through the origin on which solutions tend to 00 Consider the curve x 1 3y2 0 in R2 Suppose x0y0 lies on this curve and let xtyt be the solution satisfying this initial condition Since x0 1 3y2 0 0 this solution becomes xt 1 3y2 0e2t yt y0et Note that we have xt 1 3yt2 0 for all t so this solution always remains on this curve Moreover as t this solution tends to the equilibrium point That is we have found a stable curve through the origin on which all solutions tend to 00 Note that this curve is tangent to the yaxis at the origin See Figure 81 Can we just drop the nonlinear terms in a system The answer is as we shall see next it depends In this case however doing so is perfectly legal for we can find a change of variables that actually converts the original system to the linear system To see this we introduce new variables u and v via u x 1 3y2 v y Figure 81 Phase plane for x x y2 y y Note the stable curve tangent to the yaxis 162 Chapter 8 Equilibria in Nonlinear Systems Then in these new coordinates the system becomes 2 1 2 uxryy xry u 3 Y 3 vy y That is to say the nonlinear change of variables Fx y x y y con verts the original nonlinear system to a linear one in fact to the preceding linearized system Example In general it is impossible to convert a nonlinear system to a linear one as in the previous example since the nonlinear terms almost always make huge changes in the system far from the equilibrium point at the origin For example consider the nonlinear system 1 1 3 2 xX rxy xx yx yey le ty 1 1 3 2 yxtayy 2 2 2 Again we have an equilibrium point at the origin The linearized system is now 1 5 x 1 X 1 2 which has eigenvalues 5 1 All solutions of this system spiral away from the origin and toward oo in the counterclockwise direction as is easily checked Solving the nonlinear system looks formidable However if we change to polar coordinates the equations become much simpler We compute 1 3 rcosd rsin0 x 3 r r cosé rsind Ios I 3 a3 rsin6rcos0 y 3 r r sin rcosé from which we conclude after equating the coefficients of cos and sin6 rf rd2 0 1 We can now solve this system explicitly since the equations are decoupled Rather than do this we will proceed in a more geometric fashion From the Hirsch Ch089780123820105 2012125 1452 Page 163 5 81 Some Illustrative Examples 163 Figure 82 Phase plane for r 1 2r r3 θ 1 equation θ 1 we conclude that all nonzero solutions spiral around the ori gin in the counterclockwise direction From the first equation we see that solutions do not spiral toward Indeed we have r 0 when r 1 so all solutions that start on the unit circle stay there forever and move periodi cally around the circle Since r 0 when 0 r 1 we conclude that nonzero solutions inside the circle spiral away from the origin and toward the unit cir cle Since r 0 when r 1 solutions outside the circle spiral toward it See Figure 82 In the previous example there is no way to find a global change of coordi nates that puts the system into a linear form since no linear system has this type of spiraling toward a circle However near the origin this is still possible To see this first note that if r0 satisfies 0 r0 1 then the nonlinear vector field points outside the circle of radius r0 This follows from the fact that on any such circle r r01 r2 02 0 Consequently in backward time all solutions of the nonlinear system tend to the origin and in fact spiral as they do so We can use this fact to define a conjugacy between the linear and nonlinear system in the disk r r0 much as we did in Chapter 4 Let φt denote the flow of the nonlinear system In polar coordinates the preceding linearized system becomes r r2 θ 1 Let ψt denote the flow of this system Again all solutions of the linear system tend toward the origin in backward time We will now define a conjugacy between these two flows in the disk D given by r 1 Fix r0 with 0 r0 1 Hirsch Ch089780123820105 2012125 1452 Page 164 6 164 Chapter 8 Equilibria in Nonlinear Systems For any point rθ in D with r 0 there is a unique t trθ for which φtrθ belongs to the circle r r0 We then define the function hrθ ψtφtrθ where t trθ We stipulate also that h takes the origin to the origin Then it is straightforward to check that h φsrθ ψs hrθ for any point rθ in D so that h gives a conjugacy between the nonlinear and linear systems It is also easy to check that h takes D onto all of R2 Thus we see that while it may not always be possible to linearize a system globally we may sometimes accomplish this locally Unfortunately not even this is always possible Example Now consider the system x y ϵxx2 y2 y x ϵyx2 y2 Here ϵ is a parameter that we may take to be either positive or negative The linearized system is x y y x so we see that the origin is a center and all solutions travel in the counter clockwise direction around circles centered at the origin with unit angular speed This is hardly the case for the nonlinear system In polar coordinates this system reduces to r ϵr3 θ 1 Thus when ϵ 0 all solutions spiral away from the origin whereas when ϵ 0 all solutions spiral toward the origin The addition of the nonlinear terms no matter how small near the origin changes the linearized phase por trait dramatically we cannot use linearization to determine the behavior of this system near the equilibrium point Hirsch Ch089780123820105 2012125 1452 Page 165 7 82 Nonlinear Sinks and Sources 165 Figure 83 Phase plane for x x2y y Example Now consider one final example x x2 y y The only equilibrium solution for this system is the origin All other solutions except those on the yaxis move to the right and toward the xaxis On the yaxis solutions tend along this straight line to the origin Thus the phase portrait is as shown in Figure 83 Note that this picture is quite different from the corresponding picture for the linearized system x 0 y y for which all points on the xaxis are equilibrium points and all of the other solutions lie on vertical lines x constant The problem here as in the previous example is that the equilibrium point for the linearized system at the origin is not hyperbolic When a linear planar system has a zero eigenvalue or a center the addition of nonlinear terms often completely changes the phase portrait 82 Nonlinear Sinks and Sources As we saw in the examples of the previous section solutions of planar nonlin ear systems near equilibrium points resemble those of their linear parts only Hirsch Ch089780123820105 2012125 1452 Page 166 8 166 Chapter 8 Equilibria in Nonlinear Systems in the case where the linearized system is hyperbolic that is when neither of the eigenvalues of the system has a zero real part In this section we begin to describe the situation in the general case of a hyperbolic equilibrium point in a nonlinear system by considering the special case of a sink For simplicity we will prove the results in the following planar case although all of the results hold in Rn Let X FX and suppose that FX0 0 Let DFX0 denote the Jacobian matrix of F evaluated at X0 Then as in Chapter 7 the linear system of differential equations Y DFX0Y is called the linearized system near X0 Note that if X0 0 the linearized sys tem is obtained by simply dropping all of the nonlinear terms in F just as we did in the previous section In analogy with our work with linear systems we say that an equilibrium point X0 of a nonlinear system is hyperbolic if all of the eigenvalues of DFX0 have nonzero real parts We now specialize the discussion to the case of an equilibrium of a planar system for which the linearized system has a sink at 0 Suppose our system is x f xy y gxy with f x0y0 0 gx0y0 If we make the change of coordinates u x x0v y y0 then the new system has an equilibrium point at 00 Thus we may as well assume that x0 y0 0 at the outset We then make a fur ther linear change of coordinates that puts the linearized system in canonical form For simplicity let us assume at first that the linearized system has dis tinct eigenvalues λ µ 0 Thus after these changes of coordinates our system becomes x λx h1xy y µy h2xy where hj hjxy contains all of the higher order terms That is in terms of its Taylor expansion each hj contains terms that are quadratic or higher order in x andor y Equivalently we have lim xy00 hjxy r 0 where r2 x2 y2 Hirsch Ch089780123820105 2012125 1452 Page 167 9 82 Nonlinear Sinks and Sources 167 The linearized system is now given by x λx y µy For this linearized system recall that the vector field always points inside the circle of radius r centered at the origin Indeed if we take the dot product of the linear vector field with xy we find λxµyxy λx2 µy2 0 for any nonzero vector xy As we saw in Chapter 4 this forces all solutions to tend to the origin with strictly decreasing radial components The same thing happens for the nonlinear system at least close to 00 Let hxy denote the dot product of the vector field with xy We have hxy λx h1xyµy h2xyxy λx2 xh1xy µy2 yh2xy µx2 y2 µ λx2 xh1xy yh2xy µr2 xh1xy yh2xy since µ λx2 0 Therefore we have hxy r2 µ xh1xy yh2xy r2 As r 0 the right side tends to µ Thus it follows that hxy is negative at least close to the origin As a consequence the nonlinear vector field points into the interior of circles of small radius about 0 and so all solutions with initial conditions that lie inside these circles must tend to the origin Thus we are justified in calling this type of equilibrium point a sink just as in the linear case It is straightforward to check that the same result holds if the linearized sys tem has eigenvalues α iβ with α 0 β 0 In the case of repeated negative eigenvalues we first need to change coordinates so that the linearized system is x λx ϵy y λy where ϵ is sufficiently small We showed how to do this in Chapter 4 Then again the vector field points inside circles of sufficiently small radius We can now conjugate the flow of a nonlinear system near a hyperbolic equilibrium point that is a sink to the flow of its linearized system Indeed Hirsch Ch089780123820105 2012125 1452 Page 168 10 168 Chapter 8 Equilibria in Nonlinear Systems the argument used in the second example of the previous section goes over essentially unchanged In similar fashion nonlinear systems near a hyperbolic source are also conjugate to the corresponding linearized system This result is a special case of the following more general theorem The Linearization Theorem Suppose the ndimensional system X FX has an equilibrium point at X0 that is hyperbolic Then the nonlinear flow is conjugate to the flow of the linearized system in a neighborhood of X0 We will not prove this theorem here since the proof requires analytic tech niques beyond the scope of this book when there are eigenvalues present with both positive and negative real parts 83 Saddles We turn now to the case of an equilibrium for which the linearized system has a saddle at the origin in R2 As in the previous section we may assume that this system is in the form x λx f1xy y µy f2xy where µ 0 λ and fjxyr tends to 0 as r 0 As in the case of a linear system we call this type of equilibrium point a saddle For the linearized system the yaxis serves as the stable line with all solu tions on this line tending to 0 as t Similarly the xaxis is the unstable line As we saw in Section 81 we cannot expect these stable and unstable straight lines to persist in the nonlinear case However there does exist a pair of curves through the origin that have similar properties Let W s0 denote the set of initial conditions with solutions that tend to the origin as t Let W u0 denote the set of points with solutions that tend to the origin as t W s0 and W u0 are called the stable curve and unstable curve respectively The following theorem shows that solutions near nonlinear saddles behave much the same as in the linear case Hirsch Ch089780123820105 2012125 1452 Page 169 11 83 Saddles 169 The Stable Curve Theorem Suppose the system x λx f1xy y µy f2xy satisfies µ 0 λ and fjxyr 0 as r 0 Then there is an ϵ 0 and a curve x hsy that is defined for y ϵ and satisfies hs0 0 Furthermore 1 All solutions with initial conditions that lie on this curve remain on this curve for all t 0 and tend to the origin as t 2 The curve x hsy passes through the origin tangent to the yaxis 3 All other solutions with initial conditions that lie in the disk of radius ϵ centered at the origin leave this disk as time increases Some remarks are in order The curve x hsy is called the local stable curve at 0 We can find the complete stable curve W s0 by following solu tions that lie on the local stable curve backwards in time The function hsy is actually C at all points though we will not prove this result here There is a similar Unstable Curve Theorem that provides us with a local unstable curve of the form y hux This curve is tangent to the xaxis at the origin All solutions on this curve tend to the origin as t We begin with a brief sketch of the proof of the Stable Curve Theorem Consider the square bounded by the lines x ϵ and y ϵ for ϵ 0 suf ficiently small The nonlinear vector field points into the square along the interior of the top and bottom boundaries y ϵ since the system is close to the linear system x λx y µy which clearly has this property Similarly the vector field points outside the square along the left and right boundaries x ϵ Now consider the initial conditions that lie along the top boundary y ϵ Some of these solutions will exit the square to the left while others will exit to the right Solutions cannot do both so these sets are disjoint Moreover these sets are open So there must be some initial conditions with solutions that do not exit at all We will show first of all that each of these nonexiting solutions tends to the origin Secondly we will show that there is only one initial condition on the top and bottom boundary with a solution that behaves in this way Finally we will show that this solution lies along some graph of the form x hsy that has the required properties Now we fill in the details of the proof Let Bϵ denote the square bounded by x ϵ and y ϵ Let S ϵ denote the top and bottom boundaries of Bϵ Let CM denote the conical region given by yMx inside Bϵ Here we think of the slopes M of the boundary of CM as being large See Figure 84 170 Chapter 8 Equilibria in Nonlinear Systems y yMx x Figure 84 The cone Cy Lemma Given M 0 there exists 0 such that the vector field points outside Cy for points on the boundary of CyuN Be except of course at the origin Proof Given M choose 0 so that Ifo 1 x y r 2M21 for all xy Be Now suppose x 0 Then along the right boundary of Cy we have x hx fi x Mx rx fi x Mx x A x 7 2M21 Xr dx x 1 2M21 Xr x 0 2 Thus x 0 on this side of the boundary of the cone Similarly if y 0 we may choose 0 smaller if necessary so that we have y 0 on the edges of Cyy where y 0 Indeed choosing so that bb Inoy r poy 2M21 guarantees this exactly as before Thus on the edge of Cy that lies in the first quadrant we have shown that the vector field points down and to the right and therefore out of Cyy Similar calculations show that the vector field points outside Cy on all other edges of Cyy This proves the lemma 83 Saddles 171 It follows from the lemma that there is a set of initial conditions in SN Cy with solutions that eventually exit from Cy to the right and another set in SM Cy with solutions that exit to the left These sets are open because of continuity of solutions with respect to initial conditions see Chapter 7 Section 73 We next show that each of these sets is actually a single open interval Let Cj denote the portion of Cy lying above the xaxis and let Cj denote the portion lying below this axis Lemma Suppose M 1 Then there is ane 0 such that y 0 in Cy and y OinCy Proof In Chr we have Mx y so that 2 27 2 rs v7 Ty or r av MM As in the previous lemma we choose so that Ul Ihsy r 2M21 for all x y B We then have in Chr y bylhoyl Ul py r 2VM21 bb py SHYT ay bb S735 since M 1 This proves the result for Cy the proof for Cy is similar From this result we see that solutions that begin on Sf M Cy decrease in the ydirection while they remain in Che In particular no solution can remain in Chr for all time unless that solution tends to the origin By the Existence and Uniqueness Theorem the set of points in St that exit to the right or left must then be a single open interval The complement of these two intervals in S is therefore a nonempty closed interval on which solutions do not leave Cy and therefore tend to 0 as t oo We have similar behavior in Cy Hirsch Ch089780123820105 2012125 1452 Page 172 14 172 Chapter 8 Equilibria in Nonlinear Systems We next claim that the interval of initial conditions in S ϵ with solutions that tend to 0 is actually a single point To see this note first that if we multiply our system by a smooth realvalued function that is positive then the solution curves for the system do not change The parametrization of these curves does change but the curve itself as well as its orientation with respect to time does not Consider the preceding case where y 0 For ϵ small enough we have µy f2xy 0 in S ϵ CM So let gxy 1 µ f2xyy which is positive in this region Multiplying our system by gxy yields the new system x Hxy λx f1 µ f2y y y Taking the partial derivative of Hxy with respect to x yields H x xy λx f1xµ f2y λx f1f2x1y µ f2y2 λ µ hot as ϵ 0 Thus Hx is positive along horizontal line segments in S ϵ CM Now suppose we have two solutions of this system given by x0tϵet and x1tϵet with ϵ x00 x10 ϵ Then x1t x0t is mono tonically increasing so there can be at most one such solution that tends to the origin as t This is the solution that lies on the stable curve To check that this solution tends to the origin tangentially to the yaxis the preceding first lemma shows that given any large slope M we can find ϵ 0 such that the stable curve lies inside the thin triangle S ϵ CM Since M is arbitrary it follows that xtyt 0 as t Then we have xt yt λxt f1xtyt µyt f2xtyt λxt µyt hot 0 Hirsch Ch089780123820105 2012125 1452 Page 173 15 83 Saddles 173 as t Thus the normalized tangent vector along the stable curve becomes vertical as t This concludes the proof of the Stable Curve Theorem We conclude this section with a brief discussion of higherdimensional sad dles Suppose X FX where X Rn Suppose that X0 is an equilibrium solution for which the linearized system has k eigenvalues with negative real parts and n k eigenvalues with positive real parts Then the local stable and unstable sets are not generally curves Rather they are submanifolds of dimen sion k and n k respectively Without entering the realm of manifold theory we simply note that this means there is a linear change of coordinates in which the local stable set is given near the origin by the graph of a C function g Br Rnk which satisfies g0 0 and all partial derivatives of g van ish at the origin Here Br is the disk of radius r centered at the origin in Rk The local unstable set is a similar graph over an n kdimensional disk Each of these graphs is tangent at the equilibrium point to the stable and unstable subspaces at X0 Thus they meet only at X0 Example Consider the system x x y y z z x2 y2 The linearized system at the origin has eigenvalues 1 and 1 repeated The change of coordinates u x v y w z 1 3x2 y2 converts the nonlinear system to the linear system u u v v w w The plane w 0 for the linear system is the stable plane Under the change of coordinates this plane is transformed to the surface z 1 3x2 y2 Hirsch Ch089780123820105 2012125 1452 Page 174 16 174 Chapter 8 Equilibria in Nonlinear Systems y x z Wu0 Ws0 Figure 85 Phase portrait for x x y y z z x2 y2 which is a paraboloid passing through the origin in R3 and opening down ward All solutions tend to the origin on this surface we call this the stable surface for the nonlinear system See Figure 85 84 Stability The study of equilibria plays a central role in ordinary differential equations and their applications An equilibrium point however must satisfy a certain stability criterion to be significant physically Here as in several other places in this book we use the word physical in a broad sense in some contexts physical could be replaced by biological chemical or even economic An equilibrium is said to be stable if nearby solutions stay nearby for all future time In applications of dynamical systems one cannot usually pinpoint positions exactly but only approximately so an equilibrium must be stable to be physically meaningful More precisely suppose X Rn is an equilibrium point for the differential equation X FX Then X is a stable equilibrium if for every neighborhood O of X in Rn there is a neighborhood O1 of X in O such that every solution Xt with X0 X0 in O1 is defined and remains in O for all t 0 A different form of stability is asymptotic stability If O1 can be chosen so that in addition to the properties for stability we have limt Xt X Hirsch Ch089780123820105 2012125 1452 Page 175 17 85 Bifurcations 175 then we say that X is asymptotically stable In applications these are often the most important types of equilibria since they are visible Moreover from our previous results we have the following theorem Theorem Suppose the ndimensional system X FX has an equilibrium point at X and all of the eigenvalues of the linearized system at X have negative real parts Then X is asymptotically stable An equilibrium X that is not stable is called unstable This means there is a neighborhood O of X such that for every neighborhood O1 of X in O there is at least one solution Xt starting at X0 O1 which does not lie entirely in O for all t 0 Sources and saddles are examples of unstable equilibria An example of an equilibrium that is stable but not asymptotically stable is the origin in R2 for a linear equation X AX where A has pure imaginary eigenvalues The impor tance of this example in applications is limited despite the famed harmonic oscillator because the slightest nonlinear perturbation will destroy its char acter as we saw in Section 81 Even a small linear perturbation can make a center into a sink or a source Thus when the linearization of the system at an equilibrium point is hyper bolic we can immediately determine the stability of that point Unfortunately many important equilibrium points that arise in applications are nonhyper bolic It would be wonderful to have a technique that determined the stability of an equilibrium point that works in all cases Unfortunately we as yet have no universal way of determining stability except by actually finding all solu tions of the system which is usually difficult if not impossible We will present some techniques that allow us to determine stability in certain special cases in the next chapter 85 Bifurcations In this section we will describe some simple examples of bifurcations that occur for nonlinear systems We consider a family of systems X FaX where a is a real parameter We assume that Fa depends on a in a C fashion A bifurcation occurs when there is a significant change in the structure of the solutions of the system as a varies The simplest types of bifurcations occur when the number of equilibrium solutions changes as a varies Hirsch Ch089780123820105 2012125 1452 Page 176 18 176 Chapter 8 Equilibria in Nonlinear Systems Recall the elementary bifurcations we encountered in Chapter 1 for first order equations x fax If x0 is an equilibrium point then we have fax0 0 If f ax0 0 then small changes in a do not change the local structure near x0 that is the differential equation x faϵx has an equilibrium point x0ϵ that varies continuously with ϵ for small ϵ A glance at the increasing or decreasing graphs of faϵx near x0 shows why this is true More rigorously this is an immediate consequence of the Implicit Function Theorem see Exercise 3 at the end of this chapter Thus bifurcations for firstorder equations only occur in the nonhyperbolic case where f ax0 0 Example The firstorder equation x fax x2 a has a single equilibrium point at x 0 when a 0 Note f 00 0 but f 0 0 0 For a 0 this equation has no equilibrium points since fax 0 for all x but for a 0 this equation has a pair of equilibria Thus a bifurcation occurs as the parameter passes through a 0 This kind of bifurcation is called a saddlenode bifurcation we will see the saddle in this bifurcation a little later In a saddlenode bifurcation there is an interval about the bifurcation value a0 and another interval I on the xaxis in which the differential equation has 1 Two equilibrium points in I if a a0 2 One equilibrium point in I if a a0 3 No equilibrium points in I if a a0 Of course the bifurcation could take place the other way with no equi libria when a a0 The preceding example is actually the typical type of bifurcation for firstorder equations Theorem SaddleNode Bifurcation Suppose x fax is a firstorder differential equation for which 1 fa0x0 0 2 f a0x0 0 3 f a0x0 0 4 fa0 a x0 0 85 Bifurcations 177 Then this differential equation undergoes a saddlenode bifurcation at a ao Proof Let Gx a fax We have Gxo ao 0 Also dG a 28 509 9 0 da da so we may apply the Implicit Function Theorem to conclude that there is a smooth function a ax such that Gx ax 0 In particular x is an equilibrium point for the equation x farxx since facxx 0 Differentiating Gx ax with respect to x we find dGdx a x dGda Now 0G0xX9 ao fay xo 0 while 8G0axpa9 0 by assump tion Thus a x9 0 Differentiating once more we find aG AG 1 dG aG a x dx da dx dxda aG da Since 0 G0x x9 do 0 we have aG axe x0 ao ax 2 40 Xo 40 da since 87 Gdxxo ao fo xp 0 This implies that the graph of a ax is either concave up or concave down so we have two equilibria near xo for avalues on one side of ap and no equilibria for avalues on the other side We said earlier that such saddlenode bifurcations were the typical bifur cations involving equilibrium points for firstorder equations The reason for this is that we must have both 1 fay Xo 0 2 fzx0 0 if x fx is to undergo a bifurcation when a ag Generically in the sense of Chapter 5 Section 56 the next higherorder derivatives at xo 49 will be Hirsch Ch089780123820105 2012125 1452 Page 178 20 178 Chapter 8 Equilibria in Nonlinear Systems x a0 x0 a Figure 86 Bifurcation diagram for a saddlenode bifurcation nonzero That is we typically have 3 f a0x0 0 4 fa a x0a0 0 at such a bifurcation point But these are precisely the conditions that guaran tee a saddlenode bifurcation Recall that the bifurcation diagram for x fax is a plot of the various phase lines of the equation versus the parameter a The bifurcation diagram for a typical saddlenode bifurcation is displayed in Figure 86 The directions of the arrows and the curve of equilibria may change Example Pitchfork Bifurcation Consider x x3 ax There are three equilibria for this equation at x 0 and x a when a 0 When a 0 x 0 is the only equilibrium point The bifurcation diagram shown in Figure 87 explains why this bifurcation is so named Now we turn to some bifurcations in higher dimensions The saddlenode bifurcation in the plane is similar to its onedimensional cousin but now we see where the saddle comes from Example Consider the system x x2 a y y 85 Bifurcations 179 x a Figure 87 Bifurcation diagram for a pitchfork bifurcation Figure 88 Saddlenode bifurcation when a 0 a0 and a 0 When a 0 this is one of the systems considered in Section 81 There is a unique equilibrium point at the origin and the linearized system has a zero eigenvalue When a passes through a 0 a saddlenode bifurcation occurs When a 0 we have x 0 so all solutions move to the right the equilibrium point disappears When a 0 we have a pair of equilibria at the points a 0 The linearized equation is 2x 0 va x So we have a sink at a0 and a saddle at a 0 Note that solutions on the lines x a remain for all time on these lines since x 0 on these lines Solutions tend directly to the equilibria on these lines since y y This bifurcation is sketched in Figure 88 a A saddlenode bifurcation may have serious global implications for the behavior of solutions as the following example shows Hirsch Ch089780123820105 2012125 1452 Page 180 22 180 Chapter 8 Equilibria in Nonlinear Systems Example Consider the system given in polar coordinates by r r r3 θ sin2θ a where a is again a parameter The origin is always an equilibrium point since r 0 when r 0 There are no other equilibria when a 0 since in that case θ 0 When a 0 two additional equilibria appear at rθ 10 and rθ 1π When 1 a 0 there are four equilibria on the circle r 1 These occur at the roots of the equation sin2θ a We denote these roots by θ and θ π where we assume that 0 θ π2 and π2 θ 0 Note that the flow of this system takes the straight rays through the origin given by θ constant to other straight rays This occurs since θ depends only on θ not on r Also the unit circle is invariant in the sense that any solution that starts on the circle remains there for all time This follows since r 0 on this circle All other nonzero solutions tend to this circle since r 0 if 0 r 1 whereas r 0 if r 1 Now consider the case a 0 In this case the xaxis is invariant and all nonzero solutions on this line tend to the equilibrium points at x 1 In the upper halfplane we have θ 0 so all other solutions in this region wind counterclockwise about 0 and tend to x 1 the θcoordinate increases to θ π while r tends monotonically to 1 No solution winds more than angle π about the origin since the xaxis acts as a barrier The system behaves symmetrically in the lower halfplane When a 0 two things happen First of all the equilibrium points at x 1 disappear and now θ 0 everywhere Thus the barrier on the xaxis has been removed and all solutions suddenly are free to wind forever about the origin Secondly we now have a periodic solution on the circle r 1 and all nonzero solutions are attracted to it This dramatic change is caused by a pair of saddlenode bifurcations at a 0 Indeed when 1 a 0 we have two pair of equilibria on the unit circle The rays θ θ and θ θ π are invariant and all solutions on these rays tend to the equilibria on the circle Consider the halfplane θ θ θ π For θvalues in the interval θ θ θ we have θ 0 while θ 0 in the interval θ θ θ π Solutions behave symmetrically in the complementary halfplane Therefore all solutions that do not lie on the rays θ θ or θ θ π tend to the equilibrium points at r 1 θ θ or at r 1 θ θ π These equilibria are therefore sinks At the other equi libria we have saddles The stable curves of these saddles lie on the rays θ θ 85 Bifurcations 181 Va La ZOZIN ASIN LE PK A YS NET FS PFeKY Lx YA pf 7 fe BOTS EGE a N EE Lad Zs a Figure 89 Global effects of saddlenode bifurcations when a 0 a0 and a 0 and 6 z 64 and the unstable curves of the saddles are given by the unit circle minus the sinks See Figure 89 a The previous examples all featured bifurcations that occur when the lin earized system has a zero eigenvalue Another case where the linearized system fails to be hyperbolic occurs when the system has pure imaginary eigenvalues Example Hopf Bifurcation Consider the system x axyxxy y xtayyxy There is an equilibrium point at the origin and the linearized system is x x Aloa The eigenvalues are a i so we expect a bifurcation when a 0 To see what happens as a passes through 0 we change to polar coordinates The system becomes rf arr 61 Note that the origin is the only equilibrium point for this system since 6 4 0 For a 0 the origin is a sink since ar r 0 for all r 0 Thus all solutions tend to the origin in this case When a 0 the equilibrium becomes a source What else happens When a 0 we have r 0 if r a So the circle of radius a is a periodic solution with period 27 We also have r 0 if 0 r fa while r 0if r a Thus all nonzero solutions spiral toward this circular solution as t oo Hirsch Ch089780123820105 2012125 1452 Page 182 24 182 Chapter 8 Equilibria in Nonlinear Systems Figure 810 Hopf bifurcation for a 0 and a 0 This type of bifurcation is called a Hopf bifurcation At a Hopf bifurcation no new equilibria arise Instead a periodic solution is born at the equilibrium point as a passes through the bifurcation value See Figure 810 86 Exploration Complex Vector Fields In this exploration you will investigate the behavior of systems of differen tial equations in the complex plane of the form z Fz Throughout this section z will denote the complex number z x iy and Fz will be a poly nomial with complex coefficients Solutions of the differential equation will be expressed as curves zt xt iyt in the complex plane You should be familiar with complex functions such as exponential sine and cosine as well as with the process of taking complex square roots to comprehend fully what you see in the following Theoretically you should also have a grasp of complex analysis as well However all of the rou tine tricks from integration of functions of real variables work just as well when integrating with respect to z You need not prove this for you can always check the validity of your solutions when you have completed the integrals 1 Solve the equation z az where a is a complex number What kind of equilibrium points do you find at the origin for these differential equations 2 Solve each of the following complex differential equations and sketch the phase portrait a z z2 b z z2 1 c z z2 1 Hirsch Ch089780123820105 2012125 1452 Page 183 25 86 Exploration Complex Vector Fields 183 3 For a complex polynomial Fz the complex derivative is defined just as the real derivative Fz0 lim zz0 Fz Fz0 z z0 only this limit is evaluated along any smooth curve in C that passes through z0 This limit must exist and yield the same complex number for each such curve For polynomials this is easily checked Now write Fx iy uxy ivxy Evaluate Fz0 in terms of the derivatives of u and v by taking the limit first along the horizontal line z0 t and second along the vertical line z0 it Use this to conclude that if the derivative exists then we must have u x v y and u y v x at every point in the plane The equations are called the CauchyRiemann equations 4 Use the preceding observation to determine all possible types of equilib rium points for complex vector fields 5 Solve the equation z z z0z z1 where z0z1 C and z0 z1 What types of equilibrium points occur for different values of z0 and z1 6 Find a nonlinear change of variables that converts the previous system to w αw with α C Hint Since the original system has two equilibrium points and the linear system only one the change of variables must send one of the equilibrium points to 7 Classify all complex quadratic systems of the form z z2 az b where ab C 8 Consider the equation z z3 az with a C First use a computer to describe the phase portraits for these systems Then prove as much as you can about these systems and classify them with respect to a Hirsch Ch089780123820105 2012125 1452 Page 184 26 184 Chapter 8 Equilibria in Nonlinear Systems 9 Choose your own nontrivial family of complex functions depending on a parameter a C and provide a complete analysis of the phase portraits for each a Some interesting families to consider include aexpz asinz or z2 az2 a E X E R C I S E S 1 For each of the following nonlinear systems a Find all of the equilibrium points and describe the behavior of the associated linearized system b Describe the phase portrait for the nonlinear system c Does the linearized system accurately describe the local behavior near the equilibrium points i x sinx y cosy ii x xx2 y2 y yx2 y2 iii x x y2 y 2y iv x y2 y y v x x2 y y2 2 Find a global change of coordinates that linearizes the system x x y2 y y z z y2 3 Consider a firstorder differential equation x fax for which fax0 0 and f ax0 0 Prove that the differential equation x faϵx has an equilibrium point x0ϵ where ϵ x0ϵ is a smooth function satisfying x00 x0 for ϵ sufficiently small 4 Find general conditions on the derivatives of fax so that the equation x fax undergoes a pitchfork bifurcation at a a0 Prove that your conditions lead to such a bifurcation Hirsch Ch089780123820105 2012125 1452 Page 185 27 Exercises 185 5 Consider the system x x2 y y x y a where a is a parameter a Find all equilibrium points and compute the linearized equation at each b Describe the behavior of the linearized system at each equilibrium point c Describe any bifurcations that occur 6 Give an example of a family of differential equations is x fax for which there are no equilibrium points if a 0 a single equilibrium if a 0 and four equilibrium points if a 0 Sketch the bifurcation diagram for this family 7 Discuss the local and global behavior of solutions of r r r3 θ sin2θ a at the bifurcation value a 1 8 Discuss the local and global behavior of solutions of r r r2 θ sin2θ2 a at all of the bifurcation values 9 Consider the system r r r2 θ sinθ a a For which values of a does this system undergo a bifurcation b Describe the local behavior of solutions near the bifurcation values at before and after the bifurcation c Sketch the phase portrait of the system for all possible different cases d Discuss any global changes that occur at the bifurcations Hirsch Ch089780123820105 2012125 1452 Page 186 28 186 Chapter 8 Equilibria in Nonlinear Systems 10 Let X FX be a nonlinear system in Rn Suppose that F0 0 and that DF0 has n distinct eigenvalues with negative real parts Describe the construction of a conjugacy between this system and its linearization 11 Consider the system X FX where X Rn Suppose that F has an equilibrium point at X0 Show that there is a change of coordinates that moves X0 to the origin and converts the system to X AX GX where A is an n n matrix that is the canonical form of DFX0 and where GX satisfies lim X0 GX X 0 12 In the definition of an asymptotically stable equilibrium point we required that the equilibrium point also be stable This requirement is not vacuous Give an example of a phase portrait a sketch is sufficient that has an equilibrium point toward which all nearby solution curves eventually tend but which is not stable Hirsch Ch099780123820105 2012125 1832 Page 187 1 9 Global Nonlinear Techniques In this chapter we present a variety of qualitative techniques for analyzing the behavior of nonlinear systems of differential equations The reader should be forewarned that none of these techniques works for all nonlinear systems most work only in specialized situations which as we shall see in the ensu ing chapters nonetheless occur in many important applications of differential equations 91 Nullclines One of the most useful tools for analyzing nonlinear systems of differential equations especially planar systems is the nullcline For a system in the form x 1 f1x1xn x n fnx1xn the xjnullcline is the set of points where x j vanishes so the xjnullcline is the set of points determined by setting fjx1xn 0 Differential Equations Dynamical Systems and an Introduction to Chaos DOI 101016B9780123820105000099 c 2013 Elsevier Inc All rights reserved 187 Hirsch Ch099780123820105 2012125 1832 Page 188 2 188 Chapter 9 Global Nonlinear Techniques The xjnullclines usually separate Rn into a collection of regions in which the xjcomponents of the vector field point in either the positive or negative direction If we determine all of the nullclines then this allows us to decom pose Rn into a collection of open sets in each of which the vector field points in a certain direction This is easiest to understand in the case of a planar system x f xy y gxy On the xnullclines we have x 0 so the vector field points straight up or down and these are the only points at which this happens Therefore the xnullclines divide R2 into regions where the vector field points either to the left or to the right Similarly on the ynullclines the vector field is horizon tal so the ynullclines separate R2 into regions where the vector field points either upward or downward The intersections of the x and ynullclines yield the equilibrium points In any of the regions between the nullclines the vector field is neither ver tical nor horizontal so it must point in one of four directions northeast northwest southeast or southwest We call such regions basic regions Often a simple sketch of the basic regions allows us to understand the phase portrait completely at least from a qualitative point of view Example For the system x y x2 y x 2 the xnullcline is the parabola y x2 and the ynullcline is the vertical line x 2 These nullclines meet at 24 so this is the only equilibrium point The nullclines divide R2 into four basic regions labeled A D in Figure 91a By first choosing one point in each of these regions and then determining the direction of the vector field at that point we can decide the direction of the vector field at all points in the basic region For example the point 01 lies in region A and the vector field is 12 at this point which points toward the southeast Thus the vector field points southeast at all points in this region Of course the vector field may be nearly horizontal or nearly vertical in this region when we say southeast we mean that the angle θ of the vector field lies in the sector π2 θ 0 Continuing in this fashion we get the direction of the vector field in all four regions as in Figure 91b This also determines the horizontal and vertical directions of the vector field on the nullclines Just from the direction field 91 Nullclines 189 B AK PYRO PERS att x KY QR RR EEE ESE PL SSS ae FWY RSE EE ss aK ROP s AS 77 isxseperse etre ww gp LY pee aware Naa percepts aa ew py YR BRS Re awe ee NYRR SITTIN ss yest etoolt p CTT NAS Pf ett i e KF EKVANY panes ceccetestsseanesin weed CN cen Deere er nals aa e we ee PE El peer ae www yw yg PY pee ee ew Kw yy gf fl bee ee wee wwe YY gp fb ee ee a b Figure 91 Nullclines and direction field ix2 I wo ya Figure 92 Solutions enter the basic region B and then tend to oo alone it appears that the equilibrium point is a saddle Indeed this is the case because the linearized system at 24 is 4 1 X X which has eigenvalues 2 5 one of which is positive the other negative More important we can fill in the approximate behavior of solutions every where in the plane For example note that the vector field points into the basic region marked Bat all points along its boundary and then it points northeast erly at all points inside B Thus any solution in region B must stay in region B for all time and tend toward oo in the northeast direction See Figure 92 Similarly solutions in the basic region D stay in that region and head toward oo in the southwest direction Solutions starting in the basic regions A and C have a choice They must eventually cross one of the nullclines and enter Hirsch Ch099780123820105 2012125 1832 Page 190 4 190 Chapter 9 Global Nonlinear Techniques Figure 93 Nullclines and phase portrait for x y x2 y x 2 regions B and D and therefore we know their ultimate behavior or else they tend to the equilibrium point However there is only one curve of such solu tions in each region the stable curve at 24 Thus we completely understand the phase portrait for this system at least from a qualitative point of view See Figure 93 Example Heteroclinic Bifurcation Next consider the system that depends on a parameter a x x2 1 y xy ax2 1 The xnullclines are given by x 1 while the ynullclines are xy ax2 1 The equilibrium points are 10 Since x 0 on x 1 the vector field is actually tangent to these nullclines Moreover we have y y on x 1 and y y on x 1 So solutions tend to 10 along the vertical line x 1 and tend away from 10 along x 1 This happens for all values of a Now lets look at the case a 0 Here the system simplifies to x x2 1 y xy so y 0 along the axes In particular the vector field is tangent to the x axis and is given by x x2 1 on this line So we have x 0 if x 1 and Hirsch Ch099780123820105 2012125 1832 Page 191 5 91 Nullclines 191 x 0 if x 1 Thus at each equilibrium point we have one straightline solution tending to the equilibrium and one tending away So it appears that each equilibrium is a saddle This is indeed the case as is easily checked by linearization There is a second ynullcline along x 0 but the vector field is not tangent to this nullcline Computing the direction of the vector field in each of the basic regions determined by the nullclines yields Figure 94 from which we can deduce immediately the qualitative behavior of all solutions Note that when a 0 one branch of the unstable curve through 10 matches up exactly with a branch of the stable curve at 10 All solutions on this curve simply travel from one saddle to the other Such solutions are called heteroclinic solutions or saddle connections Typically for planar systems sta ble and unstable curves rarely meet to form such heteroclinic connections When they do however one can expect a bifurcation Now consider the case where a 0 The xnullclines remain the same at x 1 But the ynullclines change drastically as shown in Figure 95 They are given by y ax2 1x When a 0 consider the basic region denoted by A Here the vector field points southwesterly In particular the vector field points in this direction along the xaxis between x 1 and x 1 This breaks the heteroclinic con nection the right portion of the stable curve associated with 10 must now come from y in the upper half plane while the left portion of the unsta ble curve associated with 10 now descends to y in the lower half plane This opens an avenue for certain solutions to travel from y to y between the two lines x 1 Whereas when a 0 all solutions a b Figure 94 Nullclines and phase portrait for x x2 1 y xy Hirsch Ch099780123820105 2012125 1832 Page 192 6 192 Chapter 9 Global Nonlinear Techniques a A A b Figure 95 Nullclines and phase plane when a0 after the heteroclinic bifurcation remain for all time confined to either the upper or lower half plane the hete roclinic bifurcation at a 0 opens the door for certain solutions to make this transit A similar situation occurs when a 0 see Exercise 2 at the end of this chapter 92 Stability of Equilibria Determining the stability of an equilibrium point is straightforward if the equilibrium is hyperbolic When this is not the case this determination becomes more problematic In this section we develop an alternative method for showing that an equilibrium is asymptotically stable Due to the Russian mathematician Liapunov this method generalizes the notion that for a lin ear system in canonical form the radial component r decreases along solution curves Liapunov noted that other functions besides r could be used for this purpose Perhaps more important Liapunovs method gives us a grasp on the size of the basin of attraction of an asymptotically stable equilibrium point By definition the basin of attraction is the set of all initial conditions with solutions that tend to the equilibrium point Let LO R be a differentiable function defined on an open set O in Rn that contains an equilibrium point X of the system X FX Consider the function LX DLXFX 92 Stability of Equilibria 193 As we have seen if X is the solution of the system passing through X when t 0 then we have 10 4 Lod dto by the Chain Rule Consequently if LX is negative L decreases along the solution curve through X We can now state Liapunovs Stability Theorem Theorem Liapunov Stability Let X be an equilibrium point for X FX Let L OR be a differentiable function defined on an open set O containing X Suppose further that a LX 0 and LX0 if xX X b L0inO X Then X is stable Furthermore if L also satisfies c L0inOxX then X is asymptotically stable A function L satisfying a and b is called a Liapunov function for X If c also holds we call L a strict Liapunov function Note that Liapunovs theorem can be applied without solving the differen tial equation all we need to compute is DLxFX This is a real plus On the other hand there is no cutanddried method of finding Liapunov functions it is usually a matter of ingenuity or trial and error in each case Sometimes there are natural functions to try For example in the case of mechanical or electrical systems energy is often a Liapunov function as we shall see in Chapter 13 Example Consider the system of differential equations in R given by x ex2yz4 1 y xteyz1 7 7 where is a parameter The origin is the only equilibrium point for this system The linearization of the system at 000 is e 2 0 y1 e OY 0 0 0 Hirsch Ch099780123820105 2012125 1832 Page 194 8 194 Chapter 9 Global Nonlinear Techniques The eigenvalues are 0 and ϵ 2i Thus from the linearization we can only conclude that the origin is unstable if ϵ 0 This follows since when z 0 the xyplane is invariant and the system is linear on this plane When ϵ 0 all we can conclude is that the origin is not hyperbolic When ϵ 0 we search for a Liapunov function for 000 of the form Lxyz ax2 by2 cz2 with abc 0 For such an L we have L 2axx byy czz so that L2 axϵx 2yz 1 byx ϵyz 1 cz4 ϵax2 by2z 1 2a bxyz 1 cz4 For stability we want L 0 this can be arranged for example by setting a 1 b 2 and c 1 If ϵ 0 we then have L 2z4 0 so the origin is stable It can be shown see Exercise 4 at the end of this chapter that the origin is not asymptotically stable in this case If ϵ 0 then we find L2 ϵx2 2y2z 1 z4 so that L 0 in the region O given by z 1 minus the origin We con clude that the origin is asymptotically stable in this case and indeed from Exercise 4 that all solutions that start in the region O tend to the origin Example The Nonlinear Pendulum Consider a pendulum consisting of a light rod of length ℓ to which is attached a ball of mass m The other end of the rod is attached to a wall at a point so that the ball of the pendulum moves on a circle centered at this point The position of the mass at time t is completely described by the angle θt of the mass from the straightdown position and measured in the counterclockwise direction Thus the position of the mass at time t is given by ℓsinθtℓcos θt The speed of the mass is the length of the velocity vector which is ℓdθdt and the acceleration is ℓd2θdt2 We assume that the only two forces acting on the pendulum are the force of gravity and a force due to friction The gravitational force is a constant force equal to mg acting in the downward direction the component of this force tangent to the circle of motion is given by mg sinθ We take the force due to friction to be proportional to velocity and so this force is given by bℓdθdt for some constant b 0 When there is no force due to friction b 0 we have an ideal pendulum 92 Stability of Equilibria 195 Newtons Law then gives the secondorder differential equation for the pendulum ao be do nd ml bé mgsind dt dt 8 For simplicity we assume that units have been chosen so that m f g 1 Rewriting this equation as a system we introduce v d0dt and get dv v bvsiné Clearly we have two equilibrium points mod 277 the downward rest posi tion at 6 0 v 0 and the straightup position 0 z v 0 This upward position is an unstable equilibrium both from a mathematical check the linearization and physical point of view For the downward equilibrium point the linearized system is 0 1 va 4y The eigenvalues here are either pure imaginary when b 0 or else have neg ative real parts when b 0 So the downward equilibrium is asymptotically stable if b 0 as everyone on earth who has watched a reallife pendulum knows To investigate this equilibrium point further consider the function E6v sv 1cos For readers with a background in elementary mechanics this is the wellknown total energy function which we will describe further in Chapter 13 We compute Ewsin66 bv so that E0 Thus E is a Liapunov function Thus the origin is a stable equilibrium If b 0 that is there is no friction then E 0 That is E is constant along all solutions of the system Therefore we may simply plot the level curves of E to see where the solution curves reside We find the phase portrait shown in Figure 96 Note that we do not have to solve the differential equation to paint this picture knowing the level curves of E and the direc tion of the vector field tells us everything We will encounter many such very special functions that are constant along solution curves later in this chapter The solutions encircling the origin have the property that m 0t a for all t Therefore these solutions correspond to the pendulum oscillating about the downward rest position without ever crossing the upward position Hirsch Ch099780123820105 2012125 1832 Page 196 10 196 Chapter 9 Global Nonlinear Techniques 2π 2π Figure 96 Phase portrait for the ideal pendulum θ π The special solutions connecting the equilibrium points at π0 correspond to the pendulum tending to the upwardpointing equilibrium in both the forward and backward time directions You dont often see such motions Beyond these special solutions we find solutions for which θt either increases or decreases for all time in these cases the pendulum spins forever in the counterclockwise or clockwise direction We will return to the pendulum example for the case b0 later but first we prove Liapunovs theorem Proof Let δ 0 be so small that the closed ball BδX around the equilibrium point X of radius δ lies entirely in O Let α be the minimum value of L on the boundary of BδX that is on the sphere SδX of radius δ and center X Then α 0 by assumption Let U X BδX LX α Then no solution starting in U can meet SδX since L is nonincreasing on solution curves Thus every solution starting in U never leaves BδX This proves that X is stable Now suppose that assumption c in the Liapunov Stability Theorem holds as well so that L is strictly decreasing on solutions in U X Let Xt be a solution starting in U X and suppose that Xtn Z0 BδX for some sequence tn We claim that Z0 X To see this observe that LXtLZ0 for all t 0 since LXt decreases and LXtn LZ0 by continuity of L If Z0 X let Zt be the solution starting at Z0 For any s0 we have LZs LZ0 Thus for any solution Ys starting sufficiently near Z0 we have LYs LZ0 Hirsch Ch099780123820105 2012125 1832 Page 197 11 92 Stability of Equilibria 197 Setting Y0 Xtn for sufficiently large n yields the contradiction LX tn s LZ0 Therefore Z0 X This proves that X is the only possible limit point of the set Xt t 0 and completes the proof of Liapunovs theorem Figure 97 makes the theorem intuitively obvious The condition L 0 means that when a solution crosses a level surface L1c it moves inside the set where L c and can never come out again Unfortunately it is some times difficult to justify the diagram shown in this figure why should the sets L1c shrink down to X Of course in many cases Figure 97 is indeed cor rect because for example if L is a quadratic function such as ax2 by2 with ab 0 But what if the level surfaces look like those in Figure 98 It is hard to imagine such an L that fulfills all the requirements of a Liapunov function however rather than trying to rule out that possibility it is simpler to give the analytic proof as before Example Now consider the system x x3 y yx2 z2 1 z sinz L1c3 L1c2 L1c1 Figure 97 Solutions decrease through the level sets L1cj of a strict Liapunov function Hirsch Ch099780123820105 2012125 1832 Page 198 12 198 Chapter 9 Global Nonlinear Techniques L1c2 L1c1 Figure 98 Level sets of a Liapunov function may look like this The origin is again an equilibrium point It is not the only one however since 00nπ is also an equilibrium point for each n Z Thus the origin can not be globally asymptotically stable Moreover the planes z nπ for n Z are invariant in the sense that any solution that starts on one of these planes remains there for all time This occurs because z 0 when z nπ In par ticular any solution that begins in the region z π must remain trapped in this region for all time Linearization at the origin yields the system Y 0 0 0 0 1 0 0 0 1 Y which tells us nothing about the stability of this equilibrium point However consider the function Lxyz x2 y2 z2 Clearly L 0 except at the origin We compute L 2x4 2y2x2 z2 1 2z sinz Then L 0 at all points in the set z π except the origin since z sinz 0 when z 0 Thus the origin is asymptotically stable Moreover we can conclude that the basin of attraction of the origin is the entire region z π From the proof of the Liapunov Stability Theorem it follows immediately that any solution that starts inside a sphere of radius r π must tend to the origin Outside of the sphere of radius π and between the planes z π the function L is still strictly decreasing Since solutions are trapped between these two planes it follows that they too must tend to the origin Hirsch Ch099780123820105 2012125 1832 Page 199 13 92 Stability of Equilibria 199 Liapunov functions not only detect stable equilibria they can also be used to estimate the size of the basin of attraction of an asymptotically stable equilib rium as the preceding example shows The following theorem gives a criterion for asymptotic stability and the size of the basin even when the Liapunov function is not strict To state it we need several definitions Recall that a set P is called invariant if for each X P φtX is defined and in P for all t R For example the region z π in the previous example is an invariant set The set P is positively invariant if for each X P φtX is defined and in P for all t 0 The portion of the region z π inside a sphere centered at the origin in the previous example is positively invariant but not invariant Finally an entire solution of a system is a set of the form φtX t R Theorem Lasalles Invariance Principle Let X be an equilibrium point for X FX and let L U R be a Liapunov function for X where U is an open set containing X Let P U be a neighborhood of X that is closed Suppose that P is positively invariant and that there is no entire solution in P X on which L is constant Then X is asymptotically stable and P is contained in the basin of attraction of X Before proving this theorem we apply it to the equilibrium X 00 of the damped pendulum discussed earlier Recall that a Liapunov function is given by Eθv 1 2v2 1 cosθ and that E bv2 Since E 0 on v 0 this Liapunov function is not strict To estimate the basin of 00 fix a number c with 0 c 2 and define Pc θv Eθv c and θ π Clearly 00 Pc We shall prove that Pc lies in the basin of attraction of 00 Note first that Pc is positively invariant To see this suppose that θtvt is a solution with θ0v0 Pc We claim that θtvt Pc for all t 0 We clearly have Eθtvt c since E 0 If θt π then there must exist a smallest t0 such that θt0 π But then Eθt0vt0 Eπvt0 1 2vt02 2 2 However Eθt0vt0 c 2 This contradiction shows that θt0 π and so Pc is positively invariant Hirsch Ch099780123820105 2012125 1832 Page 200 14 200 Chapter 9 Global Nonlinear Techniques We now show that there is no entire solution in Pc on which E is constant except the equilibrium solution Suppose there is such a solution Then along that solution E 0 and so v 0 Thus θ 0 so θ is constant on the solution We also have v sinθ 0 on the solution Since θ π it follows that θ 0 Thus the only entire solution in Pc on which E is constant is the equilibrium point 00 Finally Pc is a closed set For if θ0v0 is a limit point of Pc then θ0 π and Eθ0v0 c by continuity of E But θ0 π implies Eθ0v0c as we showed before Thus θ0 π and so θ0v0 does belong to Pc Pc is therefore closed From the theorem we conclude that Pc belongs to the basin of attraction of 00 for each c 2 thus the set P Pc 0 c 2 is also contained in this basin Note that we may write P θv Eθv 2 and θ π Figure 99 displays the phase portrait for the damped pendulum The curves marked γc are the level sets Eθv c Note that solutions cross each of these curves exactly once and eventually tend to the origin This result is quite natural on physical grounds For if θ π then Eθ0 2 and so the solution through θ0 tends to 00 That is if we start the pendulum from rest at any angle θ except the vertical position the pendulum will eventually wind down to rest at its stable equilibrium position π γc π Figure 99 The curve γc bounds the region Pc Hirsch Ch099780123820105 2012125 1832 Page 201 15 92 Stability of Equilibria 201 There are other initial positions in the basin of 00 that are not in the set P For example consider the solution through πu where u is very small but not zero Then πu P but the solution through this point moves quickly into P and therefore eventually approaches 00 Thus πu also lies in the basin of attraction of 00 This can be seen in Figure 99 where the solutions that begin just above the equilibrium point at π0 and just below π0 quickly cross γc and then enter Pc See Exercise 5 at the end of this chapter for further examples of this We now prove the theorem Proof Imagine a solution Xt that lies in the positively invariant set P for 0 t but suppose that Xt does not tend to X as t Then there must be a point Z X in P and a sequence tn such that lim nXtn Z We may assume that the sequence tn is an increasing sequence We claim that the entire solution through Z lies in P That is φtZ is defined and in P for all t R not just t 0 This can be seen as follows First φtZ is certainly defined for all t 0 since P is positively invariant On the other hand φtXtn is defined and in P for all t in the interval tn0 Since tn is an increasing sequence we have that φtXtnk is also defined and in P for all t tn0 and all k 0 Since the points Xtnk Z as k it follows from continuous dependence of solutions on initial conditions that φtZ is defined and in P for all t tn0 Since this holds for any tn we see that the solution through Z is an entire solution lying in P Finally we show that L is constant on the entire solution through Z If LZ α then we have LXtn α and moreover lim nLXtn α More generally if sn is any sequence of times for which sn as n then LXsn α as well This follows from the fact that L is non increasing along solutions Now the sequence Xtn s converges to φsZ and so LφsZ α This contradicts our assumption that there are no entire solutions lying in P on which L is constant and proves the theorem In this proof we encountered certain points that were limits of a sequence of points on the solution through X The set of all points that are limit points of a given solution is called the set of ωlimit points or the ωlimit set of the solution Xt Similarly we define the set of αlimit points or the αlimit set of a solution Xt to be the set of all points Z such that limn Xtn Z 202 Chapter 9 Global Nonlinear Techniques for some sequence t oo The reason such as it is for this terminol ogy is that is the first letter and w the last letter of the Greek alphabet The following facts essentially proved before will be used in the following chapter Proposition The alimit set and the wlimit set of a solution that is defined for allt R are closed invariant sets O 93 Gradient Systems Now we turn to a particular type of system for which the previous material on Liapunov functions is particularly germane A gradient system on R is a system of differential equations of the form X grad VX where V R Risa C function and aV aV grad V Ox OXn The negative sign in this system is traditional The vector field grad V is called the gradient of V Note that grad VX gradVX Gradient systems have special properties that make their flows rather simple The following equality is fundamental DVxY grad VXY This says that the derivative of V at X evaluated at Y y1Vn R is given by the dot product of the vectors grad VX and Y This follows immediately from the formula n aV DVxY X yj x 95 00 jl Let Xt be a solution of the gradient system with X0 Xo and let V R R be the derivative of V along this solution That is voo4 vixen dt 93 Gradient Systems 203 Proposition The function V is strictly decreasing along nonconstant solu tions of the system X grad VX Moreover VX 0 if and only if X is an equilibrium point Proof By the Chain Rule we have VX DVxX grad VXgrad VX grad VX 0 In particular VX 0 ifand only if grad VX 0 O An immediate consequence of this is the fact that if X is an isolated critical point that is a minimum of V then X is an asymptotically stable equilibrium of the gradient system Indeed the fact that X is isolated guarantees that V 0 ina neighborhood of X not including X To understand a gradient flow geometrically we look at the level surfaces of the function V R R These are the subsets Vc with ce R If XE Vc isa regular point that is grad VX 0 then Vc looks like a surface of dimension n 1 near X To see this assume by renumbering the coordinates that 0V0xX 4 0 Using the Implicit Function Theorem we find a C function g R R such that near X the level set Vc is given by V x 6 Xp1y 8 X15 5 Xn1 c That is near X Vc looks like the graph of the function g In the special case where n 2 Vc is a simple curve through X when X is a regular point If all points in Vc are regular points then we say that c is a regular value for V In the case n 2 if c is a regular value then the level set Vc is a union of simple or nonintersecting curves If X is a nonregular point for V then grad VX 0 so X is a critical point for the function V since all partial derivatives of V vanish at X Now suppose that Y is a vector that is tangent to the level surface Vc at X Then we can find a curve yt in this level set for which y0 Y Since V is constant along y it follows that d DVxY Voyt0 dt 9 We thus have by the preceding observations that grad VXY 0 or in other words grad VX is perpendicular to every tangent vector to the level 204 Chapter 9 Global Nonlinear Techniques set Vc at X That is the vector field grad VX is perpendicular to the level surfaces Vc at all regular points of V We may summarize all of this in the following theorem Theorem Properties of Gradient Systems For the system X grad VX 1 Ifc is a regular value of V then the vector field is perpendicular to the level set V c 2 The critical points of V are the equilibrium points of the system 3 If a critical point is an isolated minimum of V then this point is an asymptotically stable equilibrium point Example Let V R Rbe the function Vx y xx 1 y Then the gradient system X FX grad VX is given by x 2xx 12x 1 y 2y There are three equilibrium points 00 120 and 10 The lineariza tions at these three points yield the following matrices 2 0 1 0O 2 0 DF00 0 DF120 DF10 0 Thus 00 and 10 are sinks while 120 is a saddle Both the x and yaxes are invariant as are the lines x 12 and x 1 Since y 2y on these vertical lines it follows that the stable curve at 120 is the line x12 while the unstable curve at 120 is the interval 01 on the xaxis a The level sets of V and the phase portrait are shown in Figure 910 Note that it appears that all solutions tend to one of the three equilibria This is no accident for we have the following Proposition Let Z be analimit point or an wlimit point of a solution of a gradient flow Then Z is an equilibrium point Proof Suppose Z is an wlimit point As in the proof of the Lasalles Invari ance Principle from Section 92 V is constant along the solution through 93 Gradient Systems 205 eS 5 WINE Figure 910 Level sets and phase portrait for the gradient system determined by Vx y x2x 1 y Z Thus VZ 0 and so Z must be an equilibrium point The case of an alimit point is similar In fact an alimit point Z of X grad VX is an wlimit point of X grad VX so that grad VZ 0 O If a gradient system has only isolated equilibrium points this result implies that every solution of the system must tend either to infinity or to an equi librium point In the preceding example we see that the sets V0c are closed bounded and positively invariant under the gradient flow Therefore each solution entering such a set is defined for all t 0 and tends to one of the three equilibria 00 10 or 120 The solution through every point does enter such a set since the solution through xy enters the set V0 co where Vx y o There is one final property that gradient systems share Note that in the preceding example all of the eigenvalues of the linearizations at the equilib ria have real eigenvalues Again this is no accident for the linearization of a gradient system at an equilibrium point X is a matrix aj where aV aig X OXj0X Since mixed partial derivatives are equal we have av aV aa JOD sO OXj0X OXjOX and so aj 4j It follows that the matrix corresponding to the linearized system is a symmetric matrix It is known that such matrices have only real 206 Chapter 9 Global Nonlinear Techniques eigenvalues For example in the 2 x 2 case a symmetric matrix assumes the form a b b c and the eigenvalues are easily seen to be a Vac 40 4 ac 2 2 both of which are real numbers A more general case is relegated to Exercise 15 at the end of this chapter We therefore have the following proposition Proposition For a gradient system X grad VX the linearized system at any equilibrium point has only real eigenvalues O 94 Hamiltonian Systems In this section we deal with another special type of system a Hamiltonian system As we shall see in Chapter 13 this is the type of system that arises in classical mechanics We shall restrict attention in this section to Hamiltonian systems in R A Hamiltonian system in R is a system of the form OH x y dy oH 37 y Ox where H R Risa C function called the Hamiltonian function Example Undamped Harmonic Oscillator Recall that this system is given by x y y kx where k 0 A Hamiltonian function for this system is 1 k Hxy y x a oy sy 5 94 Hamiltonian Systems 207 Example Ideal Pendulum The equation for this system as we saw in Section 92 is 6 v v sind The total energy function 1 Ev 5 1cosé serves as a Hamiltonian function in this case Note that we say a Hamiltonian function because we can always add a constant to any Hamiltonian function without changing the equations What makes Hamiltonian systems so important is the fact that the Hamilto nian function is a first integral or constant of the motion That is H is constant along every solution of the system or in the language of the previous sections H 0 This follows immediately from it 0H 4 0H xX Ox Oy Y 0H 0H 1 0H oH 0 ax dy ay ax Thus we have the next proposition Proposition For a Hamiltonian system in IR H is constant along every solution curve oO The importance of knowing that a given system is Hamiltonian is the fact that we can essentially draw the phase portrait without solving the system Assuming that H is not constant on any open set we simply plot the level curves Hxy constant The solutions of the system lie on these level sets all we need to do is figure out the directions of the solution curves on these level sets But this is easy since we have the vector field Note also that the equilibrium points for a Hamiltonian system occur at the critical points of H that is at points where both partial derivatives of H vanish Example Consider the system xy y x x 208 Chapter 9 Global Nonlinear Techniques Figure 911 Phase portrait for XyYx 4x A Hamiltonian function is HK 2 ep Oe 4 212 The constant value 14 is irrelevant here we choose it so that H has mini mum value 0 which occurs at 10 as is easily checked The only other equilibrium point lies at the origin The linearized system is 0 1 v2 5 x At 00 this system has eigenvalues 1 so we have a saddle At 10 the eigenvalues are 42i so we have a center at least for the linearized system Plotting the level curves of H and adding the directions at nonequilibrium points yields the phase portrait shown in Figure 911 Note that the equilib rium points at 10 remain centers for the nonlinear system Also note that the stable and unstable curves at the origin match up exactly That is we have solutions that tend to 00 in both forward and backward time Such solutions are known as homoclinic solutions or homoclinic orbits a The fact that the eigenvalues of this system assume the special forms 1 and 2i is again no accident Proposition Suppose xo yo is an equilibrium point for a planar Hamilto nian system Then the eigenvalues of the linearized system are either or id wherex ER O Hirsch Ch099780123820105 2012125 1832 Page 209 23 95 Exploration The Pendulum with Constant Forcing 209 The proof of the proposition is straightforward see Exercise 11 at the end of this chapter 95 Exploration The Pendulum with Constant Forcing Recall from Section 92 that the equations for a nonlinear pendulum are θ v v bv sinθ Here θ gives the angular position of the pendulum which we assume to be measured in the counterclockwise direction and v is its angular velocity The parameter b0 measures the damping Now we apply a constant torque to the pendulum in the counterclockwise direction This amounts to adding a constant to the equation for v so the system becomes θ v v bv sinθ k where we assume that k 0 Since θ is measured mod 2π we may think of this system as being defined on the cylinder S1 R where S1 denotes the unit circle 1 Find all equilibrium points for this system and determine their stability 2 Determine the regions in the bkparameter plane for which there are different numbers of equilibrium points Describe the motion of the pendulum in each different case 3 Suppose k1 Prove that there exists a periodic solution for this system Hint What can you say about the vector field in a strip of the form 0 v1 k sinθb v2 4 Describe the qualitative features of a Poincare map defined on the line θ 0 for this system 5 Prove that when k1 there is a unique periodic solution for this system Hint Recall the energy function Eθy 1 2y2 cosθ 1 and use the fact that the total change of E along any periodic solution must be 0 Hirsch Ch099780123820105 2012125 1832 Page 210 24 210 Chapter 9 Global Nonlinear Techniques 6 Prove that there are parameter values for which a stable equilibrium and a periodic solution coexist 7 Describe the bifurcation that must occur when the periodic solution ceases to exist E X E R C I S E S 1 For each of the following systems sketch the x and y nullclines and use this information to determine the nature of the phase portrait You may assume that these systems are defined only for xy 0 a x xy 2x 2 y yy 1 b x xy 2x 2 y yy x 3 c x x2 y 2x y y3 3y x d x x2 y 2x y y3 y 4x e x x2500 x2 y2 y y70 y x 2 Describe the phase portrait for x x2 1 y xy ax2 1 when a 0 What qualitative features of this flow change as a passes from negative to positive 3 Consider the system of differential equations x xx y 1 y yax y b where a and b are parameters with ab 0 Suppose that this system is only defined for xy 0 a Use the nullclines to sketch the phase portrait for this system for various a and b values b Determine the values of a and b at which a bifurcation occurs c Sketch the regions in the abplane where this system has quali tatively similar phase portraits and describe the bifurcations that occur as the parameters cross the boundaries of these regions Hirsch Ch099780123820105 2012125 1832 Page 211 25 Exercises 211 4 Consider the system x ϵx 2yz 1 y x ϵyz 1 z z3 a Show that the origin is not asymptotically stable when ϵ 0 b Show that when ϵ 0 the basin of attraction of the origin contains the region z 1 5 For the nonlinear damped pendulum show that for every integer n and every angle θ0 there is an initial condition θ0v0 with a solution that corresponds to the pendulum moving around the circle at least n times but not n 1 times before settling down to the rest position 6 Find a strict Liapunov function for the equilibrium point 00 of x 2x y2 y y x2 Find δ 0 as large as possible so that the open disk of radius δ and center 00 is contained in the basin of 00 7 For each of the following functions VX sketch the phase portrait of the gradient flow X gradVX Sketch the level surfaces of V on the same diagram Find all of the equilibrium points and determine their type a x2 2y2 b x2 y2 2x 4y 5 c y sinx d 2x2 2xy 5y2 4x 4y 4 e x2 y2 z f x2x 1 y2y 2 z2 8 Sketch the phase portraits for the following systems Determine if the system is Hamiltonian or gradient along the way Thats a little hint by the way a x x 2y y y b x y2 2xy y x2 2xy c x x2 2xy y y2 2xy d x x2 2xy y y2 x2 e x sin2 x siny y 2sinx cosx cosy 212 Chapter 9 Global Nonlinear Techniques 9 Let X AX bea linear system where a b A a Determine conditions on abc and d that guarantee that this system is a gradient system Give a gradient function explicitly b Repeat the previous question for a Hamiltonian system 10 Consider the planar system x fy y gx Determine explicit conditions on f and g that guarantee that this system is a gradient system or a Hamiltonian system 11 Prove that the linearization at an equilibrium point of a planar Hamilto nian system has eigenvalues that are either EA or iA where 1 R 12 Let T be the torus defined as the square 0 60 27 with opposite sides identified Let F662 cos cos6 Sketch the phase portrait for the system grad F in T Sketch a threedimensional representation of this phase portrait with T represented as the surface of a doughnut 13 Repeat the previous exercise but assume now that F is a Hamiltonian function 14 On the torus T from Exercise 12 let F602 cos 02 cos62 Sketch the phase portrait for the system gradF in T Sketch a three dimensional representation of this phase portrait with T represented as the surface of a doughnut 15 Prove that a 3 x 3 symmetric matrix has only real eigenvalues 16 A solution Xt of a system is called recurrent if Xt X0 for some sequence t oo Prove that a gradient dynamical system has no nonconstant recurrent solutions 17 Show that a closed bounded w limit set is connected Give an example of a planar system having an unbounded w limit set consisting of two parallel lines Hirsch Ch109780123820105 2012127 2341 Page 213 1 10 Closed Orbits and Limit Sets In the previous few chapters we concentrated on equilibrium solutions of systems of differential equations These are undoubtedly among the most important solutions but there are other types of solutions that are impor tant as well In this chapter we will investigate another important type of solution the periodic solution or closed orbit Recall that a periodic solution occurs for X FX if we have a nonequilibrium point X and a time τ 0 for which φτX X It follows that φtτX φtX for all t so φt is a periodic function The least such τ 0 is called the period of the solution As an example all nonzero solutions of the undamped harmonic oscillator equation are periodic solutions Like equilibrium points that are asymptot ically stable periodic solutions may also attract other solutions That is solutions may limit on periodic solutions just as they can approach equilibria In the plane the limiting behavior of solutions is essentially restricted to equilibria and closed orbits although there are a few exceptional cases We will investigate this phenomenon in this chapter in the guise of the important PoincareBendixson Theorem We will see later that in dimensions greater thantwo thelimitingbehaviorofsolutionscanbequiteabitmorecomplicated 101 Limit Sets We begin by describing the limiting behavior of solutions of systems of differential equations Recall that Y Rn is an ωlimit point for the solution Differential Equations Dynamical Systems and an Introduction to Chaos DOI 101016B9780123820105000105 c 2013 Elsevier Inc All rights reserved 213 Hirsch Ch109780123820105 2012127 2341 Page 214 2 214 Chapter 10 Closed Orbits and Limit Sets through X if there is a sequence tn such that limn φtnX Y That is the solution curve through X accumulates on the point Y as time moves forward The set of all ωlimit points of the solution through X is the ωlimit set of X and is denoted by ωX The αlimit points and the αlimit set αX are defined by replacing tn with tn in the above definition By a limit set we mean a set of the form ωX or αX Here are some examples of limit sets If X is an asymptotically stable equi librium it is the ωlimit set of every point in its basin of attraction Any equilibrium is its own α and ωlimit set A periodic solution is the αlimit and ωlimit set of every point on it Such a solution may also be the ωlimit set of many other points Example Consider the planar system given in polar coordinates by r 1 2r r3 θ 1 As we saw in Chapter 8 Section 81 all nonzero solutions of this equation tend to the periodic solution that resides on the unit circle in the plane See Figure 101 Consequently the ωlimit set of any nonzero point is this closed orbit Example Consider the system x sinx01cosx cosy y sinycosx 01cosy Figure 101 The phase plane for r 1 2r r3 θ 1 Hirsch Ch109780123820105 2012127 2341 Page 215 3 101 Limit Sets 215 00 ππ Figure 102 The ωlimit set of any solution emanating from the source at π2π2 is the square bounded by the four equilibria and the heteroclinic solutions There are equilibria that are saddles at the corners of the square 00 0π ππ and π0 as well as at many other points There are heteroclinic solu tions connecting these equilibria in the order listed See Figure 102 There is also a spiral source at π2π2 All solutions emanating from this source accumulate on the four heteroclinic solutions connecting the equilibria see exercise 4 at the end of this chapter Thus the ωlimit set of any point on these solutions is the square bounded by x 0π and y 0π In three dimensions there are extremely complicated examples of limit sets which are not very easy to describe In the plane however limit sets are fairly simple In fact Figure 102 is typical in that one can show that a closed and bounded limit set other than a closed orbit or equilibrium point is made up of equilibria and solutions joining them The PoincareBendixson Theorem discussed in Section 105 states that if a closed and bounded limit set in the plane contains no equilibria then it must be a closed orbit Recall from Chapter 9 Section 92 that a limit set is closed in Rn and is invariant under the flow We will also need the following result Proposition 1 If X and Z lie on the same solution then ωX ωZ and αX αZ 2 If D is a closed positively invariant set and Z D then ωZ D and similarly for negatively invariant sets and αlimits 3 A closed invariant set and in particular a limit set contains the αlimit and ωlimit sets of every point in it Hirsch Ch109780123820105 2012127 2341 Page 216 4 216 Chapter 10 Closed Orbits and Limit Sets Proof For 1 suppose that Y ωX and φsX Z If φtnX Y then we have φtnsZ φtnX Y Thus Y ωZ as well For 2 if φtnZ Y ωZ as tn then we have tn 0 for sufficiently large n so that φtnZ D Therefore Y D since D is a closed set Finally 3 follows immediately from 2 102 Local Sections and Flow Boxes For the rest of this chapter we restrict the discussion to planar systems In this section we describe the local behavior of the flow associated with X FX near a given point X0 that is not an equilibrium point Our goal is to construct first a local section at X0 and then a flow box neighborhood of X0 In this flow box solutions of the system behave particularly simply Suppose FX0 0 The transverse line at X0 denoted by ℓX0 is the straight line through X0 that is perpendicular to the vector FX0 based at X0 We parametrize ℓX0 as follows Let V0 be a unit vector based at X0 and perpendicular to FX0 Then define h R ℓX0 by hu X0 uV0 Since FX is continuous the vector field is not tangent to ℓX0 at least in some open interval in ℓX0 surrounding X0 We call such an open subinterval containing X0 a local section at X0 At each point of a local section S the vector field points away from S so solutions must cut across a local section In particular FX 0 for X S See Figure 103 Our first use of a local section at X0 will be to construct an associated flow box in a neighborhood of X0 A flow box gives a complete description of the X0 X0 φtX0 S Figure 103 A local section S at X0 and several representative vectors from the vector field along S 102 Local Sections and Flow Boxes 217 N S YN Figure 104 Flow box associated with S behavior of the flow in a neighborhood of a nonequilibrium point by means of a special set of coordinates An intuitive description of the flow in a flow box is simple Points move in parallel straight lines at constant speed Given a local section S at Xo we may construct a map W from a neigh borhood AN of the origin in R to a neighborhood of Xo as follows Given s u R we define Wsu pshu where h is the parametrization of the transverse line described above Note that Y maps the vertical line 0 uv in N to the local section S WY also maps horizontal lines in VV to pieces of solution curves of the system Provided that we choose NV sufficiently small the map W is then one to one on NV Also note that DW takes the constant vector field 10 in NV to vector field FX Using the language of Chapter 4 W is a local conjugacy between the flow of this constant vector field and the flow of the nonlinear system We usually take N in the form s us o where o 0 In this case we sometimes write V YN and call V the flow box at or about Xo See Figure 104 An important property of a flow box is that if X V then oX S for a unique t o If S is a local section the solution through a point Zp perhaps far from S may reach Xp S at a certain time f see Figure 105 We show that in a certain local sense this time of first arrival at S is a continuous function of Zo The following proposition shows this more precisely Proposition Let S be a local section at Xo and suppose 4Zo Xo Let W be a neighborhood of Z Then there is an open set UC W containing Zo and a continuous function t U R such that tZo to and rx X S for eachX EU Hirsch Ch109780123820105 2012127 2341 Page 218 6 218 Chapter 10 Closed Orbits and Limit Sets X Z0 X0 φt0Z0 φτXX Figure 105 Solutions crossing the local section S Proof Suppose FX0 is the vector αβ and recall that αβ 00 For Y y1y2 R2 define η R2 R by ηY Y FX0 αy1 βy2 Recall that Y belongs to the transverse line ℓX0 if and only if Y X0 V where V FX0 0 Thus Y ℓX0 if and only if ηY Y FX0 X0FX0 Now define G R2 R R by GXt ηφtX φtXFX0 We have GZ0t0 X0FX0 since φt0Z0 X0 Furthermore G t Z0t0 FX02 0 We may thus apply the Implicit Function Theorem to find a smooth function τ R2 R defined on a neighborhood U1 of Z0t0 such that τZ0 t0 and GXτX GZ0t0 X0FX0 Thus φτXX belongs to the transverse line ℓX0 If U U1 is a sufficiently small neighborhood of Z0 then φτXX S as required 103 The Poincare Map As in the case of equilibrium points closed orbits may also be stable asymp totically stable or unstable The definitions of these concepts for closed orbits Hirsch Ch109780123820105 2012127 2341 Page 219 7 103 The Poincare Map 219 are entirely analogous to those for equilibria as in Chapter 8 Section 84 How ever determining the stability of closed orbits is much more difficult than the corresponding problem for equilibria Although we do have a tool that resem bles the linearization technique that is used to determine the stability of most equilibria generally this tool is much more difficult to use in practice Here is the tool Given a closed orbit γ there is an associated Poincare map for γ some examples of which we previously encountered in Chapter 1 Section 14 and Chapter 6 Section 62 Near a closed orbit this map is defined as follows Choose X0 γ and let S be a local section at X0 We consider the first return map on S This is the function P that associates to X S the point PX φtX S where t is the smallest positive time for which φtX S Now P may not be defined at all points on S as the solutions through certain points in S may never return to S But we certainly have PX0 X0 and an application of the Implicit Function Theorem as in the previous proposition guarantees that P is defined and continuously differentiable in a neighborhood of X0 In the case of planar systems a local section is a subset of a straight line through X0 so we may regard this local section as a subset of R and take X0 0 R Thus the Poincare map is a real function taking 0 to 0 If P0 1 it follows that P assumes the form Px ax higherorder terms where a 1 Thus for x near 0 Px is closer to 0 than x This means that the solution through the corresponding point in S moves closer to γ after one passage through the local section Continuing we see that each passage through S brings the solution closer to γ and so we see that γ is asymptotically stable We have the following Proposition Let X FX be a planar system and suppose that X0 lies on a closed orbit γ Let P be a Poincare map defined on a neighborhood of X0 in some local section If PX0 1 then γ is asymptotically stable Example Consider the planar system given in polar coordinates by r r1 r θ 1 Clearly there is a closed orbit lying on the unit circle r 1 This solution is given by costsint when the initial condition is 10 Also there is a local section lying along the positive real axis since θ 1 Furthermore given any x 0 we have φ2πx0 which also lies on the positive real axis R Thus we have a Poincare map P R R Moreover P1 1 since the point x 1 y 0 is the initial condition giving the periodic solution To check the stability of this solution we need to compute P1 220 Chapter 10 Closed Orbits and Limit Sets To do this we compute the solution starting at x0 We have 0t t so we need to find r27 To compute rf we separate variables to find dr t tant t constant rlr Evaluating this integral yields 2 xe rt 1xxel Thus Px rm x r27 1xxe Differentiating we find P1 1e so that 0 P1 1 Thus the peri odic solution is asymptotically stable a The astute reader may have noticed a little scam here To determine the Poincaré map we actually first found formulas for all of the solutions starting at x0 So why on earth would we need to compute a Poincaré map Well good question Actually it is usually very difficult to compute the exact form of a Poincaré map or even its derivative along a closed orbit since in practice we rarely have a closed form expression for the closed orbit never mind the nearby solutions As we shall see the Poincaré map is usually more useful when setting up a geometric model of a specific system see the Lorenz system in Chapter 14 There are some cases where we can circumvent this problem and gain insight into the Poincaré map as we shall see when we investigate the van der Pol equation in Chapter 12 Section 123 104 Monotone Sequences in Planar Dynamical Systems Let Xo X R be a finite or infinite sequence of distinct points on the solution curve through Xp We say that the sequence is monotone along the solution if Xo Xy withO t h Let Yo Yj be a finite or infinite sequence of points on a line segment I in R2 We say that this sequence is monotone along I if Y is between Y1 and Y1 in the natural order along I for all n 1 A sequence of points may be on the intersection of a solution curve and a segment I they may be monotone along the solution curve but not along 104 Monotone Sequences 221 Xo Xo X x Figure 106 Two solutions crossing a straight line On the left Xo X1X2 is monotone along the solution but not along the straight line On the right Xo X71 X2 is monotone along both the solution and the line the segment or vice versa see Figure 106 However this is impossible if the segment is a local section in the plane Proposition Let S be a local section for a planar system of differential equa tions and let Yo Y Y2 be a sequence of distinct points in S that lie on the same solution curve If this sequence is monotone along the solution then it is also monotone along S Proof It suffices to consider three points Yo Y and Y2 in S Let be the simple closed curve made up of the part of the solution between Yo and Y and the segment T C S between Yo and Y Let D be the region bounded by X We suppose that the solution through Yj leaves D at Y see Figure 107 if the solution enters D the argument is similar Thus the solution leaves D at every point in T since T is part of the local section It follows that the complement of D is positively invariant because no solu tion can enter D at a point of T nor can it cross the solution connecting Yo and Yj by uniqueness of solutions Therefore Y R D for all t 0 In particular Y S T The set S T is the union of two halfopen intervals Ip and I with Y an endpoint of J for j 01 One can draw an arc from a point Y1 with 0 very small to a point of I without crossing Therefore I is outside D Similarly Ip is inside D It follows that Y2 I since it must be outside D This shows that Y is between Yo and Y in I proving the proposition L 222 Chapter 10 Closed Orbits and Limit Sets YY Figure 107 Solutions exit the region D through T We now come to an important property of limit points Proposition For a planar system suppose that Y wX Then the solution through Y crosses any local section at no more than one point The same is true if Y aX Proof Suppose that Y and Y are distinct points on the solution through Y and that S is a local section containing Y and Y2 Suppose Y wX the argument for aX is similar Then Y wX for k 12 Let Vy be flow boxes at Y defined by some intervals J C S we assume that J and Jp are dis joint as shown in Figure 108 The solution through X enters each VY infinitely often thus it crosses J infinitely often Therefore there is a sequence a by a2 ba a3 b3 see that is monotone along the solution through X with ay Ji by Jy for n 12 But such a sequence cannot be monotone along S since J and Jz are disjoint contradicting the previous proposition O 105 The PoincareBendixson Theorem In this section we prove a celebrated result concerning planar systems Theorem PoincaréBendixson Suppose that Q is a nonempty closed and bounded limit set of a planar system of differential equations that contains no equilibrium point Then Q is a closed orbit Hirsch Ch109780123820105 2012127 2341 Page 223 11 105 The PoincareBendixson Theorem 223 X Y1 Y2 1 2 Figure 108 The solution through X cannot cross V1 and V2 infinitely often Proof Suppose that ωX is closed and bounded and that Y ωX The case of αlimit sets is similar We show first that Y lies on a closed orbit and later that this closed orbit actually is ωX Since Y belongs to ωX we know from Section 101 that ωY is a nonempty subset of ωX Let Z ωY and let S be a local section at Z Let V be a flow box associated with S By the results of the previous section the solution through Y meets S at exactly one point On the other hand there is a sequence tn such that φtnY Z thus infinitely many φtnY belong to V We can therefore find rs R such that r s and φrYφsY S It follows that φrY φsY thus φrsY Y and r s 0 Since ωX contains no equilibria Y must lie on a closed orbit It remains to prove that if γ is a closed orbit in ωX then γ ωX For this it is enough to show that lim tdφtXγ 0 where dφtxγ is the distance from φtX to the set γ that is the distance from φtX to the nearest point of γ Let S be a local section at Y γ Let ϵ 0 and consider a flow box Vϵ associated with S Then there is a sequence t0 t1 such that 1 φtnX S 2 φtnX Y 3 φtX S for tn1 t tn n 12 Let Xn φtnX By the first proposition in the previous section Xn is a monotone sequence in S that converges to Y 224 Chapter 10 Closed Orbits and Limit Sets We claim that there exists an upper bound for the set of positive num bers ty41 ty To see this suppose Y Y where t 0 Then for Xy sufficiently near Y X Ve and thus br4tXn S for some t e Therefore tht1ty tte This provides the upper bound for ty41 th Let 6 0 be small By continuity of solutions with respect to initial con ditions there exists 6 0 such that if ZY 6 and tte then bZ Y B That is the distance from the solution Z to y is less than f for all t satisfying t t Let mo be so large that X Y 6 for all n np Then lPeXn Y B if t 7 e and n no Nowlet t t Let n no be such that ty StS thy Then AprX OrX br1 1 lott Xn 11 Y B since t t t This shows that the distance from X to y is less than 6 for all sufficiently large t This completes the proof of the Poincaré Bendixson Theorem Example Another example of an wlimit set that is neither a closed orbit nor an equilibrium is provided by a homoclinic solution Consider the system 4 2 2 rey SX FX UY V3 xy 5 5 5 x x 4 2 2 po3y XS XY Yxx A 5 5 Jr A computation shows that there are three equilibria at 00 10 and 10 The origin is a saddle while the other two equilibria are sources The Hirsch Ch109780123820105 2012127 2341 Page 225 13 106 Applications of PoincareBendixson 225 Figure 109 A pair of homoclinic solutions in the ωlimit set phase portrait of this system is shown in Figure 109 Note that solutions far from the origin tend to accumulate on the origin and a pair of homo clinic solutions each of which leaves and then returns to the origin Solutions emanating from either source have an ωlimit set that consists of just one homoclinic solution and 00 See Exercise 6 at the end of this chapter for proofs of these facts 106 Applications of PoincareBendixson The PoincareBendixson Theorem essentially determines all of the possible limiting behaviors of a planar flow We give a number of corollaries of this important theorem in this section A limit cycle is a closed orbit γ such that γ ωX or γ αX for some X γ In the first case γ is called an ωlimit cycle in the second case an αlimit cycle We deal only with ωlimit sets in this section the case of αlimit sets is handled by simply reversing time In the proof of the PoincareBendixson Theorem it was shown that limit cycles have the following property If γ is an ωlimit cycle there exists X γ such that lim tdφtXγ 0 Geometrically this means that some solution spirals toward γ as t See Figure 1010 Not all closed orbits have this property For example in the Hirsch Ch109780123820105 2012127 2341 Page 226 14 226 Chapter 10 Closed Orbits and Limit Sets γ Figure 1010 A solution spiraling toward a limit cycle case of a linear system with a center at the origin in R2 the closed orbits that surround the origin have no solutions approaching them and so are not limit cycles Limit cycles possess a kind of onesided at least stability Let γ be an ωlimit cycle and suppose φtX spirals toward γ as t Let S be a local section at Z γ Then there is an interval T S disjoint from γ bounded by φt0X and φt1X with t0 t1 and not meeting the solution through X for t0 t t1 See Figure 1011 The annular region A that is bounded on one side by γ and on the other side by the union of T and the curve φtXt0 t t1 is positively invariant as is the set B A γ It is easy to see that φtY spirals toward γ for all Y B Thus we have the following corollary Corollary 1 Let γ be an ωlimit cycle If γ ωX where X γ then X has a neighborhood O such that γ ωY for all Y O In other words the set Y ωY γ γ is open As another consequence of the PoincareBendixson Theorem suppose that K is a positively invariant set that is closed and bounded If X K then ωX must also lie in K Thus K must contain either an equilibrium point or a limit cycle Hirsch Ch109780123820105 2012127 2341 Page 227 15 106 Applications of PoincareBendixson 227 A T Z γ Figure 1011 The region A is positively invariant Corollary 2 A closed and bounded set K that is positively or negatively invariant contains either a limit cycle or an equilibrium point The next result exploits the spiraling property of limit cycles Corollary 3 Let γ be a closed orbit and let U be the open region in the interior of γ Then U contains either an equilibrium point or a limit cycle Proof Let D be the closed and bounded set U γ Then D is invariant since no solution in U can cross γ If U contains no limit cycle and no equilibrium then for any X U ωX αX γ by PoincareBendixson If S is a local section at a point Z γ there are sequences tn sn such that φtnXφsnX S and both φtnX and φsnX tend to Z as n But this leads to a contradiction of the proposition in Section 104 on monotone sequences Actually this last result can be considerably sharpened as follows Corollary 4 Let γ be a closed orbit that forms the boundary of an open set U Then U contains an equilibrium point Proof Suppose U contains no equilibrium point Consider first the case that there are only finitely many closed orbits in U We may choose the closed orbit that bounds the region with smallest area There are then no closed orbits or equilibrium points inside this region and this contradicts Corollary 3 Now suppose that there are infinitely many closed orbits in U If Xn X in U and each Xn lies on a closed orbit then X must lie on a closed orbit Otherwise the solution through X would spiral toward a limit cycle since there Hirsch Ch109780123820105 2012127 2341 Page 228 16 228 Chapter 10 Closed Orbits and Limit Sets are no equilibria in U By Corollary 1 so would the solution through some nearby Xn which is impossible Let ν 0 be the greatest lower bound of the areas of regions enclosed by closed orbits in U Let γn be a sequence of closed orbits enclosing regions of areas νn such that limn νn ν Let Xn γn Since γ U is closed and bounded we may assume that Xn X U Then if U contains no equilib rium X lies on a closed orbit β bounding a region of area ν The usual section argument shows that as n γn gets arbitrarily close to β and thus the area νn ν of the region between γn and β goes to 0 Then the previous argument provides a contradiction to Corollary 3 The following result uses the spiraling properties of limit cycles in a subtle way Corollary 5 Let H be a first integral of a planar system If H is not constant on any open set then there are no limit cycles Proof Suppose there is a limit cycle γ let c R be the constant value of H on γ If Xt is a solution that spirals toward γ then HXt c by continuity of H In Corollary 1 we found an open set with solutions that spiral toward γ thus H is constant on an open set Finally the following result is implicit in our development of the theory of Liapunov functions in Chapter 9 Section 92 Corollary 6 If L is a strict Liapunov function for a planar system then there are no limit cycles 107 Exploration Chemical Reactions that Oscillate For much of the twentieth century chemists believed that all chemical reac tions tended monotonically to equilibrium This belief was shattered in the 1950s when the Russian biochemist Belousov discovered that a certain reac tion involving citric acid bromate ions and sulfuric acid when combined with a cerium catalyst could oscillate for long periods of time before settling to equilibrium The concoction would turn yellow for a while then fade then turn yellow again then fade and on and on like this for over an hour This reaction now called the BelousovZhabotinsky reaction the BZ reaction for short was a major turning point in the history of chemical reactions Now Exercises 229 many systems are known to oscillate Some have even been shown to behave chaotically One particularly simple chemical reaction is given by a chlorine dioxide iodinemalonic acid interaction The exact differential equations modeling this reaction are extremely complicated However there is a planar nonlinear system that closely approximates the concentrations of two of the reactants The system is Axy x ax 1x y bx 1 I 3 where x and y represent the concentrations of I and ClO respectively and aand bare positive parameters 1 Begin the exploration by investigating these reaction equations numeri cally What qualitatively different types of phase portraits do you find 2 Find all equilibrium points for this system 3 Linearize the system at your equilibria and determine the type of each equilibrium 4 In the abplane sketch the regions where you find asymptotically stable or unstable equilibria 5 Identify the abvalues where the system undergoes bifurcations 6 Using the nullclines for the system together with the PoincaréBendixson Theorem find the abvalues for which a stable limit cycle exists Why do these values correspond to oscillating chemical reactions For more details on this reaction see Lengyel et al 27 The very interesting history of the BZ reaction is described in Winfree 47 The original paper by Belousov is reprinted in Field and Burger 17 EXERCISES 1 For each of the following systems identify all points that lie in either an w or an alimit set a fPr7r01 b f P 37r2r 6 1 c r sinr 0 1 d x sinxsiny y cosx cosy 230 Chapter 10 Closed Orbits and Limit Sets 2 Consider the threedimensional system rf r17r g1 Zz Compute the Poincaré map along the closed orbit lying on the unit circle given by r 1 and show that this closed orbit is asymptotically stable 3 Consider the threedimensional system rf r17r g1 Z z Again compute the Poincaré map for this system What can you now say about the behavior of solutions near the closed orbit Sketch the phase portrait for this system 4 Consider the system x sin x01 cos x cos y y sin ycosx 01 cosy Show that all solutions emanating from the source at 72 72 have wlimit sets equal to the square bounded by x 07 and y 07 5 The system r ar r r 1 depends on a parameter a Determine the phase plane for representative a values and describe all bifurcations for the system 6 Consider the system 4 2 2 roy SF LY V3 xy 1 5 5 x x 4 2 2 po3y SY Yuxx Z 5 5 y a Find all equilibrium points b Determine the types of these equilibria Hirsch Ch109780123820105 2012127 2341 Page 231 19 Exercises 231 A Figure 1012 The region A is positively invariant c Prove that all nonequilibrium solutions have ωlimit sets consisting of either one or two homoclinic solutions plus a saddle point 7 Let A be an annular region in R2 Let F be a planar vector field that points inward along the two boundary curves of A Suppose also that every radial segment of A is local section See Figure 1012 Prove there is a periodic solution in A 8 Let F be a planar vector field and again consider an annular region A as in the previous problem Suppose that F has no equilibria and that F points inward along the boundary of the annulus as before a Prove there is a closed orbit in A Notice that the hypothesis is weaker than in the previous problem b If there are exactly seven closed orbits in A show that one of them has orbits spiraling toward it from both sides 9 Let F be a planar vector field on a neighborhood of the annular region A above Suppose that for every boundary point X of A FX is a nonzero vector tangent to the boundary a Sketch the possible phase portraits in A under the further assump tion that there are no equilibria and no closed orbits besides the boundary circles Include the case where the solutions on the boundary travel in opposite directions b Suppose the boundary solutions are oppositely oriented and that the flow preserves area Show that A contains an equilibrium 10 Show that a closed orbit of a planar system meets a local section in at most one point 11 Show that a closed and bounded limit set is connected that is not the union of two disjoint nonempty closed sets Hirsch Ch109780123820105 2012127 2341 Page 232 20 232 Chapter 10 Closed Orbits and Limit Sets 12 Let X FX be a planar system with no equilibrium points Suppose the flow φt generated by F preserves area that is if U is any open set the area of φtU is independent of t Show that every solution is a closed set 13 Let γ be a closed orbit of a planar system Let λ be the period of γ Let γn be a sequence of closed orbits Suppose the period of γn is λn If there are points Xn γn such that Xn X γ prove that λn λ This result can be false for higher dimensional systems It is true however that if λn µ then µ is an integer multiple of λ 14 Consider a system in R2 having only a finite number of equilibria a Show that every limit set is either a closed orbit or the union of equi librium points and solutions φtX such that limt φtX and limt φtX are these equilibria b Show by example draw a picture that the number of distinct solutions in ωX may be infinite 15 Let X be a recurrent point of a planar system that is there is a sequence tn such that φtnX X a Prove that either X is an equilibrium or X lies on a closed orbit b Show by example that there can be a recurrent point for a nonplanar system that is not an equilibrium and does not lie on a closed orbit 16 Let X FX and X GX be planar systems Suppose that FXGX 0 for all X R2 If F has a closed orbit prove that G has an equilibrium point 17 Let γ be a closed orbit for a planar system and suppose that γ forms the boundary of an open set U Show that γ is not simultaneously the ω and αlimit set of points of U Use this fact and the PoincareBendixson Theorem to prove that U contains an equilibrium that is not a saddle Hint Consider the limit sets of points on the stable and unstable curves of saddles Hirsch Ch119780123820105 201222 1240 Page 233 1 11 Applications in Biology In this chapter we make use of the techniques developed in the previous few chapters to examine some nonlinear systems that have been used as mathe matical models for a variety of biological systems In Section 111 we utilize preceding results involving nullclines and linearization to describe several bio logical models involving the spread of communicable diseases In Section 112 we investigate the simplest types of equations that model a predatorprey ecology A more sophisticated approach is used in Section 113 to study the populations of a pair of competing species Instead of developing explicit formulas for these differential equations we instead make only qualitative assumptions about the form of the equations We then derive geometric information about the behavior of solutions of such systems based on these assumptions 111 Infectious Diseases The spread of infectious diseases such as measles or malaria may be modeled as a nonlinear system of differential equations The simplest model of this type is the SIR model Here we divide a given population into three disjoint groups The population of susceptible individuals is denoted by S the infected population by I and the recovered population by R As usual each of these Differential Equations Dynamical Systems and an Introduction to Chaos DOI 101016B9780123820105000117 c 2013 Elsevier Inc All rights reserved 233 234 Chapter 11 Applications in Biology are functions of time We assume for simplicity that the total population is constant so that S I R 0 In the most basic case we make the assumption that once an individual has been infected and subsequently has recovered that individual cannot be reinfected This is the situation that occurs for such diseases as measles mumps and smallpox among many others We also assume that the rate of transmission of the disease is proportional to the number of encounters between susceptible and infected individuals The easiest way to character ize this assumption mathematically is to put S BSI for some constant B 0 We finally assume that the rate at which infected individuals recover is proportional to the number of infected The SIR model is then SBSI I BSIvI RviI where f and v are positive parameters As stipulated we have S I R 0 so that I Risa constant This simplifies the system for if we determine St and It we then derive Rt for free Thus it suffices to consider the twodimensional system SBSI I BSIvI The equilibria for this system are given by the Saxis J 0 Linearization at S0 yields the matrix 0 S 0 BSvpP so the eigenvalues are 0 and 6S v This second eigenvalue is negative if 0 S vB and positive if S vB The Snullclines are given by the S and Iaxes On the Iaxis we have I vI so solutions simply tend to the origin along this line The Jnullclines are I 0 and the vertical line S vB Thus we have the nullcline diagram as shown in Figure 111 From this it appears that given any initial population So 1p with Sp vB and Ip 0 the susceptible population decreases mono tonically while the infected population at first rises but eventually reaches a maximum and then declines to 0 We can actually prove this analytically for we can explicitly compute a func tion that is constant along solution curves Note that the slope of the vector Hirsch Ch119780123820105 201222 1240 Page 235 3 111 Infectious Diseases 235 S I S νβ Figure 111 Nullclines and direction field for the SIR model field is a function of S alone I S βSI νI βSI 1 ν βS Thus we have dI dS dIdt dSdt 1 ν βS which we may immediately integrate to find I IS S ν β logS constant Therefore the function I S νβlogS is constant along solution curves It then follows that there is a unique solution curve connecting each equi librium point in the interval νβ S to an equilibrium point in the interval 0 S νβ as shown in Figure 112 A slightly more complicated model for infectious diseases arises when we assume that recovered individuals may lose their immunity and become rein fected with the disease Examples of this type of disease include malaria and tuberculosis We assume that reinfection occurs at a rate proportional to the population of recovered individuals This leads to the SIRS model the extra S indicating that recovered individuals may reenter the susceptible group The system becomes S βSI µR I βSI νI R νI µR 236 Chapter 11 Applications in Biology A Ss Figure 112 Phase portrait for the SIR system Again we see that the total population S I Ris a constant which we denote by t We may eliminate R from this system by setting R t SI S BSI ut SD I BSIvI Here 8 4 v and 7 are all positive parameters Unlike the SIR model we now have at most two equilibria one at t0 and the other at yp Lt 3 SI 7 Bo vu The first equilibrium point corresponds to no disease whatsoever in the pop ulation The second equilibrium point only exists when t v8 When t vB we have a bifurcation as the two equilibria coalesce at t0 The quantity v is called the threshold level for the disease The linearized system is given by pl BSu y Y pI BSv At the equilibrium point 70 the eigenvalues are z and Bt v so this equilibrium point is a saddle provided that the total population exceeds the threshold level At the second equilibrium point a straightforward computa tion shows that the trace of the matrix is negative while the determinant is positive It then follows from the results in Chapter 4 that both eigenvalues have negative real parts and so this equilibrium point is asymptotically stable 112 PredatorPrey Systems 237 Biologically this means that the disease may become established in the com munity only when the total population exceeds the threshold level We will only consider this case in what follows Note that the SIRS system is only of interest in the region given by IS 0 and I t Denote this triangular region by A of course Note that the Iaxis is no longer invariant while on the Saxis solutions increase up to the equilibrium at 70 Proposition The region A is positively invariant Proof We check the direction of the vector field along the boundary of A The field is tangent to the boundary along the lower edge I 0 as well as at 07 Along S 0 we have S xt I 0 so the vector field points inward for 0 I t Along the hypotenuse if 0 S vB we have S BSI 0 and II1BSv 0 so the vector field points inward When vB S tT we have pen 4 o l14 0 S BS so again the vector field points inward This completes the proof O The Jnullclines are given as in the SIR model by I 0 and vf The Snullcline is given by the graph of the function tTS T BS A calculus student will compute that IS 0 and IS 0 when0 S T So this nullcline is the graph of a decreasing and concave up function that passes through both 70 and 07 as shown in Figure 113 Note that in this phase portrait all solutions appear to tend to the equilibrium point S I the proportion of infected to susceptible individuals tends to a steady state To prove this however one would need to eliminate the possibility of closed orbits encircling the equilibrium point for a given set of parameters jx v and tT 112 PredatorPrey Systems We next consider a pair of species one of which consists of predators with a population that is denoted by y and the other its prey with population x We assume that the prey population is the total food supply for the predators We also assume that in the absence of predators the prey population grows 238 Chapter 11 Applications in Biology SA Ss Figure 113 Nullclines and phase portrait in A for the SIRS system Here 6 v 1 and T2 at a rate proportional to the current population That is as in Chapter 1 when y 0 we have x ax where a 0 So in this case xt xp expat When predators are present we assume that the prey population decreases at a rate proportional to the number of predatorprey encounters As in the pre vious section one simple model for this is bxy where b 0 So the differential equation for the prey population is x ax bxy For the predator population we make more or less the opposite assump tions In the absence of prey the predator population declines at a rate proportional to the current population So when x 0 we have y cy with c 0 and thus yt yo expct The predator species becomes extinct in this case When there are prey in the environment we assume that the predator population increases at a rate proportional to the predatorprey meetings or dxy We do not at this stage assume anything about overcrow ing Thus our simplified predatorprey system also called the VolterraLotka system is x ax bxy xa by y cy dxy yc dx where the parameters a bc and d are all assumed to be positive Since we are dealing with populations we only consider x y 0 As usual our first job is to locate the equilibrium points These occur at the origin and at xy cdab The linearized system is aby bx X X dy c x Hirsch Ch119780123820105 201222 1240 Page 239 7 112 PredatorPrey Systems 239 x y y ab x cd Figure 114 Nullclines and direction field for the predatorprey system so when x y 0 we have a saddle with eigenvalues a and c We know the stable and unstable curves They are the y and xaxes respectively At the other equilibrium point cdab the eigenvalues are pure imaginary iac and so we cannot conclude anything at this stage about the stability of this equilibrium point We next sketch the nullclines for this system The xnullclines are given by the straight lines x 0 and y ab while the ynullclines are y 0 and x cd The nonzero nullcline lines separate the region xy 0 into four basic regions in which the vector field points are as indicated in Figure 114 Thus the solutions wind in the counterclockwise direction about the equilibrium point From this we cannot determine the precise behavior of solutions They could possibly spiral in toward the equilibrium point spiral toward a limit cycle spiral out toward infinity and the coordinate axes or else lie on closed orbits To make this determination we search for a Liapunov function L Employing the trick of separation of variables we look for a function of the form Lxy Fx Gy Recall that L denotes the time derivative of L along solutions We com pute Lxy d dt Lxtyt dF dx x dG dy y Thus Lxy x dF dx a by y dG dy c dx Hirsch Ch119780123820105 201222 1240 Page 240 8 240 Chapter 11 Applications in Biology We obtain L 0 provided x dFdx dx c y dGdy by a Since x and y are independent variables this is possible if and only if x dFdx dx c y dGdy by a constant Setting the constant equal to 1 we obtain dF dx d c x dG dy b a y Integrating we find Fx dx c logx Gy by alogy Thus the function Lxy dx c logx by alogy is constant on solution curves of the system when xy 0 By considering the signs of Lx and Ly it is easy to see that the equilibrium point Z cdab is an absolute minimum for L It follows that L or more precisely L LZ is a Liapunov function for the system Therefore Z is a stable equilibrium We note next that there are no limit cycles this follows from corollary 5 in Chapter 10 Section 106 because L is not constant on any open set We now prove the following Theorem Every solution of the predatorprey system is a closed orbit except the equilibrium point Z and the coordinate axes Proof Consider the solution through W Z where W does not lie on the x or yaxis This solution spirals around Z crossing each nullcline infinitely often Thus there is a doubly infinite sequence t1 t0 t1 such that φtnW is on the line x cd and tn as n If W is not on a closed orbit the points φtnW are monotone along the line x cd as discussed in the previous chapter Since there are no limit cycles either Hirsch Ch119780123820105 201222 1240 Page 241 9 112 PredatorPrey Systems 241 x cd Figure 115 Nullclines and phase portrait for the VolterraLotka system φtnW Z as n or φtnW Z as n Since L is constant along the solution through W this implies that LW LZ But this contradicts minimality of LZ This completes the proof The phase portrait for this predatorprey system is shown in Figure 115 We conclude that for any given initial populations x0y0 with x0 0 and y0 0 other than Z the populations of predator and prey oscillate cyclically No matter what the populations of prey and predator are neither species will die out nor will its population grow indefinitely Now let us introduce overcrowding into the prey equation As in the logistic model in Chapter 1 the equations for prey in the absence of predators may be written in the form x ax λx2 We also assume that the predator population obeys a similar equation y cy µy2 when x 0 Incorporating the preceding assumptions yields the predatorprey equations for species with limited growth x xa by λx y yc dx µy As before the parameters abcd as well as λ and µ are all positive When y 0 we have the logistic equation x xa λx which yields equilibria at Hirsch Ch119780123820105 201222 1240 Page 242 10 242 Chapter 11 Applications in Biology the origin and at aλ0 As we saw in Chapter 1 all nonzero solutions on the xaxis tend to aλ When x 0 the equation for y is y cy µy2 Since y 0 when y 0 it follows that all solutions on this axis tend to the origin Thus we confine attention to the upperright quadrant Q where xy 0 The nullclines are given by the x and yaxes together with the lines L a by λx 0 M c dx µy 0 Along the lines L and M we have x 0 and y 0 respectively There are two possibilities according to whether these lines intersect in Q or not We first consider the case where the two lines do not meet in Q In this case we have the nullcline configuration shown in Figure 116 All solutions to the right of M head upward and to the left until they meet M between the lines L and M solutions now head downward and to the left Thus they either meet L or tend directly to the equilibrium point at aλ0 If solutions cross L they then head right and downward but they cannot cross L again Thus they too tend to aλ0 All solutions in Q therefore tend to this equilibrium point We conclude that in this case the predator population becomes extinct and the prey population approaches its limiting value aλ We may interpret the behavior of solutions near the nullclines as follows Since both x and y are never both positive it is impossible for both prey and predators to increase at the same time If the prey population is above its limiting value it must decrease After a while the lack of prey causes the M L Figure 116 Nullclines and phase portrait for a predatorprey system with limited growth when the nullclines do not meet in Q 112 PredatorPrey Systems 243 predator population to begin to decrease when the solution crosses M After that point the prey population can never increase past aA and so the predator population continues to decrease If the solution crosses L the prey popula tion increases again but not past aA while the predators continue to die off In the limit the predators disappear and the prey population stabilizes at ai Suppose now that L and M cross at a point Z x9 yo in the quadrant Q of course Z is an equilibrium The linearization of the vector field at Z is X en oe x dy LYo The characteristic polynomial has trace given by Ax yo 0 and deter minant bd Ajtxoyo 0 From the tracedeterminant plane of Chapter 4 we see that Z has eigenvalues that are either both negative or both complex with negative real parts Thus Z is asymptotically stable Note that in addition to the equilibria at Z and 00 there is still an equi librium at a40 Linearization shows that this equilibrium is a saddle its stable curve lies on the xaxis See Figure 117 It is not easy to determine the basin of Z nor do we know whether there are any limit cycles Nevertheless we can obtain some information The line L meets the xaxis at aA0 and the yaxis at 0 ab Let T be a rectangle with corners that are 00 p0 0 q and pq with p aA q ab and the point pq lying in M Every solution at a boundary point of I either enters I or is part of the boundary Therefore I is positively invariant Every point in Q is contained in such a rectangle T qd a M EY LSS p Figure 117 Nullclines and phase portrait for a predatorprey system with limited growth when the nullclines do meet in QO 244 Chapter 11 Applications in Biology By the PoincaréBendixson Theorem the wlimit set of any point xy inT with xy 0 must be a limit cycle or contain one of the three equilibria 00 Z or aA0 We rule out 00 and a40 by noting that these equilibria are saddles with stable curves that lie on the x or yaxes Therefore wx y is either Z or a limit cycle in I By Corollary 4 of the PoincaréBendixson Theorem any limit cycle must surround Z We observe further that any such rectangle I contains all limit cycles for a limit cycle like any solution must enter I and I is positively invari ant Fixing pq as before it follows that for any initial values x0 y0 there exists f 0 such that xt p yt q if t f We conclude that in the long run a solution either approaches Z or else spirals down to a limit cycle From a practical standpoint a solution that tends toward Z is indistinguish able from Z after a certain time Likewise a solution that approaches a limit cycle y can be identified with y after it is sufficiently close We conclude that any population of predators and prey that obeys these equations eventually settles down to either a constant or periodic population Furthermore there are absolute upper bounds that no population can exceed in the long run no matter what the initial populations are 113 Competitive Species We consider now two species that compete for a common food supply Instead of analyzing specific equations we follow a different procedure We consider a large class of equations about which we assume only a few qualitative features In this way considerable generality is gained and little is lost because specific equations can be very difficult to analyze Let x and y denote the populations of the two species The equations of growth of the two populations may be written in the form x Mxyx y Nuyy where the growth rates M and N are functions of both variables As usual we assume that x and y are nonnegative Thus the xnullclines are given by x 0 and Mx y 0 and the ynullclines are y 0 and Nx y 0 We make the following assumptions on M and N 1 Because the species compete for the same resources if the population of either species increases then the growth rate of the other goes down 113 Competitive Species 245 Thus oM aN 0 and 0 oy Ox 2 If either population is very large both populations decrease Thus there exists K 0 such that Mxy0 and Nxy0 if xKoryK 3 In the absence of either species the other has a positive growth rate up to a certain population and a negative growth rate beyond it Therefore there are constants ab 0 such that Mx00 for xa and Mx00 for xa NOy0 for yb and N0y0 for yb By conditions 1 and 3 each vertical line x x R meets the set w M0 exactly once if0 x aand notatall if x a By 1 and the Implicit Function Theorem ju is the graph of a nonnegative function f 0a R such that f0 a Below the curve 4 M is positive and above it M is negative In the same way the set v N0 is a smooth curve of the form yxgy where g 0 b R is a nonnegative function with g 0 b The function N is positive to the left of v and negative to the right Suppose first that w and v do not intersect and that is below v Then the phase portrait can be determined immediately from the nullclines The equilibria are 00 a0 and 0b The origin is a source while a0 is a saddle assuming that 0Mdxa0 0 The equilibrium at 0 b is a sink again assuming that 0Ndy0 b 0 All solutions with yp 0 tend to the asymptotically stable equilibrium 0 b with the exception of solutions on the xaxis See Figure 118 In case pu lies above v the situation is reversed and all solutions with xo 0 tend to the sink that now appears at a 0 Suppose now that jz and v intersect We make the assumption that uv is a finite set and at each intersection point and v cross transversely that is they have distinct tangent lines at the intersection points This assumption may be eliminated we make it only to simplify the process of determining the flow Hirsch Ch119780123820105 201222 1240 Page 246 14 246 Chapter 11 Applications in Biology a b ν μ Figure 118 Phase portrait when µ and ν do not meet The nullclines µ and ν and the coordinate axes bound a finite number of connected open sets in the upper right quadrant These are the basic regions where x 0 and y 0 They are of four types A x 0 y 0 B x 0 y 0 C x 0 y 0 D x 0 y 0 Equivalently these are the regions where the vector field points northeast northwest southwest or southeast respectively Some of these regions are indicated in Figure 119 The boundary R of a basic region R is made up of points of the following types points of µ ν called vertices points on µ or ν but not on both or on the coordinate axes called ordinary boundary points and points on the axes A vertex is an equilibrium the other equilibria lie on the axes at 00 a0 and 0b At an ordinary boundary point Z R the vector field is either vertical if Z µ or horizontal if Z ν This vector points either into or out of R since µ has no vertical tangents and ν has no horizontal tangents We call Z an inward or outward point of R accordingly Note that in Figure 119 the vector field either points inward at all ordinary points on the boundary of a basic region or else points outward at all such points This is no accident for we have this proposition Proposition Let R be a basic region for the competitive species model Then the ordinary boundary points of R are either all inward or all outward Proof There are only two ways that the curves µ and ν may intersect at a vertex P As y increases along ν the curve ν may either pass from below µ to above µ or from above to below µ These two scenarios are illustrated Hirsch Ch119780123820105 201222 1240 Page 247 15 113 Competitive Species 247 μ ν C D S R B Q b P a A Figure 119 Basic regions when the nullclines µ and ν intersect μ ν ν a b μ Figure 1110 In a ν passes from below µ to above µ as y increases The situation is reversed in b in Figures 1110a and b There are no other possibilities since we have assumed that these curves cross transversely Since x 0 below µ and x 0 above µ and since y 0 to the left of ν and y 0 to the right we have the following configurations for the vector field in these two cases See Figure 1111 In each case we see that the vector field points inward in two opposite basic regions abutting P and outward in the other two basic regions If we now move along µ or ν to the next vertex along this curve we see that adjacent basic regions must maintain their inward or outward configuration Therefore at all ordinary boundary points on each basic region the vector field either points outward or points inward as required As a consequence of the proposition it follows that each basic region and its closure is either positively or negatively invariant What are the possible 248 Chapter 11 Applications in Biology x M a v 7 v Figure 1111 Configurations of the vector field near vertices Figure 1112 All solutions must enter and then remain inT wlimit points of this system There are no closed orbits because a closed orbit must be contained in a basic region but this is impossible since xt and yt are monotone along any solution curve in a basic region Therefore all wlimit points are equilibria We note also that each solution is defined for all t 0 because any point lies in a large rectangle with corners at 00 x90 070 and x9 yo with xo a and yo b such a rectangle is positively invariant See Figure 1112 Thus we have shown the following Theorem The flow of the competitive species system has the following property for all points xy with x 0y 0 the limit lim hr x y too exists and is one of a finite number of equilibria We conclude that the populations of two competing species always tend to one of a finite number of limiting populations 113 Competitive Species 249 oe EIN Figure 1113 This configuration of 4 and v leads to an asymptotically stable equilibrium point Examining the equilibria for stability one finds the following results A ver tex where jz and v each have negative slope but yu is steeper is asymptotically stable See Figure 1113 One sees this by drawing a small rectangle with sides parallel to the axes around the equilibrium putting one corner in each of the four adjacent basic regions Such a rectangle is positively invariant since it can be arbitrarily small the equilibrium is asymptotically stable This may also be seen as follows We have slope of Me slope of v Ns 0 My Ny where M 0Mdx My 0Mdy and so on at the equilibrium Now recall that My 0 and N 0 Therefore at the equilibrium point we also have My 0 and Ny 0 Linearization at the equilibrium point yields the matrix ea tN yNx yNy The trace of this matrix is xMyN 0 while the determinant is xyMNy MNx 0 Thus the eigenvalues have negative real parts and so we have a sink A casebycase study of the different ways 2 and v can cross shows that the only other asymptotically stable equilibrium in this model is 0 b when 0 b is above 1 or a0 when a0 is to the right of v All other equilibria are unstable There must be at least one asymptotically stable equilibrium If 0 b is not one then it lies under y and if a0 is not one it lies over jz In that case yt and v cross and the first crossing to the left of a0 is asymptotically stable For example this analysis tells us that in Figure 1114 only P and 0 b are asymptotically stable all other equilibria are unstable In particular assuming Hirsch Ch119780123820105 201222 1240 Page 250 18 250 Chapter 11 Applications in Biology Z R P S Q b B B μ ν Figure 1114 Note that solutions on either side of the point Z in the stable curve of Q have very different fates that the equilibrium Q in Figure 1114 is hyperbolic then it must be a saddle because certain nearby solutions tend toward it while others tend away The point Z lies on one branch of the stable curve through Q All points in the region denoted B to the left of Z tend to the equilibrium at 0b while points to the right go to P Thus as we move across the branch of the stable curve containing Z the limiting behavior of solutions changes radically Since solutions just to the right of Z tend to the equilibrium point P it follows that the populations in this case tend to stabilize On the other hand just to the left of Z solutions tend to an equilibrium point where x 0 Thus in this case one of the species becomes extinct A small change in initial conditions has led to a dramatic change in the fate of populations Ecologically this small change could have been caused by the introduction of a new pesticide the importation of additional members of one of the species a forest fire or the like Mathematically this event is a jump from the basin of P to that of 0b 114 Exploration Competition and Harvesting In this exploration we investigate the competitive species model where we allow either harvesting emigration or immigration of one of the species We Hirsch Ch119780123820105 201222 1240 Page 251 19 115 Exploration Adding Zombies to the SIR Model 251 consider the system x x1 ax y y yb x y h Here ab and h are parameters We assume that ab 0 If h 0 then we are harvesting species y at a constant rate whereas if h 0 we add to the popula tion y at a constant rate The goal is to understand this system completely for all possible values of these parameters As usual we only consider the regime where xy 0 If yt 0 for any t 0 then we consider this species to have become extinct 1 First assume that h 0 Give a complete synopsis of the behavior of this system by plotting the different behaviors you find in the abparameter plane 2 Identify the points or curves in the abplane where bifurcations occur when h 0 and describe them 3 Now let h 0 Describe the abparameter plane for various fixed h values 4 Repeat the previous exploration for h 0 5 Describe the full threedimensional parameter space using pictures flip books 3D models movies or whatever you find most appropriate 115 Exploration Adding Zombies to the SIR Model The recent uptick in the number of zombie movies suggests that we might revise the SIR model so that the infected population now consists of zombies In this situation the zombie population again denoted by I is much more active in infecting the susceptible population One way to model this is as follows Assume that SI and R denote the fraction of the total population that is susceptible infected and recovered So S I R 1 Then I is much larger than I when I is small so the zombies are much more likely to cause a problem The new equations are S βS I I βS I νI Another possible scenario is that zombies continue to infect susceptibles until they are destroyed by a susceptible This would mean that the infected 252 Chapter 11 Applications in Biology population would now decrease at a rate that is proportional to the number of susceptibles present or S BSI IBSIyS 1 In each case describe the behavior of solutions of the corresponding system 2 What happens if we revise the SIRS model so that either of the preceding assumptions hold EXERCISES 1 For the SIRS model prove that all solutions in the triangular region A tend to the equilibrium point 70 when the total population does not exceed the threshold level for the disease 2 Sketch the phase plane for the following variant of the predatorprey system x x1xxy yy 1 x 3 A modification of the predatorprey equations is given by axy x1x x x x x41 yyy where a 0 is a parameter a Find all equilibrium points and classify them b Sketch the nullclines and the phase portraits for different values of a c Describe any bifurcations that occur as a varies 4 Another modification of the predatorprey equations is given by xy x1x x x x x4b yyy where b 0 is a parameter Hirsch Ch119780123820105 201222 1240 Page 253 21 Exercises 253 a Find all equilibrium points and classify them b Sketch the nullclines and the phase portraits for different values of b c Describe any bifurcations that occur as b varies 5 The equations x x2 x y y y3 2x y satisfy conditions 1 through 3 in Section 113 for competing species Determine the phase portrait for this system Explain why these equa tions make it mathematically possible but extremely unlikely for both species to survive 6 Consider the competing species model x xa x ay y yb bx y where the parameters a and b are positive a Find all equilibrium points for this system and determine their stability type These types will of course depend on a and b b Use the nullclines to determine the various phase portraits that arise for different choices of a and b c Determine the values of a and b for which there is a bifurcation in this system and describe the bifurcation that occurs d Record your findings by drawing a picture of the abplane and indi cating in each open region of this plane the qualitative structure of the corresponding phase portraits 7 Two species xy are in symbiosis if an increase of either population leads to an increase in the growth rate of the other Thus we assume x Mxyx y Nxyy with M y 0 and N x 0 and xy 0 We also suppose that the total food supply is limited thus Hirsch Ch119780123820105 201222 1240 Page 254 22 254 Chapter 11 Applications in Biology for some A 0B 0 we have Mxy 0 if x A Nxy 0 if y B If both populations are very small they both increase thus M00 0 and N00 0 Assuming that the intersections of the curves M10N10 are finite and that all are transverse show that a Every solution tends to an equilibrium in the region 0 x A 0 y B b There are no sources c There is at least one sink d If Mx 0 and Ny 0 there is a unique sink Z and Z is the ωlimit set for all xy with x 0y 0 8 Give a system of differential equations for a pair of mutually destruc tive species Then prove that under plausible hypotheses two mutually destructive species cannot coexist in the long run 9 Let y and x denote predator and prey populations Let x Mxyx y Nxyy where M and N satisfy the following conditions a If there are not enough prey the predators decrease Thus for some b 0 Nxy 0 if x b b An increase in the prey improves the predator growth rate thus Nx 0 c In the absence of predators a small prey population will increase thus M00 0 d Beyond a certain size the prey population must decrease thus there exists A 0 with Mxy 0 if x A e Any increase in predators decreases the rate of growth of prey thus My 0 f The two curves M10N10 intersect transversely and at only a finite number of points Exercises 255 Show that if there is some uv with Mu v 0 and Nuv 0 then there is either an asymptotically stable equilibrium or an wlimit cycle Moreover show that if the number of limit cycles is finite and positive one of them must have orbits spiraling toward it from both sides 10 Consider the following modification of the predatorprey equations x x1x ay xC y by 1 x where ab and c are positive constants Determine the region in the parameter space for which this system has a stable equilibrium with both xy 0 Prove that if the equilibrium point is unstable this system has a stable limit cycle Hirsch 16ch122572769780123820105 2012217 1913 Page 257 1 12 Applications in Circuit Theory In this chapter we first present a simple but very basic example of an electri cal circuit and then derive the differential equations governing this circuit Certain special cases of these equations are analyzed using the techniques developed in Chapters 8 9 and 10 in the next two sections these are the classical equations of Lienard and van der Pol In particular the van der Pol equation could perhaps be regarded as one of the fundamental examples of a nonlinear ordinary differential equation It possesses an oscillation or peri odic solution that is a periodic attractor Every nontrivial solution tends to this periodic solution no linear system has this property Whereas asymptoti cally stable equilibria sometimes imply death in a system attracting oscillators imply life We give an example in Section 124 of a continuous transition from one such situation to the other 121 An RLC Circuit In this section we present our first example of an electrical circuit This cir cuit is the simple but fundamental series RLC circuit shown in Figure 121 We begin by explaining what this diagram means in mathematical terms The circuit has three branches one resistor marked by R one inductor marked by Differential Equations Dynamical Systems and an Introduction to Chaos DOI 101016B9780123820105000129 c 2013 Elsevier Inc All rights reserved 257 Hirsch 16ch122572769780123820105 2012217 1913 Page 258 2 258 Chapter 12 Applications in Circuit Theory R L C α γ β Figure 121 RLC circuit L and one capacitor marked by C We think of a branch of this circuit as a certain electrical device with two terminals For example in this circuit the branch R has terminals α and β and all of the terminals are wired together to form the points or nodes αβ and γ In the circuit there is a current flowing through each branch that is mea sured by a real number More precisely the currents in the circuit are given by the three real numbers iRiL and iC where iR measures the current through the resistor and so on Current in a branch is analogous to water flowing in a pipe the corresponding measure for water would be the amount flowing in unit time or better the rate at which water passes by a fixed point in the pipe The arrows in the diagram that orient the branches tell us how to read which way the current read water is flowing for example if iR is positive then according to the arrow current flows through the resistor from β to α the choice of the arrows is made once and for all at the start The state of the currents at a given time in the circuit is thus represented a point i iRiLiC R3 But Kirchhoffs current law KCL says that in reality there is a strong restriction on which i can occur KCL asserts that the total current flowing into a node is equal to the total current flowing out of that node Think of the water analogy to make this plausible For our circuit this is equivalent to KCL iR iL iC This defines the onedimensional subspace K1 of R3 of physical current states Our choice of orientation of the capacitor branch may seem unnatural In fact these orientations are arbitrary in this example they were chosen so that the equations eventually obtained relate most directly to the history of the subject The state of the circuit is characterized by the current i iRiLiC together with the voltage or more precisely the voltage drop across each branch These voltages are denoted by vRvL and vC for the resistor branch inductor 121 An RLC Circuit 259 branch and capacitor branch respectively In the water analogy one thinks of the voltage drop as the difference in pressures at the two ends of a pipe To measure voltage one places a voltmeter imagine a water pressure meter at each of the nodes a 6 and y that reads Vq at a and so on Then vp is the difference in the reading at w and p VB Va vp The direction of the arrow tells us that vg V8 Vq rather than Va VB An unrestricted voltage state of the circuit is then a point v vp Vz VC I This time the Kirchhoff voltage law KVL puts a physical restriction on v KVL vaettvpevco0 This defines a twodimensional subspace K2 of R KVL follows immediately from our definition of the vg vr and vc in terms of voltmeters that is ve vi vo VB Va Va Viv VB Viv 9 The product space R x R is called the state space for the circuit Those states iv eR x R satisfying Kirchhoffs laws form a threedimensional subspace of the state space Now we give a mathematical definition of the three kinds of electrical devices in the circuit First consider the resistor element A resistor in the R branch imposes a functional relationship on ig and ver In our exam ple we take this relationship to be defined by a function f R R so that vr f ir If R is a conventional linear resistor then f is linear and so f ip kig This relation is known as Ohms law Nonlinear functions yield a generalized Ohms law The graph of f is called the characteristic of the resistor A couple of examples of characteristics are given in Figure 122 A characteristic like that in Figure 122b occurs in a tunnel diode A physical state i v R x R is a point that satisfies KCL KVL and also f ir vr These conditions define the set of physical states C R x R Thus is the set of points ip iz ic VR VL VC in R x R that satisfy l igitic KCL 2 ve ttvpvc0 KVL 3 firvr generalized Ohms law Now we turn to the differential equations governing the circuit The induc tor which we think of as a coil it is hard to find a water analogy specifies 260 Chapter 12 Applications in Circuit Theory VR VR ip ip a b Figure 122 Several possible characteristics for a resistor that di t a vzt Faradays law where L is a positive constant called the inductance On the other hand the capacitor which may be thought of as two metal plates separated by some insulator in the water model it is a tank imposes the condition dvct C icp Ti ct where C is a positive constant called the capacitance Lets summarize the development so far A state of the circuit is given by the six numbers ip izic vr VL Vc that is a point in the state space Rx R These numbers are subject to three restrictions Kirchhoffs current law Kirchhoffs voltage law and the resistor characteristic or generalized Ohms law Therefore the set of physical states is a certain subset C R x R The way a state changes in time is determined by the preceding two differential equations Next we simplify the set of physical states X by observing that i and vc determine the other four coordinates This follows since ig iz and ic iy by KCL ve f ir f iz by the generalized Ohms law and vy vc ve vc f iz by KVL Therefore we can use IR as the state space with coordi nates given by iz vc Formally we define a map 7 R x R R which sends iv R x R to iz vc Then we set 2 B the restriction of z to The map zy R is one to one and onto its inverse is given by the map y R where WL vc iz iL izf iz VC fiz Vc 122 The Liénard Equation 261 It is easy to check that iz vc satisfies KCL KVL and the generalized Ohms law so w does map R into D It is also easy to see that 7 and w are inverse to each other We therefore adopt R as our state space The differential equations gov erning the change of state must be rewritten in terms of our new coordinates it Vc diz L v vo ft oT vo fi dvc Cic ij dt Cc L For simplicity and since this is only an example we set L C 1 If we write x ip and y vc we then have a system of differential equations in the plane of the form dx a fx dy x dt This is one form of the equation known as the Liénard equation We analyze this system in the following section 122 The Lienard Equation In this section we begin the study of the phase portrait of the Liénard system from the circuit of the previous section dx a fx dy x dt In the special case where fx x x this system is called the van der Pol equation First consider the simplest case where f is linear Suppose f x kx where k 0 Then the Liénard system takes the form Y AY where k 1 a 262 Chapter 12 Applications in Circuit Theory The eigenvalues of A are given by A4 k k 422 Since Ax is either negative or else has a negative real part the equilibrium point at the origin is a sink It is a spiral sink if k 2 For any k 0 all solutions of the system tend to the origin physically this is the dissipative effect of the resistor Note that we have ylx ytkeyhby so that the system is equivalent to the secondorder equation y ky y 0 which is often encountered in elementary differential equations courses Next we consider the case of a general characteristic f There is a unique equilibrium point for the Liénard system that is given by 0 f0 Lineariza tion yields the matrix f0 1 1 Op with eigenvalues given by 1 ALS 5 f0 y f 4 J We conclude that this equilibrium point is a sink if f0 0 and a source if f0 0 In particular for the van der Pol equation where fx x x the unique equilibrium point is a source To analyze the system further we define the function W R R by Wx y 5 x y Then we have W xy fx yx 2f x In particular if f satisfies fx 0 if x 0 fx 0 if x 0 and f0 0 then W is a strict Liapunov function on all of R2 It follows that in this case all solutions tend to the unique equilibrium point lying at the origin In circuit theory a resistor is called passive if its characteristic is contained in the set consisting of 00 and the interior of the first and third quadrant Therefore in the case of a passive resistor xfx is negative except when x 0 and so all solutions tend to the origin Thus the word passive correctly describes the dynamics of such a circuit Hirsch 16ch122572769780123820105 2012217 1913 Page 263 7 123 The van der Pol Equation 263 123 The van der Pol Equation In this section we continue the study of the Lienard equation in the special case where f x x3 x This is the van der Pol equation dx dt y x3 x dy dt x Let φt denote the flow of this system In this case we can give a fairly complete phase portrait analysis Theorem There is one nontrivial periodic solution of the van der Pol equa tion and every other solution except the equilibrium point at the origin tends to this periodic solution The system oscillates We know from the previous section that this system has a unique equilib rium point at the origin and that this equilibrium is a source since f 0 0 The next step is to show that every nonequilibrium solution rotates in a cer tain sense around the equilibrium in a clockwise direction To see this note that the xnullcline is given by y x3 x and the ynullcline is the yaxis We subdivide each of these nullclines into two pieces given by v xy y 0x 0 v xy y 0x 0 g xy x 0y x3 x g xy x 0y x3 x These curves are disjoint together with the origin they form the boundaries of the four basic regions ABCD shown in Figure 123 From the configuration of the vector field in the basic regions it appears that all nonequilibrium solutions wind around the origin in the clockwise direction This is indeed the case Proposition Solution curves starting on v cross successively through g v and g before returning to v Proof Any solution starting on v immediately enters the region A since x0 0 In A we have y 0 so this solution must decrease in the ydirection Since the solution cannot tend to the source it follows that this 264 Chapter 12 Applications in Circuit Theory ve gt D A 7 c B a v x Figure 123 Basic regions and nullclines for the van der Pol system solution must eventually meet g On g we have x 0 and y 0 Conse quently the solution crosses g and then enters the region B Once inside B the solution heads southwest Note that the solution cannot reenter A since the vector field points straight downward on gt There are thus two possibilities Either the solution crosses v or the solution tends to oo in the ydirection and never crosses v We claim that the latter cannot happen Suppose that it does Let x9 yo be a point on this solution in region B and consider xo yo xt y4 Since xt is never 0 it follows that this solution curve lies for all time in the strip S given by 0 x x y yo and we have yt oo as t f for some fo We first observe that in fact f9 oo To see this note that t t yt yo ys ds xs ds 0 0 But 0 xs xo so we may only have yt oo0 if t ow Now consider xt for 0 t 00 We have x y x x Since the quan tity x x is bounded in the strip S and yt oo as t on it follows that t xt x x s ds oo 0 as t oo as well But this contradicts our assumption that xt 0 Thus this solution must cross v Now the vector field is skewsymmetric about the origin That is if Gxy is the van der Pol vector field then Gx y Gxy Exploiting this symmetry it follows that solutions must then pass through the regions C and D in similar fashion C As a consequence of this result we may define a Poincaré map P on the halfline v Given 0 yp v we define Pyp to be the ycoordinate of the 123 The van der Pol Equation 265 ve g Py otuy g ve Figure 124 The Poincaré map on vt first return of 0 yo to vt with t 0 See Figure 124 As in Chapter 10 Section 103 P isa C function that is one to one The Poincaré map is also onto To see this simply follow solutions starting on vt backward in time until they reintersect v as they must by the proposition Let P Po P denote the nfold composition of P with itself Our goal now is to prove this theorem Theorem The Poincaré map has a unique fixed point in v Further more the sequence Pyo tends to this fixed point as n oo for any nonzero YoEv Clearly any fixed point of P lies on a periodic solution On the other hand if Pyo yo then the solution through 0 yo can never be periodic Indeed if Pyo yo then the successive intersections of 0 yo with v is a mono tone sequence as in Chapter 10 Section 104 Thus the solution crosses v in an increasing sequence of points and so the solution can never meet itself The case Pyo yo is analogous We may define a semiPoincaré map a vt v7 by letting ay be the ycoordinate of the first point of intersection of 0 y with v where t 0 Also define 1 2 42 5y 5 0 9 Note for later use that there is a unique point 0 y v and time t such that 1 0 y A for0 t t 2 b0y 10 gt See Figure 125 Hirsch 16ch122572769780123820105 2012217 1913 Page 266 10 266 Chapter 12 Applications in Circuit Theory g 10 y y αy ν ν Figure 125 SemiPoincare map The theorem will now follow directly from the following rather delicate result Proposition The function δy satisfies the following 1 δy 0 if 0 y y 2 δy decreases monotonically to as y for y y We will prove the proposition shortly first we use it to complete the proof of the theorem We exploit the fact that the vector field is skewsymmetric about the origin This implies that if xtyt is a solution curve then so is xtyt Proof Part of the graph of δy is shown schematically in Figure 126 The intermediate value theorem and the proposition imply that there is a unique y0 v with δy0 0 Consequently αy0 y0 and it follows from the skewsymmetry that the solution through 0y0 is periodic Since δy 0 except at y0 we have αy y for all other yvalues and so it follows that φt0y0 is the unique periodic solution We next show that all other solutions except the equilibrium pont tend to this periodic solution Toward that end we define a map β v v sending each point of v to the first intersection of the solution for t 0 with v By symmetry we have βy αy Note also that Py β αy We identify the yaxis with the real numbers in the ycoordinate Thus if y1y2 v v we write y1 y2 if y1 is above y2 Note that α and β reverse this ordering while P preserves it Hirsch 16ch122572769780123820105 2012217 1913 Page 267 11 123 The van der Pol Equation 267 y y0 y δy Figure 126 Graph of δy Now choose y v with y y0 Since αy0 y0 we have αy y0 and Py y0 On the other hand δy 0 which implies that αy y Therefore Py βαy y We have shown that y y0 implies y Py y0 Similarly Py PPy y0 and by induction Pny Pn1y y0 for all n 0 The decreasing sequence Pny has a limit y1 y0 in v Note that y1 is a fixed point of P for by continuity of P we have Py1 y1 lim nPPny y1 y1 y1 0 Since P has only one fixed point we have y1 y0 This shows that the solu tion through y spirals toward the periodic solution as t The same is true if y y0 the details are left to the reader Since every solution except the equilibrium meets v the proof of the theorem is complete See Figure 127 Figure 127 Phase portrait of the van der Pol equation 268 Chapter 12 Applications in Circuit Theory Finally we turn to the proof of the proposition We adopt the following notation Let y ab R be a smooth curve in the plane and let F R R We write yt x4 y and define b Fxy Fxty0 dt y a If it happens that xt 4 0 for a t b then along y y is a function of x so we may write y yx In this case we can change variables b xb dt rome dt Fx yx Tk dx a xa Therefore xb Fx yx Fxy dx dxdt Y xa We have a similar expression if yt 4 0 Now recall the function 1 Wisy 5 x y introduced in the previous section Let p v Suppose ap p Let yt xt yt for 0 t t 1p be the solution curve joining p vt to ap v By definition 1 2 2 5p 5 v 0 WxT yT Wx0 y0 Thus T d 5p GVO dt 0 Recall from Section 122 that we have W xf x xx x 123 The van der Pol Equation 269 Thus we have T 5p xtxt xt dt 0 T xora xt dt 0 This immediately proves part 1 of the proposition because the integrand is positive for 0 xt 1 We may rewrite the last equality as 5p J 1x Y We restrict attention to points p vt with p y We divide the correspond ing solution curve y into three curves 7172 y3 as shown in Figure 128 The curves y and y3 are defined for 0 x 1 while the curve y is defined for yi y 2 Then 5p 61p 52p 63p where 5p x 1x i123 Vi Yo NY igt Figure 128 Curves y172 and y3 shown on the closed orbit through yo 270 Chapter 12 Applications in Circuit Theory Notice that along 1 yt may be regarded as a function of x Thus we have 1 51p x 1 x d Kp dxdt 0 1 x 1 x dx y Fx 0 where fx x x As p moves up the yaxis y fx increases for x y on y Thus 5p decreases as p increases Similarly 53p decreases as p increases On y2 xt may be regarded as a function of y that is defined for y y1 y2 and x 1 Therefore since dydt x we have v bain f xy 1x09 ay xy 1 xy dy y so that 52p is negative As p increases the domain y 72 of integration becomes steadily larger The function y xy depends on p so we write it as xpy As p increases the curves y2 move to the right thus xpy increases and so xpy1 xpy decreases It follows that 52p decreases as p increases and evidently limp oo 62p oo Consequently 5p also decreases and tends to oo as p o This completes the proof of the proposition 124 A Hopf Bifurcation We now describe a more general class of circuit equations where the resistor characteristic depends on a parameter jz and is denoted by f Perhaps ju is the temperature of the resistor The physical behavior of the circuit see Figure129 is then described by the system of differential equations on R dx an fu dy x dt 124 A Hopf Bifurcation 271 u05 H01 u005 u02 Figure 129 Hopf bifurcation in the system xX yxX 4x Y x Consider as an example the special case where f is described by fulx x x and the parameter yp lies in the interval 11 When 4 1 we have the van der Pol system from the previous section As before the only equilibrium point lies at the origin The linearized system is bh 1 r te and the eigenvalues are 1 At 3 ut vie 3 Thus the origin is a spiral sink for 1 w 0 and a spiral source for 0 uw 1 Indeed when 1 ys 0 the resistor is passive as the graph of f lies in the first and third quadrants Therefore all solutions tend to the origin in this case This holds even in the case where yz 0 and the linearization yields Hirsch 16ch122572769780123820105 2012217 1913 Page 272 16 272 Chapter 12 Applications in Circuit Theory a center The circuit is physically dead in that after a period of transition all the currents and voltages stay at 0 or as close to 0 as we want However as µ becomes positive the circuit becomes alive It begins to oscil late This follows from the fact that the analysis of Section 123 applies to this system for all µ in the interval 01 We therefore see the birth of a unique periodic solution γµ as µ increases through 0 see Exercise 4 at the end of this chapter As just shown this solution attracts all other nonzero solutions As in Chapter 8 Section 85 this is an example of a Hopf bifurcation Further elaboration of the ideas in Section 123 can be used to show that γµ 0 as µ 0 with µ 0 Review Figure 129 for some phase portraits associated with this bifurcation 125 Exploration Neurodynamics One of the most important developments in the study of the firing of nerve cells or neurons was the development of a model for this phenomenon in giant squid in the 1950s by Hodgkin and Huxley 23 They developed a fourdimensional system of differential equations that described the electro chemical transmission of neuronal signals along the cell membrane a work for which they later received the Nobel Prize Roughly speaking this system is similar to systems that arise in electrical circuits The neuron consists of a cell body or soma that receives electrical stimuli These stimuli are then conducted along the axon which can be thought of as an electrical cable that connects to other neurons via a collection of synapses Of course the motion is not really electrical as the current is not really made up of electrons but rather ions predominantly sodium and potassium See EdelsteinKeshet 15 or Murray 34 for a primer on the neurobiology behind these systems The fourdimensional HodgkinHuxley system is difficult to deal with primarily because of the highly nonlinear nature of the equations An impor tant breakthrough from a mathematical point of view was achieved by Fitzhugh 18 and Nagumo et al 35 who produced a simpler model of the HodgkinHuxley model Although this system is not as biologically accu rate as the original system it nevertheless does capture the essential behavior of nerve impulses including the phenomenon of excitability alluded to in the following The FitzhughNagumo system of equations is given by x y x x3 3 I y x a by Hirsch 16ch122572769780123820105 2012217 1913 Page 273 17 Exercises 273 where a and b are constants satisfying 0 3 21 a b 1 and I is a parameter In these equations x is similar to the voltage and rep resents the excitability of the system the variable y represents a combination of other forces that tend to return the system to rest The parameter I is a stimulus parameter that leads to excitation of the system I is like an applied current Note the similarity of these equations with the van der Pol equation of Section 123 1 First assume that I 0 Prove that this system has a unique equilibrium point x0y0 Hint Use the geometry of the nullclines for this rather than explicitly solving the equations Also remember the restrictions placed on a and b 2 Prove that this equilibrium point is always a sink 3 Now suppose that I 0 Prove that there is still a unique equilibrium point xIyI and that xI varies monotonically with I 4 Determine values of xI for which the equilibrium point is a source and show that there must be a stable limit cycle in this case 5 When I 0 the point x0y0 is no longer an equilibrium point Nonetheless we can still consider the solution through this point Describe the qualitative nature of this solution as I moves away from 0 Explain in mathematical terms why biologists consider this phenomenon the excitement of the neuron 6 Consider the special case where a I 0 Describe the phase plane for each b 0 no longer restricted to b 1 as completely as possible Describe any bifurcations that occur 7 Now let I vary as well and again describe any bifurcations that occur Describe in as much detail as possible the phase portraits that occur in the Ibplane with b 0 8 Extend the analysis of the previous problem to the case b 0 9 Now fix b 0 and let a and I vary Sketch the bifurcation plane the Ia plane in this case E X E R C I S E S 1 Find the phase portrait for the differential equation x y f x f x x2 y x Hint Exploit the symmetry about the yaxis Hirsch 16ch122572769780123820105 2012217 1913 Page 274 18 274 Chapter 12 Applications in Circuit Theory 2 Let f x 2x 3 if x 1 x if 1 x 1 2x 3 if x 1 Consider the system x y f x y x a Sketch the phase plane for this system b Prove that this system has a unique closed orbit 3 Let fax 2x a 2 if x 1 ax if 1 x 1 2x a 2 if x 1 Consider the system x y fax y x a Sketch the phase plane for this system for various values of a b Describe the bifurcation that occurs when a 0 4 Consider the system described in Section 124 x y x3 µx y x where the parameter µ satisfies 0 µ 1 Fill in the details of the proof that a Hopf bifurcation occurs at µ 0 5 Consider the system x µy x3 x µ 0 y x Prove that this system has a unique nontrivial periodic solution γµ Show that as µ γµ tends to the closed curve consisting of two horizontal line segments and two arcs on y x3 x as shown in Figure 1210 This Hirsch 16ch122572769780123820105 2012217 1913 Page 275 19 Exercises 275 Figure 1210 type of solution is called a relaxation oscillation When µ is large there are two quite different time scales along the periodic solution When moving horizontally we have x very large and so the solution makes this transit very quickly On the other hand near the cubic nullcline x 0 while y is bounded Thus this transit is comparatively much slower 6 Find the differential equations for the network shown in Figure 1211 where the resistor is voltage controlled that is the resistor characteristic is the graph of a function g R R iR gvR 7 Show that the LC circuit consisting of one inductor and one capacitor wired in a closed loop oscillates 8 Determine the phase portrait of the following differential equation and in particular show there is a unique nontrivial periodic solution x y f x y gx where all of the following are assumed a gx gx and xgx 0 for all x 0 b f x f x and f x 0 for 0 x a R C L Figure 1211 Hirsch 16ch122572769780123820105 2012217 1913 Page 276 20 276 Chapter 12 Applications in Circuit Theory c for x af x is positive and increasing d f x as x 9 Consider the system x y y a1 x4y x a Find all equilibrium points and classify them b Sketch the phase plane c Describe the bifurcation that occurs when a becomes positive d Prove that there exists a unique closed orbit for this system when a 0 e Show that all nonzero solutions of the system tend to this closed orbit when a 0 Hirsch Ch139780123820105 2012127 1713 Page 277 1 13 Applications in Mechanics We turn our attention in this chapter to the earliest important examples of differential equations that in fact are connected with the origins of calculus These equations were used by Newton to derive and unify the three laws of Kepler In this chapter we give a brief derivation of two of Keplers laws and then discuss more general problems in mechanics The equations of Newton our starting point have retained importance throughout the history of modern physics and lie at the root of that part of physics called classical mechanics The examples here provide us with con crete examples of historical and scientific importance Furthermore the case we consider most thoroughly here that of a particle moving in a central force gravitational field is simple enough so that the differential equations can be solved explicitly using exact classical methods just calculus However with an eye toward the more complicated mechanical systems that cannot be solved in this way we also describe a more geometric approach to this problem 131 Newtons Second Law We will be working with a particle moving in a force field F Mathematically F is just a vector field on the configuration space of the particle which in our case will be Rn From the physical point of view FX is the force exerted on a particle located at position X Differential Equations Dynamical Systems and an Introduction to Chaos DOI 101016B9780123820105000130 c 2013 Elsevier Inc All rights reserved 277 Hirsch Ch139780123820105 2012127 1713 Page 278 2 278 Chapter 13 Applications in Mechanics The example of a force field we will be most concerned with is the gravita tional field of the sun FX is the force on a particle located at X which attracts the particle to the sun We go into details of this system in Section 133 The connection between the physical concept of force field and the mathe matical concept of differential equation is Newtons second law F ma This law asserts that a particle in a force field moves in such a way that the force vector at the location X of the particle at any instant equals the accelera tion vector of the particle times the mass m That is Newtons law gives the secondorder differential equation mX FX As a system this equation becomes X V V 1 mFX where V Vt is the velocity of the particle This is a system of equations on Rn Rn This type of system is often called a mechanical system with n degrees of freedom A solution Xt Rn of the secondorder equation is said to lie in config uration space The solution of the system XtVt Rn Rn lies in the phase space or state space of the system Example Recall the simple undamped harmonic oscillator from Chapter 2 In this case the mass moves in one dimension and its position at time t is given by a function xt where x R R As we saw the differential equation governing this motion is mx kx for some constant k 0 That is the force field at the point x R is given by kx Example The twodimensional version of the harmonic oscillator allows the mass to move in the plane so the position is now given by the vector Xt x1tx2t R2 As in the onedimensional case the force field is FX kX so the equations of motion are the same mX kX 131 Newtons Second Law 279 with solutions in configuration space given by x1 t c cosokmt sinkmt x2t 03 coskmt c4sin kmt for some choices of the cj as is easily checked using the methods in Chapter 6 a Before dealing with more complicated cases of Newtons Law we need to recall a few concepts from multivariable calculus Recall that the dot product or inner product of two vectors X Y R is denoted by XY and defined by n XY xiv il where X xX Thus XX X If XY I R are smooth func tions then a version of the product rule yields XY XY 4XY as can be easily checked using the coordinate functions x and yj Recall also that if g R R the gradient of g denoted grad g is defined by dg dg dgX XX J grad gX Fe ax As we saw in Chapter 9 grad g is a vector field on R Next consider the composition of two smooth functions go F where F R Rand g R R The chain rule applied to g 0 F yields d qe grad gFtF 1 ag dF Ft 0 2g FO GO 11 We will also use the cross product or vector product U x V of vectors UV I By definition U x V tv3 03V2 43 V1 U1 V3 UpV2 UV R 280 Chapter 13 Applications in Mechanics Recall from multivariable calculus that we have Ux VVxUUIVNsind where N is a unit vector perpendicular to U and V with the orientations of the vectors UV and N given by the righthand rule Here is the angle between U and V Note that U x V 0 if and only if one vector is a scalar multiple of the other Also if U x V 0 then U x V is perpendicular to the plane containing U and V If U and V are functions of t in R then another version of the product rule asserts that d aU VUxVUxV as one can again check by using coordinates 132 Conservative Systems Many force fields appearing in physics arise in the following way There is a smooth function U R R such that dU dU dU PX X X X Ox 0x2 OXn grad UX The negative sign is traditional Such a force field is called conservative The associated system of differential equations XV 1 Vv grad UX m is called a conservative system The function U is called the potential energy of the system More properly U should be called a potential energy since adding a constant to it does not change the force field grad UX Example The preceding planar harmonic oscillator corresponds to the force field FX kX This field is conservative with potential energy 1 2 UX 5 X 132 Conservative Systems 281 For any moving particle Xt of mass m the kinetic energy is defined to be 1 2 KmVp 2 Note that the kinetic energy depends on velocity while the potential energy is a function of position The total energy or sometimes simply energy is defined on phase space by E K U The total energy function is important in mechanics because it is constant along any solution curve of the system That is in the language of Chapter 9 Section 94 Eis a constant of the motion or a first integral for the flow a Theorem Conservation of Energy Let Xt Vt be the solution curve of a conservative system Then the total energy E is constant along this solution curve Proof To show that EXf is constant in t we compute b Lmivene vex m dt 2 mVV grad UX Vgrad U grad UV 0 We remark that we may also write this type of system in Hamiltonian form Recall from Chapter 9 that a Hamiltonian system on R x R is a system of the form OH x dYi OH y Ox where H R x R R is the Hamiltonian function As we have seen the function H is constant along solutions of such a system To write the conser vative system in Hamiltonian form we make a simple change of variables We introduce the momentum vector Y mV and then set 2 HKU sy i Ux1505p This puts the conservative system in Hamiltonian form as is easily checked 282 Chapter 13 Applications in Mechanics 133 Central Force Fields A force field F is called central if FX points directly toward or away from the origin for every X In other words the vector FX is always a scalar multiple of X FX MXX where the coefficient AX depends on X We often tacitly exclude from con sideration a particle at the origin many central force fields are not defined or are infinite at the origin We deal with these types of singularities in Sec tion 137 Theoretically the function 1X could vary for different values of X on a sphere given by X constant However if the force field is conservative this is not the case Proposition Let F be a conservative force field Then the following state ments are equivalent 1 F is central 2 FX fXX 3 FX grad UX and UX gX Proof Suppose 3 is true To prove 2 we find from the Chain Rule dU d 7 4 2 2 12 gx Dx g 5 x17 x2 3 x 8X X This proves 2 with f X gXX It is clear that 2 implies 1 To show that 1 implies 3 we must prove that U is constant on each sphere Sq X RX a 0 Since any two points in Sy can be connected by a curve in Sy it suffices to show that U is constant on any curve in Sy Thus if J C R is an interval and y J Sq is a smooth curve we must show that the derivative of the composition U o y is identically 0 This derivative is d Uy grad Uyty Hirsch Ch139780123820105 2012127 1713 Page 283 7 133 Central Force Fields 283 as in Section 131 Now gradUX FX λXX since F is central Thus we have d dt Uγ t λγ tγ tγ t λγ t 2 d dt γ t2 0 because γ t α Consider now a central force field not necessarily conservative defined on R3 Suppose that at some time t0 P R3 denotes the plane containing the position vector Xt0 the velocity vector Vt0 and the origin assuming for the moment that the position and velocity vectors are not collinear Suppose that the force vector FXt0 also lies in P This makes it plausible that the particle stays in the plane P for all time In fact this is true Proposition A particle moving in a central force field in R3 always moves in a fixed plane Proof Suppose Xt is the path of a particle moving under the influence of a central force field We have d dt X V V V X V X X 0 because X is a scalar multiple of X Therefore Y Xt Vt is a constant vector If Y 0 this means that X and V always lie in the plane orthogonal to Y as asserted If Y 0 then Xt gtXt for some real function gt This means that the velocity vector of the moving particle is always directed along the line through the origin and the particle as is the force on the particle This implies that the particle always moves along the same line through the origin To prove this let x1tx2tx3t be the coordinates of Xt Then we have three separable differential equations dxk dt gtxkt for k 123 284 Chapter 13 Applications in Mechanics Integrating we find t xt ex0 where ht 0 ds 0 Therefore Xt is always a scalar multiple of X0 and so Xt moves in a fixed line and thus in a fixed plane O The vector mX x V is called the angular momentum of the system where m is the mass of the particle By the proof of the preceding proposition this vector is also conserved by the system Corollary Conservation of Angular Momentum Angular momentum is constant along any solution curve in a central force field We now restrict attention to a conservative central force field Because of the previous proposition the particle remains for all time in a plane which we may take to be x3 0 In this case angular momentum is given by the vector 00 mx v2 x2v1 Let Mx v2 x21 Thus the function is also constant along solutions In the planar case we also call the angular momentum Introducing polar coordinates x rcos and x2 rsin we find vj x r cos rsindd x rsind rcosde Then X1V2 Xv rcosOr sind rcos6 6 rsinr cosé rsin 6 17 cos sin 00 r Thus in polar coordinates mr6 We can now prove one of Keplers laws Let At denote the area swept out by the vector Xt in the time from to t In polar coordinates we have dA 5r d We define the areal velocity to be 1 2 At 5 t0t Hirsch Ch139780123820105 2012127 1713 Page 285 9 134 The Newtonian Central Force System 285 the rate at which the position vector sweeps out the area Kepler observed that the line segment joining a planet to the sun sweeps out equal areas in equal times which we interpret to mean A constant We have therefore proved more generally that this is true for any particle moving in a conservative central force field We now have found two constants of the motion or first integrals for a con servative system generated by a central force field total energy and angular momentum In the nineteenth century the idea of solving a differential equa tion was tied to the construction of a sufficient number of such constants of the motion In the twentieth century it became apparent that first integrals do not exist for differential equations very generally the culprit here is chaos which we will discuss in the next two chapters Basically chaotic behavior of solutions of a differential equation in an open set precludes the existence of first integrals in that set 134 The Newtonian Central Force System We now direct the discussion to the Newtonian central force system This sys tem deals with the motion of a single planet orbiting around the sun We assume that the sun is fixed at the origin in R3 and that the relatively small planet exerts no force on the sun The sun exerts a force on a planet given by Newtons law of gravitation which is also called the inverse square law This law states that the sun exerts a force on a planet located at X R3 with a magni tude that is gmsmpr2 where ms is the mass of the sun mp is the mass of the planet and g is the gravitational constant The direction of the force is toward the sun Therefore Newtons law yields the differential equation mpX gmsmp X X3 For clarity we change units so that the constants are normalized to one and so the equation becomes more simply X FX X X3 where F is now the force field As a system of differential equations we have X V V X X3 286 Chapter 13 Applications in Mechanics This system is called the Newtonian central force system Our goal in this sec tion is to describe the geometry of this system in the next section we derive a complete analytic solution of this system Clearly this is a central force field Moreover it is conservative since x xp grad UX where the potential energy U is given by UX X Observe that FX is not defined at 0 indeed the force field becomes infinite as the moving mass approaches collision with the stationary mass at the origin As in the previous section we may restrict attention to particles moving in the plane R Thus we look at solutions in the configuration space C IR 0 We denote the phase space by P IR 0 x R We visualize phase space as the collection of all tangent vectors at each point X C Let Ty X V V R Ty is the tangent plane to the configuration space at X Then P JIx XeC is the tangent space to the configuration space which we may naturally identify with a subset of R The dimension of phase space is four However we can cut this dimension in half by making use of the two known first integrals total energy and angular momentum Recall that energy is constant along solutions and is given by BXV KV UO SIVP 2 XI Let denote the subset of P consisting of all points X V with EX V h Xn is called an energy surface with total energy h If h 0 then Ly meets each Tx in a circle of tangent vectors satisfying 1 lv 2n aa X The radius of these circles in the tangent planes at X tends to oo as X O and decreases to 2h as X tends to oo 134 The Newtonian Central Force System 287 When h 0 the structure of the energy surface Uy is different If X 1h then there are no vectors in Ty M Xp When X 1h only the zero vector in Tx lies in Xp The circle r 1h in configuration space is therefore known as the zero velocity curve If X lies inside the zero velocity curve then Tx meets the energy surface in a circle of tangent vectors as before Figure 131 gives a caricature of Ly in the case h 0 We now introduce polar coordinates in configuration space and new variables v vg in the tangent planes via cos sind V Yr Se cos0 We have VexXer cosé 4 76 sind sin cosé so that r v and 6 vgr Differentiating once more we find 1 cosé x 1 cosd r2 Sy x3 v9 sind Vr Vo sin 0 r os es Zero velocity curve e Figure 131 Over each nonzero point inside the zero velocity curve Tx meets the energy surface Xp in a circle of tangent vectors 288 Chapter 13 Applications in Mechanics Therefore in the new coordinates 19 v vg the system becomes r Vr 6 v9r 1 w fo 4 8 r2 r yam 0 r In these coordinates total energy is given by 1 1 2442 5 r h and angular momentum is given by rvg Let consist of all points in phase space with total energy h and angular momentum For simplicity we will restrict attention to the case where h 0 If 0 we must have vg 0 So if X lies inside the zero velocity curve the tangent space at X meets Xp0 in precisely two vectors of the form cosé vy Se both of which lie on the line connecting 0 and X one pointing toward 0 the other pointing away On the zero velocity curve only the zero vector lies in Xp0 Thus we see immediately that each solution in Xp lies on a straight line through the origin The solution leaves the origin and travels along a straight line until reaching the zero velocity curve after which time it recedes back to the origin In fact since the vectors in Xy9 have magnitude tending to oo as X 0 these solutions reach the singularity in finite time in both directions Solutions of this type are called collisionejection orbits When 4 0 a different picture emerges Given X inside the zero velocity curve we have vg r so that from the total energy formula ry 2hr 2r 2 The quadratic polynomial in r on the right in Equation must therefore be nonnegative so this puts restrictions on which rvalues can occur for X Upg The graph of this quadratic polynomial is concave down since h 0 It has no real roots if 2 12h Therefore the space Dp is empty in this case If 22 12h we have a single root that occurs at r 12h Thus this is the only allowable rvalue in Xp in this case In the tangent plane at 70 134 The Newtonian Central Force System 289 we have v 0 vg 2h so this represents a circular closed orbit traversed clockwise if 0 counterclockwise if 0 If 2 12h then this polynomial has a pair of distinct roots at aw 6 with a 12h B Note that a 0 Let Ayg be the annular regiona r Bin configuration space We therefore have that motion in configuration space is confined to Agg Proposition Suppose h 0 and 7 12h Then ne CP is a two dimensional torus Proof We compute the set of tangent vectors lying in Ty 1 Dj for each X Agp If X lies on the boundary of the annulus the quadratic term on the right of Equation vanishes and so v 0 while vg r Thus there is a unique tangent vector in Ty Xpj when X lies on the boundary of the annulus When X is in the interior of Agg we have ck 1 2 2 vp V 2hr 2r07 v9 Er r so that we have a pair of vectors in Tx M Xp in this case Note that these vectors all point either clockwise or counterclockwise in Agg since vg has the same sign for all X See Figure 132 Thus we can think of Xj as being given by a pair of graphs over Agg a positive graph given by v and a negative graph given by v that are joined together along the boundary circles ra and r Of course the real picture is a subset of R This yields the torus O It is tempting to think that the two curves in the torus given by r a and r B are closed orbits for the system but this is not the case This follows 6 Qo Figure 132 A selection of vectors in Xp 290 Chapter 13 Applications in Mechanics Figure 133 Solutions in Xp that meet ra or r since when r a we have 1 wool Y5a 0 a a a However since the right side of Equation vanishes at a we have 2ha 2a 7 0 so that a 0 2ha la Since a 12h it follows that r vj 0 when r a so the rcoordinate of solutions in reaches a minimum when the curve meets r a Similarly along r B the rcoordinate reaches a maximum Thus solutions in Ag must behave as shown in Figure 133 One can eas ily show incidentally that these curves are preserved by rotations about the origin so all of these solutions behave symmetrically More however can be said Each of these solutions actually lies on a closed orbit that traces out an ellipse in configuration space To see this we need to turn to analysis 135 Keplers First Law For most nonlinear mechanical systems the geometric analysis of the previous section is just about all we can hope for In the Newtonian central force system however we get lucky As has been known for centuries we can write down explicit solutions for this system 135 Keplers First Law 291 Consider a particular solution curve of the differential equation We have two constants of the motion for this system namely the angular momentum and total energy E The case 0 yields collisionejection solutions as we saw before Thus we assume 4 0 We will show that in polar coordinates in configuration space a solution with nonzero angular momentum lies on a curve given by r1 cos x where and x are constants This equation defines a conic section as can be seen by rewriting this equation in Cartesian coordinates This fact is known as Keplers first law To prove this recall that r0 is constant and nonzero Thus the sign of 6 remains constant along each solution curve and so is always increasing or always decreasing in time Therefore we may also regard r as a function of 0 along the curve Let Wt 1rt then W is also a function of 0 Note that W U The following proposition gives a convenient formula for kinetic energy Proposition The kinetic energy is given by cue ae Ww 2 do Proof In polar coordinates we have 1 K5 r 10 Since r 1 W we also have ldw dw r Ww do do Finally ro LW r Substitution into the formula for K then completes the proof L Now we find a differential equation relating W and 6 along the solution curve Observe that K E U E W From the proposition we get WW pw 2eew xe 40 do 0 292 Chapter 13 Applications in Mechanics Differentiating both sides with respect to 6 dividing by 2dWd0 and using dEd 0 conservation of energy we obtain wow do oF where 1 is a constant Note that this equation is just the equation for a harmonic oscillator with constant forcing 1 From elementary calculus solutions of this second order equation may be written in the form 1 W p Acosé Bsin or equivalently 1 W p Ccos0 49 where the constants C and 6 are related to A and B If we substitute this expression into Equation and solve for C at say 6 69 72 we find 1 C RV 14267E Inserting this into the preceding solution we find 1 W0 1 V1 2B0 cos 05 There is no need to consider both signs in front of the radical since cos6 6 7 cos9 Oo Moreover by changing the variable 0 to 9 6 we can put any particular solution into the form 1 B 1 1 286 cose 136 The TwoBody Problem 293 This looks pretty complicated However recall from analytic geometry or from Exercise 2 at the end of this chapter that the equation of a conic in polar coordinates is 1 1 1cos r K Here x is the latus rectum and 0 is the eccentricity of the conic The origin is a focus and the three cases 1 1 ande 1 correspond respectively to a hyperbola parabola and ellipse The case 0 is a circle In our case we have V142E0 so the three different cases occur when E 0 E 0 or E 0 We have proved the following Theorem Keplers First Law The path of a particle moving under the influence of Newtons law of gravitation is a conic of eccentricity V12E0 This path lies along a hyperbola parabola or ellipse according to whether E 0 E0orE 0 136 The TwoBody Problem We now turn our attention briefly to what at first appears to be a more difficult problem the twobody problem In this system we assume that we have two masses that move in space acording to their mutual graviational attraction Let Xj Xz denote the positions of particles of mass m m in R So X x1 xx4 and X x7 353 From Newtons law of gravitation we find the equations of motion mX gmym 2a 1A gm In 3 2 XP mX gmym aT a 2A gm m2 IX X2 Hirsch Ch139780123820105 2012127 1713 Page 294 18 294 Chapter 13 Applications in Mechanics Lets examine these equations from the perspective of a viewer living on the first mass Let X X2 X1 We then have X X 2 X 1 gm1 X1 X2 X1 X23 gm2 X2 X1 X1 X23 gm1 m2 X X3 But this is just the Newtonian central force problem with a different choice of constants So to solve the twobody problem we first determine the solution of Xt of this central force problem This then determines the right side of the differ ential equations for both X1 and X2 as functions of t and so we may simply integrate twice to find X1t and X2t Another way to reduce the twobody problem to the Newtonian central force is as follows The center of mass of the twobody system is the vector Xc m1X1 m2X2 m1 m2 A computation shows that X c 0 Therefore we must have Xc At B where A and B are fixed vectors in R3 This says that the center of mass of the system moves along a straight line with constant velocity We now change coordinates so that the origin of the system is located at Xc That is we set Yj Xj Xc for j 12 Therefore m1Y1t m2Y2t 0 for all t Rewriting the differential equations in terms of the Yj we find Y 1 gm3 2 m1 m23 Y1 Y13 Y 2 gm3 1 m1 m23 Y2 Y23 which yields a pair of central force problems However since we know that m1Y1t m2Y2t 0 we need only solve one of them 137 Blowing up the Singularity The singularity at the origin in the Newtonian central force problem is the first time we have encountered such a situation Usually our vector fields have been 137 Blowing up the Singularity 295 well defined on all of R In mechanics such singularities can sometimes be removed by a combination of judicious changes of variables and time scalings In the Newtonian central force system this may be achieved using a change of variables introduced by McGehee 32 We first introduce scaled variables u r y ug rl v9 In these variables the system becomes far Uy r ug 321 2 2 ur su ug 1 2 1 uy r uptyg 0 2 rig We still have a singularity at the origin but note that the last three equations are all multiplied by r We can remove these terms by simply multiplying the vector field by r In doing so solution curves of the system remain the same but are parametrized differently More precisely we introduce a new time variable t via the rule at p2 dt By the Chain Rule we have dr dr dt dt dtdt and similarly for the other variables In this new time scale the system becomes r ru 6 u 1 uy zr u51 1 lig U Uo 8 2 re 296 Chapter 13 Applications in Mechanics where the dot now indicates differentiation with respect to t Note that when ris small dtdt is close to zero so time t moves much more slowly than time t near the origin This system no longer has a singularity at the origin We have blown up the singularity and replaced it with a new set given by r 0 with 0 u ug arbi trary On this set the system is now perfectly well defined Indeed the set r 0 is an invariant set for the flow since 0 when r 0 We have thus introduced a fictitious flow on r 0 Although solutions on r 0 mean nothing in terms of the real system by continuity of solutions they can tell us a lot about how solutions behave near the singularity We need not concern ourselves with all of r0 since the total energy relation in the new variables becomes Lio 2 hr 5 ur M6 1 On the set r 0 only the subset A defined by ue up 20arbitrary matters A is called the collision surface for the system how solutions behave on A dictate how solutions move near the singularity since any solution that approaches r 0 necessarily comes close to A in our new coordinates Note that A is a twodimensional torus It is formed by a circle in the 6direction and a circle in the uugplane On A the system reduces to 6 ug 1 uy up 1 ug Ur Up where we have used the energy relation to simplify u This system is easy to analyze We have u 0 provided ug 40 Thus the ucoordinate must increase along any solution in A with ug 0 On the other hand when up 0 the system has equilibrium points There are two circles of equilibria one given by ug 0 u V2 and arbitrary the other by ug 0 u 2 and 0 arbitrary Let C denote these two circles with u 2 on C All other solutions must travel from C to Ct since vg increases along solutions 137 Blowing up the Singularity 297 To fully understand the flow on A we introduce the angular variable y in each uugplane via up V2siny ug V2cos Wy The torus is now parametrized by 6 and w In 6wcoordinates the system becomes 6 2cos w b cosy cosp J2 The circles C are now given by wy 72 Eliminating time from this equation we find dw 1 do 2 Thus all nonequilibrium solutions have constant slope 12 when viewed in Ow coordinates See Figure 134 Now recall the collisionejection solutions described in Section 134 Each of these solutions leaves the origin and then returns along a ray 6 6 in configuration space The solution departs with v 0 and so u 0 and returns with v 0 wu 0 In our new fourdimensional coordinate system ie 6 EEE EEE c AE 6 Figure 134 Solutions on A in 6wcoordinates Recall that 6 and yw are both defined mod 27 so opposite sides of this square are identified to form a torus 298 Chapter 13 Applications in Mechanics CO A Uu 6 Ue a LJ Figure 135 Accollisionejection solution in the region r 0 leaving and returning to A and a connecting orbit on the collision surface it follows that this solution forms an unstable curve associated with the equi librium point 00 20 and a stable curve associated with 00 20 See Figure 135 What happens to nearby noncollision solutions Well they come close to the lower equilibrium point with 6 6u 2 then follow one of two branches of the unstable curve through this point up to the upper equilibrium point 6 u 42 and then depart near the unstable curve leaving this equilibrium point Interpreting this motion in con figuration space we see that each nearcollision solution approaches the origin and then retreats after 6 either increases or decreases by 27 units Of course we know this already since these solutions whip around the origin in tight ellipses 138 Exploration Other Central Force Problems In this exploration we consider the nonNewtonian central force problem where the potential energy is given by UX x where v 1 The primary goal is to understand nearcollision solutions 139 Exploration Classical Limits of OMSs 299 1 Write this system in polar coordinates 10 v vg and state explicitly the formulas for total energy and angular momentum 2 Using a computer investigate the behavior of solutions of this system when h 0 and 40 3 Blow up the singularity at the origin via the change of variables up Py ug 1 V6 and an appropriate change of time scale write down the new system 4 Compute the vector field on the collision surface A determined in r 0 by the total energy relation 5 Describe the bifurcation that occurs on A when v 2 6 Describe the structure of Dy for all v 1 7 Describe the change of structure of Uy that occurs when v passes through the value 2 8 Describe the behavior of solutions in Xy when v 2 9 Suppose 1 v 2 Describe the behavior of solutions as they pass close to the singularity at the origin 10 Using the fact that solutions of this system are preserved by rotations about the origin describe the behavior of solutions in when h 0 and 40 139 Exploration Classical Limits of Quantum Mechanical Systems In this exploration we investigate the anisotropic Kepler problem This is a classical mechanical system with two degrees of freedom that depends on a parameter jz When jz 1 the system reduces to the Newtonian central force system discussed in Section 134 When jz 1 some anisotropy is introduced into the system so that we no longer have a central force field We still have some collisionejection orbits as in the central force system but the behavior of nearby orbits is quite different from those when pw 1 The anisotropic Kepler problem was first introduced by Gutzwiller as a clas sical mechanical approximation to certain quantum mechanical systems In particular this system arises naturally when one looks for bound states of an electron near a donor impurity of a semiconductor Here the potential is due to an ordinary Coulomb field while the kinetic energy becomes anisotropic because of the electronic band structure in the solid Equivalently we can view this system as having an anisotropic potential energy function Gutzwiller suggests that this situation is akin to an electron with mass in one direction 300 Chapter 13 Applications in Mechanics that is larger than in other directions For more background on the quantum mechanical applications of this work we refer to Gutzwiller 20 The anisotropic Kepler system is given by x TUX x y3 ye y 1 uxt yp where jz is a parameter that we assume is greater than 1 1 Show that this system is conservative with potential energy given by 1 Uy 3 Vu ty write down an explicit formula for total energy 2 Describe the geometry of the energy surface for energy h 0 3 Restricting to the case of negative energy show that the only solutions that meet the zero velocity curve and are straightline collisionejection orbits for the system lie on the x and yaxes in configuration space 4 Show that angular momentum is no longer an integral for this system 5 Rewrite this system in polar coordinates 6 Using a change of variables and time rescaling as for the Newtonian central force problem Section 137 blow up the singularity and write down a new system without any singularities at r 0 7 Describe the structure of the collision surface A the intersection of X with r 0 in the scaled coordinates In particular why would someone call this surface a bumpy torus 8 Find all equilibrium points on A and determine the eigenvalues of the linearized system at these points Determine which equilibria on A are sinks sources and saddles 9 Explain the bifurcation that occurs on A when pu 98 10 Find a function that is nondecreasing along all nonequilibrium solutions in A 11 Determine the fate of the stable and unstable curves of the saddle points in the collision surface Hint Rewrite the equation on this surface to eliminate time and estimate the slope of solutions as they climb up A 12 When yp 98 describe in qualitative terms what happens to solutions that approach collision close to the collisionejection orbits on the xaxis In particular how do they retreat from the origin in the con figuration space How do solutions approach collision when traveling near the yaxis Hirsch Ch139780123820105 2012127 1713 Page 301 25 Exercises 301 1310 Exploration Motion of a Glider In this exploration we investigate the motion of a glider moving in an xyplane where x measures the horizontal direction and y the vertical direc tion Let v 0 be the velocity and θ the angle of the nose of the plane with θ 0 indicating the horizontal direction Besides gravity there are two other forces that determine the motion of the glider the drag which is parallel to the velocity vector but in the opposite direction and the lift which is perpendicular to the velocity vector 1 Assuming that both the drag and the lift are proportional to v2 use Newtons second law to show that the system of equations governing the motion of the glider may be written as θ v2 cosθ v v sinθ Dv2 where D 0 is a constant 2 Find all equilibrium solutions for this system and use linearization to determine their types 3 When D 0 show that v3 3v cosθ is constant along solution curves Sketch the phase plane in this case and describe the corresponding motion of the glider in the xyplane 4 Describe what happens when D becomes positive E X E R C I S E S 1 Which of the following force fields on R2 are conservative a Fxy x2 2y2 b Fxy x2 y2 2xy c Fxy x 0 2 Prove that the equation 1 r 1 h1 ϵ cosθ determines a hyperbola parabola and ellipse when ϵ 1 ϵ 1 and ϵ 1 respectively 302 Chapter 13 Applications in Mechanics 3 Consider the case of a particle moving directly away from the origin at time t0 in the Newtonian central force system Find a specific for mula for this solution and discuss the corresponding motion of the particle For which initial conditions does the particle eventually reverse direction 4 In the Newtonian central force system describe the geometry of Xp when h 0 andh0 5 Let FX bea force field on R Let Xo Xi be points in R and let Ys be a path in R with sp s s parametrized by arc length s from Xo to X The work done in moving a particle along this path is defined to be the integral 51 Fysys ds 50 where Ys is the unit tangent vector to the path Prove that the force field is conservative if and only if the work is independent of the path In fact if F grad V then the work done is VX VXo 6 Describe solutions to the nonNewtonian central force system given by x x X 7 Discuss solutions of the equation yr X3 This equation corresponds to a repulsive rather than attractive force at the origin 8 The following three problems deal with the twobody problem Let the potential energy be mm y mm X2 X and dU dU OU gradU ax Ax ax Show that the equations for the twobody problem may be written mX gradU Exercises 303 9 Show that the total energy K U of the system is a constant of the motion where 1 2 2 K 5 m Vil m2V2I 10 Define the angular momentum of the system by mX x Vi m2X x V2 and show that is also a first integral Hirsch 18ch143053289780123820105 2012217 1928 Page 305 1 14 The Lorenz System So far in all of the differential equations we have studied we have not encoun tered any chaos The reason is simple The linear systems of the first few chapters always have straightforward predictable behavior OK we may see solutions wrap densely around a torus as in the oscillators of Chapter 6 but this is not chaos Also for the nonlinear planar systems of the last few chap ters the PoincareBendixson Theorem completely eliminates any possibility of chaotic behavior So to find chaotic behavior we need to look at nonlinear higherdimensional systems In this chapter we investigate the system that is without doubt the most famous of all chaotic differential equations the Lorenz system from meteorol ogy First formulated in 1963 by E N Lorenz as a vastly oversimplified model of atmospheric convection this system possesses what has come to be known as a strange attractor Before the Lorenz model started making headlines the only types of stable attractors known in differential equations were equilibria and closed orbits The Lorenz system truly opened up new horizons in all areas of science and engineering as many of the phenomena present in the Lorenz system have later been found in all of the areas we have previously investigated biology circuit theory mechanics and elsewhere In the ensuing nearly 50 years much progress has been made in the study of chaotic systems Be forewarned however that the analysis of the chaotic behavior of particular systems such as the Lorenz system is usually extremely difficult Most of the chaotic behavior that is readily understandable arises from geometric models for particular differential equations rather than from Differential Equations Dynamical Systems and an Introduction to Chaos DOI 101016B9780123820105000142 c 2013 Elsevier Inc All rights reserved 305 Hirsch 18ch143053289780123820105 2012217 1928 Page 306 2 306 Chapter 14 The Lorenz System the actual equations themselves Indeed this is the avenue we pursue here We shall present a geometric model for the Lorenz system which can be com pletely analyzed using tools from discrete dynamics Although this model has been known for some 30 years it is interesting to note the fact that this model was only shown to be equivalent to the Lorenz system in the year 1999 141 Introduction In 1963 E N Lorenz 29 attempted to set up a system of differential equa tions that would explain some of the unpredictable behavior of the weather Most viable models for weather involve partial differential equations Lorenz sought a much simpler and easiertoanalyze system The Lorenz model may be somewhat inaccurately thought of as follows Imagine a planet with an atmosphere that consists of a single fluid particle As on earth this particle is heated from below and thus rises and cooled from above so then falls back down Can a weather expert predict the weather on this planet Sadly the answer is no which raises a lot of questions about the possibility of accurate weather prediction down here on earth where we have quite a few more particles in our atmosphere A little more precisely Lorenz looked at a twodimensional fluid cell that was heated from below and cooled from above The fluid motion can be described by a system of differential equations involving infinitely many vari ables Lorenz made the tremendous simplifying assumption that all but three of these variables remained constant The remaining independent variables then measured roughly speaking the rate of convective overturning x and the horizontal and vertical temperature variation y and z respectively The resulting motion led to a threedimensional system of differential equa tions which involved three parameters the Prandtl number σ the Rayleigh number r and another parameter b that is related to the physical size of the system When all of these simplifications were made the system of differential equations involved only two nonlinear terms and was given by x σy x y rx y xz z xy bz In this system all three parameters are assumed to be positive and more over σ b 1 We denote this system by X LX In Figure 141 we have displayed the solution curves through two different initial conditions Hirsch 18ch143053289780123820105 2012217 1928 Page 307 3 141 Introduction 307 x y z P1 P2 Figure 141 The Lorenz attractor Two solutions with initial conditions P1 020 and P2 020 P1 020 and P2 020 when the parameters are σ 10 b 83 and r 28 These are the original parameters that led to Lorenz discovery Note how both solutions start out very differently but eventually have more or less the same fate They both seem to wind around a pair of points alter nating at times which point they encircle This is the first important fact about the Lorenz system All nonequilibrium solutions tend eventually to the same complicated set the socalled Lorenz attractor There is another important ingredient lurking in the background here In Figure 141 we started with two relatively far apart initial conditions Had we started with two very close initial conditions we would not have observed the transient behavior apparent in Figure 141 Rather more or less the same picture would have resulted for each solution This however is mislead ing When we plot the actual coordinates of the solutions we see that these two solutions actually move quite far apart during their journey around the Lorenz attractor This is illustrated in Figure 142 where we have graphed the xcoordinates of two solutions that start out nearby one at 020 the other in gray at 02010 These graphs are nearly identical for a certain time period but then they dif fer considerably as one solution travels around one of the lobes of the attractor while the other solution travels around the other No matter how close two solutions start they always move apart in this manner when they are close to the attractor This is sensitive dependence on initial conditions one of the main features of a chaotic system 308 Chapter 14 The Lorenz System x My Figure 142 The xt graphs for two nearby initial conditions P 020 and P 02010 We will describe in detail the concept of an attractor and chaos in this chap ter But first we need to investigate some of the more familiar features of the system 142 Elementary Properties of the Lorenz System As usual to analyze this system we begin by finding the equilibria Some easy algebra yields three equilibrium points the origin and Qt 47 Or 1 4V br 1 r 1 The last two equilibria only exist when r 1 so already we see that we have a bifurcation when r 1 Linearizing we find the system o Oo 0 Yrz 1 xY y x b At the origin the eigenvalues of this matrix are b and 1 he5 c 1 4412 400 142 Elementary Properties of the Lorenz System 309 Note that both A are negative when 0 r 1 Thus the origin is a sink in this case The Lorenz vector field X possesses a symmetry If we let Sx yz xyz then we have DSLX LSX That is reflection through the zaxis preserves the vector field In particular if xt ytzf is a solution of the Lorenz equations then so is xt yt zt When x y 0 we have x y 0 so the zaxis is invariant On this axis we have simply z bz so all solutions tend to the origin on this axis In fact the solution through any point in R tends to the origin when r 1 for we have the following Proposition Suppose r 1 Then all solutions of the Lorenz system tend to the equilibrium point at the origin Proof We construct a strict Liapunov function on all of R Let Lx yZ x oy oz Then we have L2o0 x y1 rxy 20 bz We therefore have L 0 away from the origin provided that gx xy nrxy0 for xy 4 00 This is clearly true along the yaxis Along any other straightline y mx in the plane we have gx Mx x Mm lrnm1 But the quadratic term m 11rm1 is positive for all m if r 1 as is easily checked Thus gx y 0 for x y 4 00 O When r increases through 1 two things happen First the eigenvalue A at the origin becomes positive so the origin is now a saddle with a two dimensional stable surface and an unstable curve Second the two equilibria Qs are born at the origin when r 1 and move away as r increases Proposition The equilibrium points Qs are sinks provided b3 lr ro2 ob1 310 Chapter 14 The Lorenz System Proof From the linearization we calculate that the eigenvalues at Qy satisfy the cubic polynomial fra VR b40A7 4 bo rd 42bor1 0 When r 1 the polynomial f has distinct roots at 0 b and o 1 These roots are distinct since 0 b 1 so 0 1lo01b0 Thus for r close to 1 f has three real roots close to these values Note that fA 0 for 1 0 and r 1 Looking at the graph of f it follows that at least for r close to 1 the three roots of f must be real and negative We now let r increase and ask what is the lowest value of r for which f has an eigenvalue with zero real part Note that this eigenvalue must in fact be of the form i with w 0 since f is a real polynomial that has no roots equal to 0 when r 1 Solving fiw 0 by equating both real and imaginary parts to zero then yields the result recall that we have assumed o b 1 O We remark that a Hopf bifurcation is known to occur at r but proving this is beyond the scope of this book When r 1 it is no longer true that all solutions tend to the origin How ever we can say that solutions that start far from the origin do at least move closer in To be precise let VxyZ rx oy oaz 2r Note that Vx yz v 0 defines an ellipsoid in R centered at 002r We will show the following Proposition There exists v such that any solution that starts outside the ellipsoid V v eventually enters this ellipsoid and then remains trapped therein for all future time Proof We compute V 20 rx y b2 2rz 20 rx y bzr br The equation me ybizrr p 142 Elementary Properties of the Lorenz System 311 also defines an ellipsoid when jz 0 When ju br we have V 0 Thus we may choose v large enough so that the ellipsoid V v strictly contains the ellipsoid ne ybz1 br in its interior Then V 0 for all v v C As a consequence all solutions starting far from the origin are attracted to a set that sits inside the ellipsoid V v Let A denote the set of all points with solutions that remain for all time forward and backward in this ellip soid Then the wlimit set of any solution of the Lorenz system must lie in A Theoretically A could be a large set perhaps bounding an open region in R However for the Lorenz system this is not the case To see this recall from calculus that the divergence of a vector field FX on R is given by AF divF 5 xX Ox X 11 The divergence of F measures how fast volumes change under the flow of F Suppose D is a region in IR with a smooth boundary and let Dt D the image of D under the time ft map of the flow Let Vt be the volume of Dt Then Liouvilles Theorem asserts that dV divFdxdydz dt Y Da For the Lorenz system we compute immediately that the divergence is the constant o 1 b so that volume decreases at a constant rate v o1bV 0 dt Solving this simple differential equation yields Vit e OTH VO so that any volume must shrink exponentially fast to 0 In particular we have this proposition Proposition The volume of A is zero L The natural question is what more can we say about the structure of the attractor A In dimension 2 such a set would consist of a collection of 312 Chapter 14 The Lorenz System limit cycles equilibrium points and solutions connecting them In higher dimensions these attractors may be much stranger as we show in the next section 143 The Lorenz Attractor The behavior of the Lorenz system as the parameter r increases is the subject of much contemporary research we are decades if not centuries away from rig orously understanding all of the fascinating dynamical phenomena that occur as the parameters change Sparrow has written an entire book devoted to this subject 44 In this section we will deal with one specific set of parameters where the Lorenz system has an attractor Roughly speaking an attractor for the flow is an invariant set that attracts all nearby solutions The following definition is more precise Definition Let X FX be a system of differential equations in R with flow A set A is called an attractor if 1 A is compact and invariant 2 There is an open set U containing A such that for each X U X U ands06U A 3 Transitivity Given any points YY A and any open neighbor hoods U about Y in U there is a solution curve that begins in U and later passes through U The transitivity condition in this definition may seem a little strange Basi cally we include it to guarantee that we are looking at a single attractor rather than a collection of dynamically different attractors For example the transitivity condition rules out situations such as that given by the planar system x x2x yy The phase portrait of this system is shown in Figure 143 Note that any solution of this system enters the set marked U and then tends to one of the three equilibrium points either to one of the sinks at 10 or to the sad dle 00 The forward intersection of the flow applied to U is the interval 143 The Lorenz Attractor 313 SWE Figure 143 The interval on the xaxis between the two sinks is not an attractor for this system despite the fact that all solutions enter U 1 x 1 This interval meets conditions 1 and 2 in the definition but condition 3 is violated as there are no solution curves passing close to points in both the left and right half of this interval We choose not to consider this set an attractor since most solutions tend to one of the two sinks We really have two distinct attractors in this case As a remark there is no universally accepted definition of an attractor in mathematics some people choose to say that a set A that meets only condi tions 1 and 2 is an attractor while if A also meets condition 3 it is called a transitive attractor For planar systems condition 3 is usually easily veri fied in higher dimensions however this can be much more difficult as we shall see For the rest of this chapter we restrict attention to the very special case of the Lorenz system where the parameters are given by o 10 b 83 and r 28 Historically these are the values Lorenz used when he first encoun tered chaotic phenomena in this system Thus the specific Lorenz system we consider is 10y x X LX 28xyxz xy 83z As in the previous section we have three equilibria the origin and Qy 62 62 27 At the origin we find eigenvalues 4 83 and 11 1201 A 32 Hirsch 18ch143053289780123820105 2012217 1928 Page 314 10 314 Chapter 14 The Lorenz System x z y Figure 144 Linearization at the origin for the Lorenz system For later use note that these eigenvalues satisfy λ λ λ1 0 λ The linearized system at the origin is then Y λ 0 0 0 λ 0 0 0 λ1Y The phase portrait of the linearized system is shown in Figure 144 Note that all solutions in the stable plane of this system tend to the origin tangentially to the zaxis At Q a computation shows that there is a single negative real eigenvalue and a pair of complex conjugate eigenvalues with positive real parts In Figure 145 we have displayed a numerical computation of a portion of the left and right branches of the unstable curve at the origin Note that the right portion of this curve comes close to Q and then spirals away The left portion behaves symmetrically under reflection through the zaxis In Figure 146 we have displayed a significantly larger portion of these unstable curves Note that they appear to circulate around the two equilibria some times spiraling around Q sometimes around Q In particular these curves continually reintersect the portion of the plane z 27 containing Q in which the vector field points downward This suggests that we may construct a Poincare map on a portion of this plane Hirsch 18ch143053289780123820105 2012217 1928 Page 315 11 143 The Lorenz Attractor 315 Q Q 000 Figure 145 Unstable curve at the origin 000 Figure 146 More of the unstable curve at the origin As we have seen before computing a Poincare map is often impossible and this case is no different So we will content ourselves with building a simplified model that exhibits much of the behavior we find in the Lorenz system As we shall see in the following section this model provides a computable means to assess the chaotic behavior of the system 316 Chapter 14 The Lorenz System 144 A Model for the Lorenz Attractor In this section we describe a geometric model for the Lorenz attractor origi nally proposed by Guckenheimer and Williams 21 Tucker 46 showed that this model does indeed correspond to the Lorenz system for certain parame ters Rather than specify the vector field exactly we give instead a qualitative description of its flow much as we did in Chapter 11 The specific numbers we use are not that important only their relative sizes matter We will assume that our model is symmetric under the reflection x yz xy Zz as is the Lorenz system We first place an equilibrium point at the origin in R and assume that in the cube S given by x y z 5 the system is linear Rather than use the eigenvalues A and A from the actual Lorenz system we simplify the computations a bit by assuming that the eigenvalues are 12 and 3 and that the system is given in the cube by x 3x y2y Zz Note that the phase portrait of this system agrees with that in Figure 144 and that the relative magnitudes of the eigenvalues are the same as in the Lorenz case We need to know how solutions make the transit near 000 Consider a rectangle R in the plane z 1 given by x 1 O0ye 1 As time moves forward all solutions that start in 7 eventually reach the rectangle RR in the plane y 1 defined by x 1 0 z 1 Thus we have a function h Ry Rz defined by following solution curves as they pass from 7 to Ry We leave it as an exercise to check that this function assumes the form GGGr y Na ye po It follows that h takes lines y c in R to lines z c in Ry Also since x xz we have that h maps lines x c to curves of the form x CZ Each of these image curves meet the xyplane perpendicularly as shown in Figure 147 Mimicking the Lorenz system we place two additional equilibria in the plane z 27 one at Q 10 2027 and the other at Q 10 2027 We assume that the lines given by y 20 z 27 form portions of the stable lines at Q4 and that the other two eigenvalues at these points are complex with positive real parts 144 A Model for the Lorenz Attractor 317 a hx y y x Figure 147 Solutions making the transit near 000 Let denote the square xy 20 z 27 We assume that the vector field points downward in the interior of this square Thus solutions spiral away from Q in the same manner as in the Lorenz system We also assume that the stable surface of 0 00 first meets in the line of intersection of the xzplane and Let denote the two branches of the unstable curve at the origin We assume that these curves make a passage around and then enter this square as shown in Figure 148 We denote the first point of intersection of with by p 4x Fy Now consider a straight line y v in If v 0 all solutions beginning at points on this line tend to the origin as time moves forward Thus these solutions never return to We assume that all other solutions originating in do return to X as time moves forward How these solutions return leads to our major assumptions about this model These assumptions are 1 Return condition Let LD UNy0 and Y UN y 0 We assume that the solutions through any point in L return to in forward time Thus we have a Poincaré map X4U X LX We assume that the images X are as shown in Figure 148 By symmetry we have xy Pxy 2 Contracting direction For each v 40 we assume that maps the line y v in into the line y gv for some function g Moreover we assume that contracts this line in the xdirection 3 Expanding direction We assume that stretches X and Y in the y direction by a factor greater than 2 so that gy J2 318 Chapter 14 The Lorenz System fS i ee fo o ct y x Figure 148 Solutions and their intersection with in the model for the Lorenz attractor 4 Hyperbolicity condition Besides the expansion and contraction we assume that D maps vectors tangent to X with slopes that are 1 to vectors with slopes of a magnitude larger than pz 1 Analytically these assumptions imply that the map assumes the form Px y Fy8 where gy V2 and 0 dfdx c1 The hyperbolicity condition implies that a a go nbs 2 ox oy Geometrically this condition implies that the sectors in the tangent planes given by y x are mapped by D strictly inside a sector with steeper slopes Note that this condition holds if dfdy and c are sufficiently small throughout 4 Technically x0 is not defined but we do have lim xy p y0F where we recall that p is the first point of intersection of with D We call p the tip of X In fact our assumptions on the eigenvalues guarantee that gy oo as y 0 see Exercise 3 at the end of this chapter 144 A Model for the Lorenz Attractor 319 y Hyy J R wyy y Figure 149 Poincaré map on RP To find the attractor we may restrict attention to the rectangle R C given by y y where we recall that y is the ycoordinate of the tips p Let Rt RN ZX It is easy to check that any solution starting in the interior of X but outside R must eventually meet R so it suffices to consider the behavior of on R A planar picture of the action of on R is displayed in Figure 149 Note that R C R Let denote the nth iterate of and let Co A R n0 Here U denotes the closure of the set U The set A will be the intersection of the attractor for the flow with R That is let A U oa U000 teR We add the origin here so that A will be a closed set We will prove the following theorem Theorem A is an attractor for the model Lorenz system Proof The proof that A is an attractor for the flow follows immediately from the fact that A is an attractor for the mapping where an attractor for a map ping is defined completely analogously to that for a flow Clearly A is closed Technically A itself is not invariant under since is not defined along y 0 However for the flow the solutions through all such points do lie in A and so A is invariant If we let O be the open set given by x 20y 20 320 Chapter 14 The Lorenz System for an with y 20e then for any xy O there is an n such that xy R Thus lim xy CA n Oo for all xy O By definition A Nys0R and so A Nys0PO as well Therefore conditions 1 and 2 in the definition of an attractor hold for It remains to show the transitivity property We need to show that if P and Py are points in A and W are open neighborhoods of P in O then there exists an m 0 such that WN W 4 Given a set U C R let I1U denote the projection of U onto the yaxis Also let U denote the length of I1U which we call the ylength of U In the following U will be a finite collection of connected sets so U is well defined We need a lemma Lemma For any open set W C R there exists n 0 such that yW is the open interval y y Equivalently W meets each line y c in the interior of R Proof First suppose that W contains a connected piece W that extends from one of the tips to y 0 Thus W y Then W is connected and we have W J2y Moreover W also extends from one of the tips but now crosses y 0 since its ylength exceeds y Now apply again W contains two pieces one of which extends to ot the other to p Moreover W 2y Thus it follows that T1W yy and so we are done in this special case For the general case suppose first that W is connected and does not cross y 0 Then we have W 26W as before so the ylength of W grows by a factor of more than J2 If W does cross y 0 then we may find a pair of connected sets W with W Cc RN W and W UW W The images W extend to the tips p If either of these sets also meets y0 then we are done according to the preceding If neither 6W nor W7 crosses y 0 then we may apply again Both WT and W7 are connected sets and we have 2W 2W Thus for one of W or W we have 62W4 W and again the ylength of W grows under iteration Thus if we continue to iterate or and choose the appropriate largest subset of W at each stage as above then we see that the ylengths of these images grow without bound This completes the proof of the lemma We now complete the proof of the theorem 145 The Chaotic Attractor 321 Proof We must find a point in W with an image under an iterate of that lies in W Toward that end note that ox1y On 9 cx x9 since Ox y and D xp y lie on the same straight line parallel to the xaxis for each j and since contracts distances in the xdirection by a factor of cl We may assume that W is a disk of diameter Recalling that the width of R in the xdirection is 40 we choose m such that 40c Consider P Note that P2 is defined since Pz NyxoPR Say BP2 E57 From the lemma we know that there exists n such that TyW1 yy Thus we may choose a point 17 W Say 17 xy where xy Wj so that xy and P2 have the same ycoordinate Then we have b X 7 Po E1m Po 1n E 40c e We have found a point xy W with a solution that passes through W This concludes the proof Note that in the preceding proof the solution curve that starts near P and comes close to P need not lie in the attractor However it is possible to find such a solution that does lie in A see Exercise 4 at the end of this chapter 145 The Chaotic Attractor In the previous section we reduced the study of the behavior of solutions of the Lorenz system to the analysis of the dynamics of the Poincaré map In the process we dropped from a threedimensional system of differential equations to a twodimensional mapping But we can do better According to our assumptions two points that share the same ycoordinate in are mapped to two new points with ycoordinates given by gy and thus are again the same 322 Chapter 14 The Lorenz System Moreover the distance between these points is contracted It follows that under iteration of we need not worry about all points on a line y con stant we need only keep track of how the ycoordinate changes under iteration of g Then as we shall see the Poincaré map is completely determined by the dynamics of the onedimensional function g defined on the interval y y Indeed iteration of this function completely determines the behav ior of all solutions in the attractor In this section we begin to analyze the dynamics of this onedimensional discrete dynamical system In Chapter 15 we plunge more deeply into this topic Let I be the interval y y Recall that g is defined on I except at y 0 and satisfies gy gy From the results of the previous section we have gy V20 gly y and y gy 0 Also lim Fy yt Thus the graph of g resembles that shown in Figure 1410 Note that all points in the interval gy gy have two preimages while points in the inter vals y gy and gy y have only one The endpoints of I namely y have no preimages in I since g0 is undefined Let yo I We will investigate the structure of the set AN y yo We define the forward orbit of yo to be the set yo V1 a Where Vy gVn1 g yo For each yo the forward orbit of yo is uniquely determined though it terminates if gyo 0 A backward orbit of yo is a sequence of the form yo y1y2 where gyk yk41 Unlike forward orbits of g there are infinitely many distinct backward orbits for a given yo except in the case where yop ty since these two points have no preimages in I To see this suppose first that yo does not lie on the forward orbit of ty Then each point y must have either one or two distinct preimages since y ty If y has only one preimage yx1 I y AL y LV Figure 1410 Graph of the onedimensional function g on l y y 145 The Chaotic Attractor 323 then y lies in either ygy or gy y But then the graph of g shows that yx must have two preimages so no two consecutive points in a given backward orbit can have only one preimage This shows that yo has infinitely many distinct backward orbits If we happen to have y ty for some k 0 then this backward orbit stops since y has no preimage in I However yz4 must have two preim ages one of which is the endpoint and the other is a point in I that does not equal the other endpoint Thus we can continue taking preimages of this second backward orbit as before thereby generating infinitely many distinct backward orbits as in the preceding We claim that each of these infinite backward orbits of yo corresponds to a unique point in AN y yo To see this consider the line J given by yyin R Then J is a closed subinterval of Jo for each k Note that J41 C Jx since y yx is a proper subinterval of y y Thus the nested intersection of the sets J is nonempty and any point in this intersection has backward orbit yo y1y2 by construction Further more the intersection point is unique since each application of contracts the intervals y y by a factor of c 1 In terms of our model we therefore see that the attractor A is a complicated set We have proved this proposition Proposition The attractor A for the model Lorenz system meets each of the lines y yo y in R at infinitely many distinct points In forward time all of the solution curves through each point on this line either 1 Meet the line y0 in which case the solution curves all tend to the equilibrium point at 000 or 2 Continually reintersect R and the distances between these intersection points on the line y yx tends to 0 as time increases O Now we turn to the dynamics of in R We first discuss the behavior of the onedimensional function g and then use this information to understand what happens for Given any point yo I note that nearby forward orbits of g move away from the orbit of yo since g 2 More precisely we have the following proposition Proposition Let0v y Let y Iyy Given anye 0 we may find uovo I with up yo and vo yo and n 0 such that Ig uo gvo 2v Proof Let J be the interval of length 2 centered at yo Each iteration of g expands the length of J by a factor of at least 2 so there is an iteration for which gJ contains 0 in its interior Then gn J contains points arbitrarily 324 Chapter 14 The Lorenz System close to both y and thus there are points in gh J where the distance from each other is at least 2v This completes the proof O Lets interpret the meaning of this proposition in terms of the attractor A Given any point in the attractor we may always find points arbitrarily nearby with forward orbits that move apart just about as far as they possibly can This is the hallmark of a chaotic system We call this behavior sensitive dependence on initial conditions A tiny change in the initial position of the orbit may result in drastic changes in the eventual behavior of the orbit Note that we must have a similar sensitivity for the flow in A certain nearby solution curves in A must also move far apart This is the behavior we witnessed earlier in Figure 142 This should be contrasted with the behavior of points in A that lie on the same line y constant with y y y As we saw before there are infinitely many such points in A Under iteration of the successive images of all of these points move closer together rather than separating Recall now that a subset of I is dense if its closure is all of I Equivalently a subset of I is dense if there are points in the subset arbitrarily close to any point whatsoever in I Also a periodic point for g is a point yo for which gyo yo for some n 0 Periodic points for g correspond to periodic solutions of the flow Proposition The periodic points of g are dense in I Proof As in the proof in the last section that A is an attractor given any subinterval J of I 0 we may find n so that g maps some subinterval J Cc J in onetoone fashion over either y0 or 0 y Thus either gJ con tains J or the next iteration gg contains J In either case the graphs of g or g cross the diagonal line y x over J This yields a periodic point for gin J O Now let us interpret this result in terms of the attractor A We claim that periodic points for are also dense in A To see this let Pe A and U be an open neighborhood of P We assume that U does not cross the line y 0 otherwise just choose a smaller neighborhood nearby that is disjoint from y 0 For small enough 0 we construct a rectangle W C U centered at P and having width 2e in the xdirection and height in the ydirection Let W C W bea smaller square centered at P with sidelength 2 By the transitivity result of the previous section we may find a point Q W such that 6Q Q W By choosing a subset of W if necessary we may assume that n 4 and furthermore that n is so large that c 8 It follows that the image of W not W crosses through the interior of W nearly vertically and extends beyond its top and bottom boundaries as shown in Figure 1411 This fact uses the hyperbolicity condition 145 The Chaotic Attractor 325 on Ww tl N YCo ry eveo Figure 1411 maps Wacross itself Now consider the lines y c in W These lines are mapped to other such lines in R by Since the vertical direction is expanded some of the lines must be mapped above W and some below It follows that one such line y a must be mapped inside itself by and therefore there must be a fixed point for on this line Since this line is contained in W we have produced a periodic point for in W This proves density of periodic points in A In terms of the flow a solution beginning at a periodic point of is a closed orbit Thus the set of points lying on closed orbits is a dense subset of A The structure of these closed orbits is quite interesting from a topological point of view as many of these closed curves are actually knotted See Birman and Williams 10 and exercise 10 at the end of this chapter Finally we say that a function g is transitive on I if for any pair of points y1 and y2 in I and neighborhoods Uj of y we can find y U and n such that g Up Just as in the proof of density of periodic points we may use the fact that g is expanding in the ydirection to prove the following Proposition The function g is transitive on I We leave the details to the reader In terms of we almost proved the corre sponding result when we showed that A was an attractor The only detail we did not provide was the fact that we could find a point in A with an orbit that made the transit arbitrarily close to any given pair of points in A For this detail we refer to Exercise 4 at the end of this chapter Thus we can summarize the dynamics of on the attractor A of the Lorenz model as follows UO Theorem Dynamics of the Lorenz Model The Poincaré map restricted to the attractor A for the Lorenz model has the following properties 1 has sensitive dependence on initial conditions 2 Periodic points of are dense in A 3 is transitive on A Hirsch 18ch143053289780123820105 2012217 1928 Page 326 22 326 Chapter 14 The Lorenz System We say that a mapping with the preceding properties is chaotic We caution the reader that just as in the definition of an attractor there are many defini tions of chaos around Some involve exponential separation of orbits others involve positive Liapunov exponents and others do not require density of peri odic points It is an interesting fact that for continuous functions of the real line density of periodic points and transitivity are enough to guarantee sensi tive dependence See Banks et al 8 We will delve more deeply into chaotic behavior of discrete systems in the next chapter 146 Exploration The R ossler Attractor In this exploration we investigate a threedimensional system similar in many respects to the Lorenz system The Rossler system 36 is given by x y z y x ay z b zx c where ab and c are real parameters For simplicity we will restrict attention to the case where a 14 b 1 and c ranges from 0 to 7 As with the Lorenz system it is difficult to prove specific results about this system so much of this exploration will center on numerical experimentation and the construction of a model 1 First find all equilibrium points for this system 2 Describe the bifurcation that occurs at c 1 3 Investigate numerically the behavior of this system as c increases What bifurcations do you observe 4 In Figure 1412 we have plotted a single solution for c 55 Compute other solutions for this parameter value and display the results from other viewpoints in R3 What conjectures do you make about the behavior of this system 5 Using techniques described in this chapter devise a geometric model that mimics the behavior of the Rossler system for this parameter value 6 Construct a model mapping on a twodimensional region with dynamics that might explain the behavior observed in this system 7 As in the Lorenz system describe a possible way to reduce this function to a mapping on an interval 8 Give an explicit formula for this onedimensional model mapping What can you say about the chaotic behavior of your model 9 What other bifurcations do you observe in the Rossler system as c rises above 55 Exercises 327 aN x Figure 1412 Rossler attractor EXERCISES 1 Consider the system x 3x y2y Z2Z Recall from Section 144 that there is a function h R FR where Ry is given by x 1 0 ye 1 and z1 and Rz is given by x 10 z1and y 1 Show that h is given by x XI xy h 12 y 2 y 2 Suppose that the roles of x and z are reversed in the previous problem That is suppose x x and z 3z Describe the image of hx y in Ry in this case 3 For the Poincaré map xy fx ygy for the model attractor use the results of Exercise 1 to show that gy coas y 0 4 Show that it is possible to verify the transitivity condition for the Lorenz model with a solution that actually lies in the attractor 5 Prove that arbitrarily close to any point in the model Lorenz attrac tor there is a solution that eventually tends to the equilibrium point at 0 00 6 Prove that there is a periodic solution y of the model Lorenz system that meets the rectangle R in precisely two distinct points 328 Chapter 14 The Lorenz System MK Figure 1413 maps R completely across itself 7 Prove that arbitrarily close to any point in the model Lorenz attractor there is a solution that eventually tends to the periodic solution y from the previous exercise 8 Consider a map on a rectangle R as shown in Figure 1413 where has properties similar to the model Lorenz How many periodic points of period n does have 9 Consider the system x 10y x y 28xyxz z xy 83z Show that this system is not chaotic in the region where xy and z are all positive Note the xz term in the equation for ys Hint Show that most solutions tend to 00 10 A simple closed curve in R is knotted if it cannot be continuously deformed into the unknot the unit circle in the xyplane without hav ing selfintersections along the way Using the model Lorenz attractor sketch a curve that follows the dynamics of so it should approximate a real solution and is knotted You might want to use some string for this 11 Use a computer to investigate the behavior of the Lorenz system as r increases from 1 to 28 with o 10 and b 83 Describe in qualitative terms any bifurcations you observe Hirsch Ch159780123820105 201229 1441 Page 329 1 15 Discrete Dynamical Systems Our goal in this chapter is to begin the study of discrete dynamical systems As we have seen at several stages in this book it is sometimes possible to reduce the study of the flow of a differential equation to that of an iterated function namely a Poincare map This reduction has several advantages First and fore most the Poincare map lives on a lowerdimensional space which therefore makes visualization easier Second we do not have to integrate to find solu tions of discrete systems Rather given the function we simply iterate the function over and over to determine the behavior of the orbit which then dictates the behavior of the corresponding solution Given these two simplifications it then becomes much easier to com prehend the complicated chaotic behavior that often arises for systems of differential equations Although the study of discrete dynamical systems is a topic that could easily fill this entire book we will restrict attention here pri marily to the portion of this theory that helps us understand chaotic behavior in one dimension In the next chapter we will extend these ideas to higher dimensions 151 Introduction Throughout this chapter we will work with real functions f R R As usual we assume throughout that f is C although there will be several special examples where this is not the case Differential Equations Dynamical Systems and an Introduction to Chaos DOI 101016B9780123820105000154 c 2013 Elsevier Inc All rights reserved 329 Hirsch Ch159780123820105 201229 1441 Page 330 2 330 Chapter 15 Discrete Dynamical Systems Let f n denote the nth iterate of f That is f n is the nfold composition of f with itself Given x0 R the orbit of x0 is the sequence x0 x1 f x0 x2 f 2x0 xn f nx0 The point x0 is called the seed of the orbit Example Let f x x2 1 Then the orbit of the seed 0 is the sequence x0 0 x1 1 x2 2 x3 5 x4 26 xn big xn1 bigger and so forth so we see that this orbit tends to as n In analogy with equilibrium solutions of systems of differential equations fixed points play a central role in discrete dynamical systems A point x0 is called a fixed point if f x0 x0 Obviously the orbit of a fixed point is the constant sequence x0x0x0 The analogue of closed orbits for differential equations is given by periodic points of period n These are seeds x0 for which f nx0 x0 for some n 0 As a consequence like a closed orbit a periodic orbit repeats itself x0x1xn1x0x1xn1x0 Periodic orbits of period n are also called ncycles We say that the peri odic point x0 has minimal period n if n is the least positive integer for which f nx0 x0 Example The function f x x3 has fixed points at x 01 The func tion gxx3 has a fixed point at 0 and a periodic point of period 2 at x 1 since g1 1 and g1 1 so g21 1 The function hx 2 x3x 12 has a 3cycle given by x0 0x1 1x2 2x3 x0 0 Hirsch Ch159780123820105 201229 1441 Page 331 3 151 Introduction 331 A useful way to visualize orbits of onedimensional discrete dynamical sys tems is via graphical iteration In this picture we superimpose the curve y f x and the diagonal line y x on the same graph We display the orbit of x0 as follows Begin at the point x0x0 on the diagonal and draw a vertical line to the graph of f reaching the graph at x0f x0 x0x1 Then draw a horizontal line back to the diagonal ending at x1x1 This procedure moves us from a point on the diagonal directly over the seed x0 to a point directly over the next point on the orbit x1 Then we continue from x1x1 First go vertically to the graph to the point x1x2 then hori zontally back to the diagonal at x2x2 On the xaxis this moves us from x1 to the next point on the orbit x2 Continuing we produce a series of pairs of lines each of which terminates on the diagonal at a point of the form xnxn In Figure 151a graphical iteration shows that the orbit of x0 tends to the fixed point z0 under iteration of f In Figure 151b the orbit of x0 under g lies on a 3cycle x0x1x2x0x1 As in the case of equilibrium points of differential equations there are dif ferent types of fixed points for a discrete dynamical system Suppose that x0 is a fixed point for f We say that x0 is a sink or an attracting fixed point for f if there is a neighborhood U of x0 in R having the property that if y0 U then f ny0 U for all n and moreover f ny0 x0 as n Similarly x0 is a source or a repelling fixed point if all orbits except x0 leave U under iteration of f A fixed point is called neutral or indifferent if it is neither attracting nor repelling For differential equations we saw that it is the derivative of the vector field at an equilibrium point that determines the type of the equilibrium point This is also true for fixed points although the numbers change a bit y x y x x0 x0 x1 x2 x1 x2 z0 y fx y gx a b Figure 151 The orbit of x0 tends to the fixed point at z0 under iteration of f while the orbit of x0 lies on a 3cycle under iteration of g Hirsch Ch159780123820105 201229 1441 Page 332 4 332 Chapter 15 Discrete Dynamical Systems Proposition Suppose f has a fixed point at x0 Then 1 x0 is a sink if f x0 1 2 x0 is a source if f x0 1 3 We get no information about the type of x0 if f x0 1 Proof We first prove case 1 Suppose f x0 ν 1 Choose K with ν K 1 Since f is continuous we may find δ 0 so that f x K for all x in the interval I x0 δx0 δ We now invoke the Mean Value Theorem Given any x I we have f x x0 x x0 f x f x0 x x0 f c for some c between x and x0 Thus we have f x x0 Kx x0 It follows that f x is closer to x0 than x and so f x I Applying this result again we have f 2x x0 Kf x x0 K2x x0 and continuing we find f nx x0 Knx x0 so that f nx x0 in I as required since 0 K 1 The proof of case 2 follows similarly In case 3 we note that each of the functions 1 f x x x3 2 gx x x3 3 hx x x2 has a fixed point at 0 with f 0 1 But graphical iteration see Figure 152 shows that f has a source at 0 g has a sink at 0 and 0 is attracting from one side and repelling from the other for the function h Note that at a fixed point x0 for which f x0 0 the orbits of nearby points jump from one side of the fixed point to the other at each iteration See Figure 153 This is the reason why the output of graphical iteration is often called a web diagram Hirsch Ch159780123820105 201229 1441 Page 333 5 151 Introduction 333 f x xx3 gxxx3 hx x x2 Figure 152 In each case the derivative at 0 is 1 but f has a source at 0 g has a sink and h has neither x0 z0 Figure 153 Since 1 f z0 0 the orbit of x0 spirals toward the attracting fixed point at z0 Since a periodic point x0 of period n for f is a fixed point of f n we may classify these points as sinks or sources depending on whether f nx0 1 or f nx0 1 One may check that f nx0 f nxj for any other point xj on the periodic orbit so this definition makes sense see Exercise 6 at the end of this chapter Example The function f x x2 1 has a 2cycle given by 0 and 1 One checks easily that f 20 0 f 21 so this cycle is a sink In Figure 154 we show graphical iteration of f with the graph of f 2 superim posed Note that 0 and 1 are attracting fixed points for f 2 Hirsch Ch159780123820105 201229 1441 Page 334 6 334 Chapter 15 Discrete Dynamical Systems y x y fx y f 2x 1 Figure 154 Graphs of fx x2 1 and f 2 showing that 0 and 1 lie on an attracting 2cycle for f 152 Bifurcations Discrete dynamical systems undergo bifurcations when parameters are varied just as differential equations do We deal in this section with several types of bifurcations that occur for onedimensional systems Example Let fcx x2 c where c is a parameter The fixed points for this family are given by solving the equation x2 c x which yields p 1 2 1 4c 2 Thus there are no fixed points if c 14 a single fixed point at x 12 when c 14 and a pair of fixed points at p when c 14 Graphical iter ation shows that all orbits of fc tend to if c 14 When c 14 the fixed point at x 12 is neutral as is easily seen by graphical iteration See Figure 155 When c 14 we have f c p 1 1 4c 1 so p is always repelling A straightforward computation also shows that 1 f c p 1 provided 34 c 14 For these cvalues p is attracting When 34 c 14 all orbits in the interval pp tend to p though technically the orbit of p is eventually fixed since it maps directly onto p as do the orbits of certain other points in this interval when c 0 Thus as c decreases through the bifurcation value c 14 we see the birth of a single neutral fixed point which then immediately splits into two fixed points one attracting and one repelling This is an example of a saddlenode or tangent bifurcation Hirsch Ch159780123820105 201229 1441 Page 335 7 152 Bifurcations 335 c 035 c025 c 015 Figure 155 Saddlenode bifurcation for fcx x2 c at c 14 Graphically this bifurcation is essentially the same as its namesake for first order differential equations as described in Chapter 8 See Figure 155 Note that in this example at the bifurcation point the derivative at the fixed point equals 1 This is no accident for we have this theorem Theorem The Bifurcation Criterion Let fλ be a family of functions depending smoothly on the parameter λ Suppose that fλ0x0 x0 and f λ0x0 1 Then there are intervals I about x0 and J about λ0 and a smooth function p J I such that pλ0 x0 and fλpλ pλ Moreover fλ has no other fixed points in I Proof Consider the function defined by Gxλ fλx x By hypothesis Gx0λ0 0 and G x x0λ0 f λ0x0 1 0 By the Implicit Function Theorem there are intervals I about x0 and J about λ0 and a smooth function p J I such that pλ0 x0 and Gpλλ 0 for all λ J Moreover Gxλ 0 unless x pλ This concludes the proof As a consequence of this result fλ may undergo a bifurcation of fixed points only if fλ has a fixed point with derivative equal to 1 The typical bifurcation that occurs at such parameter values is the saddlenode bifurcation see Exer cises 18 and 19 at the end of this chapter However there are many other types of bifurcations of fixed points that may occur Example Let fλx λx1 x Note that fλ0 0 for all λ We have f λ0 λ so we have a possible bifurcation at λ 1 There is a sec ond fixed point for fλ at xλ λ 1λ When λ 1 xλ is negative and Hirsch Ch159780123820105 201229 1441 Page 336 8 336 Chapter 15 Discrete Dynamical Systems when λ 1 xλ is positive When λ 1 xλ coalesces with the fixed point at 0 so there is a single fixed point for f1 A computation shows that 0 is repelling and xλ is attracting if λ 1 and λ3 while the reverse is true if λ1 For this reason this type of bifurcation is known as an exchange bifurcation Example Consider the family of functions fµx µx x3 When µ 1 we have f10 0 and f 10 1 so we have the possibility for a bifurcation The fixed points are 0 and 1 µ so we have three fixed points when µ 1 but only one fixed point when µ 1 so a bifurcation does indeed occur as µ passes through 1 The only other possible bifurcation value for a onedimensional discrete system occurs when the derivative at the fixed or periodic point is equal to 1 since at these values the fixed point may change from a sink to a source or from a source to a sink At all other values of the derivative the fixed point simply remains a sink or source and there are no other periodic orbits nearby Certain portions of a periodic orbit may come close to a source but the entire orbit cannot lie close by see Exercise 7 at the end of this chapter In the case of derivative 1 at the fixed point the typical bifurcation is a perioddoubling bifurcation Example As a simple example of this type of bifurcation consider the family fλx λx near λ0 1 There is a fixed point at 0 for all λ When 1λ1 0 is an attracting fixed point and all orbits tend to 0 When λ 1 0 is repelling and all nonzero orbits tend to When λ1 0 is a neutral fixed point and all nonzero points lie on 2cycles As λ passes through 1 the type of the fixed point changes from attracting to repelling meanwhile a family of 2cycles appears Generally when a perioddoubling bifurcation occurs the 2cycles do not all exist for a single parameter value A more typical example of this bifurcation is provided by the following example Example Again consider fcx x2 c this time with c near c 34 There is a fixed point at p 1 2 1 4c 2 We have seen that f 34p 1 and that p is attracting when c is slightly larger than 34 and repelling when c is less than 34 Graphical iteration Hirsch Ch159780123820105 201229 1441 Page 337 9 153 The Discrete Logistic Model 337 c 065 c 075 c 085 Figure 156 Perioddoubling bifurcation for fcx x2 c at c 34 The fixed point is attracting for c 075 and repelling for c 075 shows that more happens as c descends through 34 We see the birth of an attracting 2cycle as well This is the perioddoubling bifurcation See Figure 156 Indeed one can easily solve for the period 2 points and check that they are attracting for 54 c 34 see Exercise 8 at the end of this chapter 153 The Discrete Logistic Model In Chapter 1 we introduced one of the simplest nonlinear firstorder differen tial equations the logistic model for population growth x ax1 x In this model we took into account the fact that there is a carrying capacity for a typical population and we saw that the resulting solutions behave quite simply All nonzero solutions tend to the ideal population Now something about this model may have bothered you way back then Populations generally are not continuous functions of time A more natural type of model would measure populations at specific times say every year or every generation Here we introduce just such a model the discrete logistic model for population growth Suppose we consider a population where members are counted each year or at other specified times Let xn denote the population at the end of year n If we assume that no overcrowding can occur then one such population model is the exponential growth model where we assume that xn1 kxn 338 Chapter 15 Discrete Dynamical Systems for some constant k0 That is the next years population is directly proportional to this years Thus we have x1 kxo x2 kx k x0 B kx2 kx Clearly x kxp so we conclude that the population explodes if k 1 becomes extinct if 0 k 1 or remains constant if k 1 This is an example of a firstorder difference equation which is an equa tion that determines x based on the value of x A secondorder difference equation would give x based on x and x2 From our point of view the successive populations are given by simply iterating the function fx kx with the seed xo A more realistic assumption about population growth is that there is a max imal population M such that if the population exceeds this amount then all resources are used up and the entire population dies out in the next year One such model that reflects these assumptions is the discrete logistic population model Here we assume that the populations obey the rule Xn Xn kxn 1 where k and M are positive parameters Note that if x M then x41 0 so the population does indeed die out in the ensuing year Rather than deal with actual population numbers we will instead let x denote the fraction of the maximal population so that 0 x 1 The logistic difference equation then becomes Xn AXn 1 Xn where A 0 is a parameter We may therefore predict the fate of the initial population xp by simply iterating the quadratic function f x Ax1 x also called the logistic map Sounds easy right Well suffice it to say that this simple quadratic iteration was only completely understood in the late 1990s thanks to the work of hundreds of mathematicians Well see why the discrete logistic model is so much more complicated than its cousin the logistic differential equation in a moment but first lets do some simple cases We will only consider the logistic map on the unit interval I We have fi 0 0 so 0 is a fixed point The fixed point is attracting in I for0 A 1 Hirsch Ch159780123820105 201229 1441 Page 339 11 153 The Discrete Logistic Model 339 and repelling thereafter The point 1 is eventually fixed since fλ1 0 There is a second fixed point xλ λ 1λ in I for λ 1 The fixed point xλ is attracting for 1λ 3 and repelling for λ 3 At λ 3 a perioddoubling bifurcation occurs see Exercise 4 at the end of this chapter For λvalues between 3 and approximately 34 the only periodic points present are the two fixed points and the 2cycle When λ 4 the situation is much more complicated Note that f λ12 0 and that 12 is the only critical point for fλ for each λ When λ 4 we have f412 1 so f 2 4 12 0 Therefore f4 maps each of the halfintervals 012 and 121 onto the entire interval I Consequently there exist points y0 012 and y1 121 such that f4yj 12 and thus f 2 4 yj 1 Therefore we have f 2 4 0y0 f 4 2 y012 I and f 2 4 12y1 f 4 2 y11 I Since the function f 2 4 is a quartic it follows that the graph of f 2 4 is as shown in Figure 157 Continuing in this fashion we find 23 subintervals of I that are mapped onto I by f 3 4 24 subintervals mapped onto I by f 4 4 and so forthWe therefore see that f4 has two fixed points in I f 2 4 has four fixed points in I f 3 4 has 23 fixed points in I and inductively f n 4 has 2n fixed points in I The fixed points for f4 occur at 0 and 34 The four fixed points for f 2 4 include these two fixed points plus a pair of periodic points of period 2 Of the eight fixed points for f 3 4 two must be the fixed points and the other six must lie on a pair of 3cycles Among the 16 fixed points for f 4 4 are two fixed points two periodic points of period 2 and 12 periodic points of period 4 Clearly a lot has changed as λ has varied from 34 to 4 12 12 12 y0 y1 y0 y1 Figure 157 Graphs of the logistic function fλx λx1 x as well as f 2 λ and f 3 λ over the interval I Hirsch Ch159780123820105 201229 1441 Page 340 12 340 Chapter 15 Discrete Dynamical Systems a b Figure 158 Orbit of the seed 0123 under f4 using a 200 iterations and b 500 iterations On the other hand if we choose a random seed in the interval I and plot the orbit of this seed under iteration of f4 using graphical iteration we rarely see any of these cycles In Figure 158 we have plotted the orbit of 0123 under iteration of f4 using 200 and 500 iterations Presumably there is something chaotic going on 154 Chaos In this section we introduce several quintessential examples of chaotic one dimensional discrete dynamical systems Recall that a subset U W is said to be dense in W if there are points in U arbitrarily close to any point in the larger set W As in the Lorenz model we say that a map f that takes an interval I αβ to itself is chaotic if 1 Periodic points of f are dense in I 2 f is transitive on I that is given any two subintervals U1 and U2 in I there is a point x0 U1 and an n 0 such that f nx0 U2 3 f has sensitive dependence in I that is there is a sensitivity constant β such that for any x0 I and any open interval U about x0 there is some seed y0 U and n 0 such that f nx0 f ny0 β It is known that the transitivity condition is equivalent to the existence of an orbit that is dense in I Clearly a dense orbit implies transitivity for 154 Chaos 341 such an orbit repeatedly visits any open subinterval in I The other direction relies on the Baire Category Theorem from analysis so we will not prove this here Curiously for maps of an interval condition 3 in the definition of chaos is redundant 8 This is somewhat surprising since the first two conditions in the definition are topological in nature while the third is a metric property it depends on the notion of distance Now we move on to discussion of several classical examples of chaotic one dimensional maps Example The Doubling Map Define the discontinuous function D 01 01 by Dx 2x mod 1 That is 2x if0x12 Dx 2x1 ifl2x1 An easy computation shows that Dx 2x mod 1 so the graph of D con sists of 2 straight lines with slope 2 each extending over the entire interval 01 See Figure 159 To see that the doubling function is chaotic on 0 1 note that D maps any interval of the form k2k 12 for k012 2 onto the interval 01 Thus the graph of D crosses the diagonal y x at some point in this interval and so there is a periodic point in any such interval Since the lengths of these intervals are 12 it follows that periodic points are dense in 0 1 Transitivity also follows since given any open interval J we may always find an interval of the form k2k 12 inside J for sufficiently large n Thus D maps J onto all of 0 1 This also proves sensitivity where we choose the sensitivity constant 12 a Figure 159 Graph of the doubling map D and its higher iterates D and D on 01 342 Chapter 15 Discrete Dynamical Systems We remark that it is possible to write down all of the periodic points for D explicitly see Exercise 5a at the end of this chapter It is also interest ing to note that if you use a computer to iterate the doubling function then it appears that all orbits are eventually fixed at 0 which of course is false See Exercise 5c at the end of this chapter for an explanation of this phenomenon Example The Tent Map Now consider a continuous cousin of the doubling map given by 2x if0x12 Tx 2x42 if12x1 T is called the tent map See Figure 1510 The fact that T is chaotic on 01 follows exactly as in the case of the doubling function using the graphs of T see exercise 15 at the end of this chapter Looking at the graphs of the tent function T and the logistic function fax 4x1 x that we discussed in Section 153 it appears that they should share many of the same properties under iteration Indeed this is the case To understand this we need to reintroduce the notion of conjugacy this time for discrete systems Suppose I and J are intervals and f I I and g J J We say that f and g are conjugate if there isa homeomorphism h I J such that h satisfies the conjugacy equation ho f goh Just as in the case of flows a conjugacy takes orbits of f to orbits of g This follows since we have hfx ghx for all xe I so h takes the nth point on the orbit of x under f to the nth point on the orbit of hx under g Similarly h takes orbits of g to orbits of f a Figure 1510 Graph of the tent map T and its higher iterates T2 and T on 01 Hirsch Ch159780123820105 201229 1441 Page 343 15 154 Chaos 343 Example Consider the logistic function f4 01 01 and the quadratic function g 22 22 given by gx x2 2 Let hx 4x 2 and note that h takes 01 to 22 Moreover we have h4x1 x hx2 2 so h satisfies the conjugacy equation and f4 and g are conjugate From the point of view of chaotic systems conjugacies are important since they map one chaotic system to another Proposition Suppose f I I and g J J are conjugate via h where both I and J are are closed intervals in R of finite length If f is chaotic on I then g is chaotic on J Proof Let U be an open subinterval of J and consider h1U I Since periodic points of f are dense in I there is a periodic point x h1U for f Say x has period n Then gnhx hf nx hx by the conjugacy equation This gives a periodic point hx for g in U and shows that periodic points of g are dense in J If U and V are open subintervals of J then h1U and h1V are open intervals in I By transitivity of f there exists x1 h1U such that f mx1 h1V for some m But then hx1U and we have gmhx1 hf mx1 V so g is transitive also For sensitivity suppose that f has sensitivity constant β Let I α0α1 We may assume that β α1 α0 For any x α0α1 β consider the function hx β hx This is a continuous function on α0α1 β which is positive Thus it has a minimum value β It follows that h takes intervals of length β in I to intervals of length at least β in J Then it is easy to check that β is a sensitivity constant for g This completes the proof It is not always possible to find conjugacies between functions with equiva lent dynamics However we can relax the requirement that the conjugacy be one to one and still salvage the preceding proposition A continuous function h that is at most n to one and that satisfies the conjugacy equation f h h g is called a semiconjugacy between g and f It is easy to check that a semi conjugacy also preserves chaotic behavior on intervals of finite length see exercise 12 at the end of this chapter A semiconjugacy need not preserve the minimal periods of cycles but it does map cycles to cycles 344 Chapter 15 Discrete Dynamical Systems Example The tent function T and the logistic function f4 are semi conjugate on the unit interval To see this let 1 hx 5 1 cos27 x Then h maps the interval 0 1 in twotoone fashion over itself except at 12 which is the only point mapped to 1 Then we compute 1 hTx 5 1 cos47x 1 1 2cos 27x 1 2 2 1cos 21x 1 1 1 1 4 cos27xcos27x 2 2 2 2 fahx Thus h is a semiconjugacy between T and fy As a remark recall that we may find arbitrarily small subintervals that are mapped onto all of 01 by T Thus f4 maps the images of these intervals under h onto all of 01 Since his continuous the images of these intervals may be chosen arbitrarily small Thus we may choose 12 as a sensitivity constant for fy as well We have proven the following proposition a Proposition The logistic function fx 4x1 x is chaotic on the unit interval O 155 Symbolic Dynamics We turn now to one of the most useful tools for analyzing chaotic sys tems symbolic dynamics We give just one example of how to use symbolic dynamics here several more are included in the next chapter Consider the logistic map f x Ax1 x where 4 4 Graphical itera tion seems to imply that almost all orbits tend to oo See Figure 1511 Of course this is not true as we have fixed points and other periodic points for this function In fact there is an unexpectedly large set called a Cantor set that is filled with chaotic behavior for this function as we shall see Unlike the case 4 4 the interval I 01 is no longer invariant when A 4 Certain orbits escape from I and then tend to oo Our goal is to understand the behavior of the nonescaping orbits Let A denote the set of 155 Symbolic Dynamics 345 ital AC firpe Ate 7 4 Figure 1511 Typical orbits for the logistic function f with 4 4 seem to tend to o after wandering around the unit interval for a while Ay Ao Ay Io Ao h a b Figure 1512 a The exit set in consists of a collection of disjoint open intervals b The intervals Io and lie to the left and right of Ap points in I with orbits that never leave I As shown in Figure 1512a there is an open interval Ap on which f 1 Thus fx 0 for any x Ag and as a consequence the orbits of all points in Ap tend to oo Note that any orbit that leaves I must first enter Ag before departing toward oo Also the orbits of the endpoints of Ag are eventually fixed at 0 so these endpoints are contained in A Now let A denote the preimage of Apo in I Aj consists of two open intervals in I one on each side of Ao All points in A are mapped into Ag by f and thus their orbits also tend to oo Again the endpoints of A are eventual fixed points Continuing we see that each of the two open intervals in A 346 Chapter 15 Discrete Dynamical Systems has as preimage a pair of disjoint intervals so there are four open intervals that consist of points where the first iteration lies in Aj and the second in Ag and so again all of these points have orbits that tend to oo Call these four intervals Ap In general let A denote the set of points in I where the nth iterate lies in Ap Ay consists on 2 disjoint open intervals in J Any point where the orbit leaves I must lie in one of the A Thus we see that CO AzI U An n0 To understand the dynamics of f on I we introduce symbolic dynamics Toward that end let Jo and J denote the left and right closed intervals respec tively in I Ap See Figure 1512b Given xp A the entire orbit of xo lies in Ip U I Thus we may associate an infinite sequence Sxp 595152 consisting of 0s and 1s to the point xp via the rule sj k ifand only if filo Ik That is we simply watch how fi xp bounces around Ip and Jj assigning a 0 or at the jth stage depending on which interval fx lies in The sequence Sxo is called the itinerary of xo Example The fixed point 0 has itinerary S0 000 The fixed point x in J has itinerary Sx 111 The point x9 1 is eventually fixed and has itinerary S1 1000 A 2cycle that hops back and forth between Ig and I has itinerary 01 or 10 where 01 denotes the infinitely repeating sequence consisting of repeated blocks 01 Let denote the set of all possible sequences of 0s and 1s A point in the space is therefore an infinite sequence of the form s s95 To visualize X we need to tell how far apart different points in X are To do this let s spsjs and t tot be points in LY A distance function or metric on is a function d ds t that satisfies 1 dst Oand ds t 0 if and only ifs t 2 dst dts 3 The triangle inequality ds u ds t dt u Since is not naturally a subset of a Euclidean space we do not have a Euclidean distance to use on Thus we must concoct one of our own Here is the distance function we choose si til 1 fy ds t S ie i0 155 Symbolic Dynamics 347 Note that this infinite series converges The numerators in this series are always either 0 or 1 and so this series converges by comparison to the geometric series Co 1 1 dsth 2 Os 2 112 10 It is straightforward to check that this choice of d satisfies the three require ments to be a distance function see Exercise 13 at the end of this chapter Although this distance function may look a little complicated at first it is often easy to compute a Example D 7 co 01 co 1 1 d0 Vo SH LE 2 2 d0110 Do 5 2 a1 7 1 1 4 3 d011 Vixo ai 1144 3 a The importance of having a distance function on is that we now know when points are close together or far apart In particular we have this proposition Proposition Suppose s s95152 and t ftih ED 1 Ifs forj0n then dst 12 2 Conversely if dst 12 then s t forj0n Proof In case 1 we have SIsi sil yo Isic 1 91 1 4 t ta i0 in1 1wol 0 gntl i i0 1 If on the other hand dst 12 then we must have s 4 for any j n for otherwise ds t s ti2 12 12 O Now that we have a notion of closeness in we are ready to prove the main theorem of this chapter 348 Chapter 15 Discrete Dynamical Systems Theorem The itinerary function S A is a homeomorphism provided A Proof Actually we will only prove this in case A is sufficiently large so that If x K 1 for some K and for all x Ip UI The reader may check that 2 24 5 suffices for this For the more complicated proof in case 4 4 24 5 see Kraft 25 We first show that S is one to one Let xy A and suppose Sx Sy Then for each n fx and fy lie on the same side of 12 This implies that f is monotone on the interval between fx and fy Consequently all points in this interval remain in Jo U I when we apply f Now f K 1 at all points in this interval so as in Section 151 each iteration of f expands this interval by a factor of K Thus the distance between fx and fy grows without bound and so these two points must eventually lie on opposite sides of Ao This contradicts the fact that they have the same itinerary To see that S is onto we first introduce the following notation Let J C I be a closed interval Let fr x eT fi J so that f J is the preimage of J under f A glance at the graph of f when i 4shows that if J C I isa closed interval then f J consists of two closed subintervals one in Jp and one in Jj Now let s sos 52 We must produce x A with Sx s To that end we define Tyo 1008 AX 1x Igy fix I x Ls 1 Nf Us 0 Us We claim that the I form a nested sequence of nonempty closed intervals Note that T5051 Is Nf Usy5 By induction we may assume that I is a nonempty subinterval so that by the preceding observation fi Jss consists of two closed intervals one in Ig and one in J Thus I Nf Uss is a single closed interval These intervals are nested because T59 sp Ts9 051 NA Us C Therefore we conclude that CO a T5551 Sn n0 156 The Shift Map 349 is nonempty Note that if x Nysolsyss then x Is fax Is and so on Thus Sx sos This proves that S is onto Observe that Nyo1s9s5 Consists of a unique point This follows immedi ately from the fact that S is one to one In particular we have that the diameter of I5ss tends to 0 as n oo To prove continuity of S we choose xe A and suppose that Sx spsS2 Let 0 Pick n so that 12 Consider the closed subin tervals Ijt defined before for all possible combinations ft t These subintervals are all disjoint and A is contained in their union There are 2 such subintervals and I5s is one of them Thus we may choose 6 such that xy 6 and ye A implies that y I5 Therefore Sy agrees with Sx in the first n 1 terms So by the previous proposition we have 1 dSx Sy an This proves the continuity of S It is easy to check that S is also continuous Thus S is a homeomorphism 156 The Shift Map We now construct a map 0 X with the following properties 1 o is chaotic 2 o is conjugate to f on A 3 o is completely understandable from a dynamical systems point of view The meaning of this last item will become clear as we proceed We define the shift mapa by O 595152 51 5253 That is the shift map simply drops the first digit in each sequence in X Note that o is a twotoone map onto This follows since if sos 52 X then we have 0 0595152 O 1595152 5951 S2 Proposition The shift mapo is continuous Proof Let s spsjs2 and let 0 Choose n so that 12 Let 6 12 Suppose that ds t 5 where t tot h Then we have 5 tfori0n1 350 Chapter 15 Discrete Dynamical Systems Now ot 5152 SnSn41tn42 so that dosot 12 This proves that o is continuous O Note that we can easily write down all of the periodic points of any period for the shift map Indeed the fixed points are 0 and 1 The 2 cycles are 01 and 10 In general the periodic points of period n are given by repeating sequences that consist of repeated blocks of length n 5951 Note how much nicer o is compared to f just try to write down explicitly all of the peri odic points of period n for f someday They are there and we know roughly where they are because we have the following Theorem The itinerary function S A provides a conjugacy between fy and the shift map o Proof In the previous section we showed that S is a homeomorphism Thus it suffices to show that Sof a 0S To that end let xA and suppose that Sx sosis Then we have xo Is fx0 Is5 fe x0 eI and so forth But then the fact that f xo I fe 0 I and so on says that Sfi 515253 80 SC 0 a Sx which is what we wanted to prove Now not only can we write down all periodic points for 0 but we can in fact write down explicitly a point in that has a dense orbit Here is such a point s 01 00011011000001 eee eee 1blocks 2blocks 3blocks Ablocks The sequence s is constructed by successively listing all possible blocks of 0s and 1s of length 1 length 2 length 3 and so forth Clearly some iterate of o applied to s yields a sequence that agrees with any given sequence in an arbitrarily large number of initial places That is given t fotih X we may find k so that the sequence os begins to ta Snt1 Sn2 and so dost 12 Thus the orbit of s comes arbitrarily close to every point in This proves that the orbit of s under o is dense in and so a is transitive Note that we may construct a multitude of other points with dense orbits in by just rearranging the blocks in the sequence s Again think about how 157 The Cantor MiddleThirds Set 351 difficult it would be to identify a seed with an orbit under a quadratic function such as f4 that is dense in 01 This is what we meant when we said earlier that the dynamics of o are completely understandable The shift map also has sensitive dependence Indeed we may choose the sensitivity constant to be 2 which is the largest possible distance between two points in The reason for this is if s sos 52 and 3 denotes not sj that is if s 0 then 3 1 or if s1 then 5 0 then the point s SoS1 SuSn15n42 satisfies 1 dss 12 but 2 dosot1s 2 As a consequence we have proved this theorem Theorem The shift map o is chaotic on X and so by the conjugacy in the previous theorem the logistic map fy is chaotic on A when k 4 Thus symbolic dynamics provides us with a computable model for the dynamics of f on the set A despite the fact that f is chaotic on A 157 The Cantor MiddleThirds Set We mentioned earlier that A was an example of a Cantor set Here we describe the simplest example of such a set the Cantor middlethirds set C As we shall see this set has some unexpectedly interesting properties To define C we begin with the closed unit intervalthat is I 01 The rule is each time we see a closed interval we remove its open middle third Thus at the first stage we remove 1323 leaving two closed intervals 013 and 231 We now repeat this step by removing the middlethirds of these two intervals What we are left with is four closed intervals 01929 132379 and 891 Removing the open middlethirds of these intervals leaves us with 2 closed intervals each of length 13 Continuing in this fashion at the nth stage we are left with 2 closed inter vals each of length 13 The Cantor middlethirds set C is what is left when we take this process to the limit as n oo Note how similar this con struction is to that of A in Section 155 In fact it can be proved that A is homeomorphic to C see exercises 16 and 17 at the end of this chapter What points in I are left in C after removing all of these open intervals Certainly 0 and 1 remain in C as do the endpoints 13 and 23 of the first removed interval Indeed each endpoint of a removed open interval lies in C for such a point never lies in an open middlethird subinterval At first glance Hirsch Ch159780123820105 201229 1441 Page 352 24 352 Chapter 15 Discrete Dynamical Systems it appears that these are the only points in the Cantor set but in fact that is far from the truth Indeed most points in C are not endpoints To see this we attach an address to each point in C The address will be an infinite string of Ls or Rs determined as follows At each stage of the construction our point lies in one of two small closed intervals one to the left of the removed open interval or one to its right So at the nth stage we may assign an L or R to the point depending on its location left or right of the interval removed at that stage For example we associate LLL with 0 and RRR with 1 The endpoints 13 and 23 have addresses LRRR and RLLL respectively At the next stage 19 has address LLRRR since 19 lies in 013 and 019 at the first two stages but then always lies in the right interval Similarly 29 has address LRLLL while 79 and 89 have addresses RLRRR and RRLLL Notice what happens at each endpoint of C As the previous examples indi cate the address of an endpoint always ends in an infinite string of all Ls or all Rs But there are plenty of other possible addresses for points in C For example there is a point with address LRLRLR This point lies in 013 2913 29727 2081727 Note that this point lies in the nested intersection of closed intervals of length 13n for each n and it is the unique such point that does so This shows that most points in C are not endpoints for the typical address will not end in all Ls or all Rs We can actually say quite a bit more The Cantor middlethirds set contains uncountably many points Recall that an infinite set is countable if it can be put in onetoone correspondence with the natural numbers otherwise the set is uncountable Proposition The Cantor middlethirds set is uncountable Proof Suppose that C is countable This means that we can pair each point in C with a natural number in some fashion say as 1 LLLLL 2 RRRR 3 LRLR 4 RLRL 5 LRRLRR and so forth But now consider the address where the first entry is the opposite of the first entry of sequence 1 and the second entry is the opposite of the 157 The Cantor MiddleThirds Set 353 second entry of sequence 2 and so forth This is a new sequence of Ls and Rs which in the preceding example began with RLRRL Thus we have created a sequence of Ls and Rs that disagrees in the nth spot with the nth sequence on our list This sequence is therefore not on our list and so we have failed in our construction of a onetoone correspondence with the natural numbers This contradiction establishes the result L We can actually determine the points in the Cantor middlethirds set in a more familiar way To do this we change the address of a point in C from a sequence of Ls and Rs to a sequence of 0s and 2s that is we replace each L with a 0 and each R with a 2 To determine the numerical value of a point x C we approach x from below by starting at 0 and moving s3 units to the right for each n where s 0 or 2 depending on the nth digit in the address forn123 For example 1 has address RRR or 222 so 1 is given by 2 2 2 241 2 1 44 1 3 32 38 3 5 sn Similarly 13 has address LRRR or 0222 which yields 0 2 2 241 2321 to442 t 3 32 38 9 d 3 92 3 Finally the point with address LRLRLR or 020202 is 0 2 0 2 21 2 1 1 staotatat Has G4I177 7 3 32 3334 5 a5 5 i 4 Note that this is one of the nonendpoints in C referred to earlier The astute reader will recognize that the address of a point x in C with 0s and 2s gives the ternary expansion of x A point x I has ternary expansion ay ajza3 if 1 ye i1 where each gq is either 0 1 or 2 Thus we see that points in the Cantor middle thirds set have ternary expansions that may be written with no 1s among the digits We should be a little careful here The ternary expansion of 13 is 1000 However 13 also has ternary expansion 0222 as we saw before So 13 may 354 Chapter 15 Discrete Dynamical Systems be written in ternary form in a way that contains no 1s In fact every endpoint in C has a similar pair of ternary representations one of which contains no s We have shown that C contains uncountably many points but we can say even more as shown in the following proposition Proposition The Cantor middlethirds set contains as many points as the interval 01 Proof C consists of all points where the ternary expansion agq a2 contains only Os or 2s Take this expansion and change each 2 to a 1 and then think of this string as a binary expansion We get every possible binary expansion in this manner We have therefore made a correspondence at most two to one between the points in C and the points in 01 since every such point has a binary expansion O Finally we note this proposition Proposition The Cantor middlethirds set has length 0 Proof We compute the length of C by adding up the lengths of the intervals removed at each stage to determine the length of the complement of C These removed intervals have successive lengths 13 29 427 and so the length of I Cis 12 4 12 statateHs 1 3 9 27 3 5 LU n0 This fact may come as no surprise since C consists of a scatter of points But now consider the Cantor middlefifths set obtained by removing the open middlefifth of each closed interval in similar fashion to the construc tion of C The length of this set is nonzero yet it is homeomorphic to C These Cantor sets have as we said earlier unexpectedly interesting prop erties And remember the set A on which fy is chaotic is just this kind of object 158 Exploration Cubic Chaos In this exploration you will investigate the behavior of the discrete dynamical system given by the family of cubic functions fx Axx You should attempt to prove rigorously everything outlined in the following 159 Exploration The Orbit Diagram 355 1 Describe the dynamics of this family of functions for all A 1 2 Describe the bifurcation that occurs at A 1 Hint Note that f is an odd function In particular what happens when the graph of f crosses the line y x 3 Describe the dynamics of f when 1 A 1 4 Describe the bifurcation that occurs at A 1 5 Find aAvalue A for which f has a pair of invariant intervals 0 x on each of which the behavior of f mimics that of the logistic function 4x1 x 6 Describe the change in dynamics that will occur when A increases through 4 7 Describe the dynamics of f when A is very large Describe the set of points A with orbits that do not escape to oo in this case 8 Use symbolic dynamics to set up a sequence space and a corresponding shift map for A large Prove that f is chaotic on Aj 9 Find the parameter value A 4 above which the results of the previous two investigations hold true 10 Describe the bifurcation that occurs as A increases through A 159 Exploration The Orbit Diagram Unlike the previous exploration this exploration is primarily experimental It is designed to acquaint you with the rich dynamics of the logsitic family as the parameter increases from 0 to 4 Using a computer and whatever software seems appropriate construct the orbit diagram for the logistic family f x Ax1 x as follows Choose N equally spaced Avalues AA2AN in the interval 0 Aj 4 For example let N 800 and set Aj 0005 For each Ajp compute the orbit of 05 under f and plot this orbit as follows Let the horizontal axis be the Aaxis and let the vertical axis be the xaxis Over each i plot the points Oph 05 for say 50 k 250 That is com pute the first 250 points on the orbit of 05 under f but display only the last 200 points on the vertical line over A 4 Effectively you are displaying the fate of the orbit of 05 in this way You will need to maginfy certain portions of this diagram one such magni fication is displayed in Figure 1513 where we have displayed only that portion of the orbit diagram for A in the interval 3 A 4 1 The region bounded by 0 357 is called the period 1 win dow Describe what you see as A increases in this window What type of bifurcations occur 2 Near the bifurcations in the previous question you sometimes see a smear of points What causes this Hirsch Ch159780123820105 201229 1441 Page 356 28 356 Chapter 15 Discrete Dynamical Systems 3 x 4 λ 357 383 Figure 1513 Orbit diagram for the logistic family with 3 λ 4 3 Observe the period 3 window bounded approximately by 3828 λ 3857 Investigate the bifurcation that gives rise to this window as λ increases 4 There are many other period n windows named for the least period of the cycle in that window Discuss any pattern you can find in how these windows are arranged as λ increases In particular if you magnify por tions between the period 1 and period 3 windows how are the larger windows in each successive enlargement arranged 5 You observe darker curves in this orbit diagram What are these Why does this happen E X E R C I S E S 1 Find all periodic points for each of the following maps and classify them as attracting repelling or neither a Qx x x2 b Qx 2x x2 c Cx x3 1 9x d Cx x3 x e Sx 1 2 sinx f Sx sinx g Ex ex1 h Ex ex i Ax arctanx j Ax π 4 arctanx 2 Discuss the bifurcations that occur in the following families of maps at the indicated parameter value Exercises 357 a Syx Asinx A1 b Cyx x b 1 Hint Exploit the fact that C is an odd function c Gx xsinxv v1 d Ex Ae A1e e Ex Ae Ae f Agx Aarctanx A1 g AxAarctanx A1 3 Consider the linear maps fx kx Show that there are four open sets of parameters for which the behavior of orbits of f is similar Describe what happens in the exceptional cases 4 For the function f x Ax1 x defined on R a Describe the bifurcations that occur at A 1 and A 3 b Find all period 2 points c Describe the bifurcation that occurs at A 175 5 For the doubling map D on 0 1 a List all periodic points explicitly b List all points with orbits that end up landing on 0 and are thereby eventually fixed c Letx 01 and suppose that x is given in binary form as aga q where each qj is either 0 or 1 First give a formula for the binary representation of Dx Then explain why this causes orbits of D generated by a computer to end up eventually fixed at 0 6 Show that if x lies on a cycle of period n then n1 f x0 fi i0 Conclude that F x0 F forj1n1 7 Prove that if f has a fixed point at xo with fio xo 1 then there is an interval I about xp and an interval J about Ao such that if A J then f has a unique fixed source in J and no other orbits that lie entirely in I 8 Verify that the family fx x c undergoes a perioddoubling bifur cation at c 34 by a Computing explicitly the period 2 orbit b Showing that this orbit is attracting for 54 c 34 358 Chapter 15 Discrete Dynamical Systems 9 Show that the family fx x c undergoes a second perioddoubling bifurcation at c 54 by using the graphs of f and f 10 Find an example of a bifurcation in which more than three fixed points are born 11 Prove that f3x 3x1 x on I is conjugate to fx x 34ona certain interval in R Determine this interval 12 Suppose fg 01 01 and that there is a semiconjugacy from f to g Suppose that f is chaotic Prove that g is also chaotic on 0 1 13 Prove that the function ds t on satisfies the three properties required for d to be a distance function or metric 14 Identify the points in the Cantor middlethirds set C with the address a LLRLLRLLR b LRRLLRRLLIRRL 15 Consider the tent map 2x if0x12 Tx Pete if12x1 Prove that T is chaotic on 01 16 Consider a different tent function defined on all of R by 3x ifx 12 Ta ae if 12 x Identify the set of points A with orbits that do not go to oo What can you say about the dynamics on this set 17 Use the results of the previous exercise to show that the set A in Section 155 is homeomorphic to the Cantor middlethirds set 18 Prove the following saddlenode bifurcation theorem Suppose that fi depends smoothly on the parameter A and satisfies a fig x0 x0 b fi 1 c fi xo 40 d oh 0 o 4 Then there is an interval I about x and a smooth function w I R satisfying 4xo Ao and such that fu OD Moreover j4xo 0 and ju xo 4 0 Hint Apply the Implicit Function Theorem to GxA fx x at x9A0 Hirsch Ch159780123820105 201229 1441 Page 359 31 Exercises 359 y y gy gy Figure 1514 Graph of the onedimensional function g on y y 19 Discuss why the saddlenode bifurcation is the typical bifurcation involving only fixed points 20 Recall that comprehending the behavior of the Lorenz system in Chap ter 14 could be reduced to understanding the dynamics of a certain onedimensional function g on an interval yy the graph is shown in Figure 1514 Recall also that gy 1 for all y 0 and that g is undefined at 0 Suppose now that g3y 0 as displayed in this graph By symmetry we also have g3y 0 Let I0 y0 and I1 0y and define the usual itinerary map on yy a Describe the set of possible itineraries under g b What are the possible periodic points for g c Prove that g is chaotic on yy Hirsch Ch169780123820105 201222 1253 Page 361 1 16 Homoclinic Phenomena In this chapter we investigate several other threedimensional systems of dif ferential equations that display chaotic behavior These systems include the Shilnikov system and the double scroll attractor As with the Lorenz sys tem our principal means of studying these systems involves reducing them to lowerdimensional discrete dynamical systems and then invoking sym bolic dynamics In these cases the discrete system is a planar map called the horseshoe map This was one of the first chaotic systems to be analyzed completely 161 The Shilnikov System In this section we investigate the behavior of a nonlinear system of differential equations that possesses a homoclinic solution to an equilibrium point that is a spiral saddle Although we deal primarily with a model system here the work of Shilnikov and others 6 40 41 shows that the phenomena described in the following hold in many actual systems of differential equations Indeed in the exploration at the end of this chapter we investigate the system of differential equations governing the Chua circuit which for certain parameter values has a pair of such homoclinic solutions For this example we do not specify the full system of differential equa tions Rather we first set up a linear system of differential equations Differential Equations Dynamical Systems and an Introduction to Chaos DOI 101016B9780123820105000166 c 2013 Elsevier Inc All rights reserved 361 Hirsch Ch169780123820105 201222 1253 Page 362 2 362 Chapter 16 Homoclinic Phenomena in a certain cylindrical neighborhood of the origin This system has a twodimensional stable surface in which solutions spiral toward the origin and a onedimensional unstable curve We then make the simple but crucial dynamical assumption that one of the two branches of the unstable curve is a homoclinic solution and thus eventually enters the stable surface We do not write down a specific differential equation having this behavior Although it is possible to do so having the equations is not particularly useful for understanding the global dynamics of the system In fact the phe nomena we study here depend only on the qualitative properties of the linear system described previously a key inequality involving the eigenvalues of this linear system and the homoclinic assumption The first portion of the system is defined in the cylindrical region S of R3 given by x2 y2 1 and z 1 In this region consider the linear system X 1 1 0 1 1 0 0 0 2 X The associated eigenvalues are 1 i and 2 Using the results of Chapter 6 the flow φt of this system is easily derived xt x0et cost y0et sint yt x0et sint y0et cost zt z0e2t Using polar coordinates in the xyplane solutions in S are given more succinctly by rt r0et θt θ0 t zt z0e2t This system has a twodimensional stable plane the xyplane and a pair of unstable curves ζ lying on the positive and negative zaxis respectively There is incidentally nothing special about our choice of eigenvalues for this system Everything that follows works fine for eigenvalues α iβ and λ where α 0β 0 and λ 0 subject only to the important condition that λ α The boundary of S consists of three pieces the upper and lower disks D given by z 1 r 1 and the cylindrical boundary C given by r 1 z 1 The stable plane meets C along the circle z 0 and divides C 161 The Shilnikov System 363 into two pieces the upper and lower halves given by Ct and C7 on which z 0 and z 0 respectively We may parametrize D by r and 6 and C by 6 and z We will concentrate in this section on C Any solution of this system that starts in Ct must eventually exit from S through Dt Thus we can define a map wy Ct D given by following solution curves that start in C until they first meet Dt Given 69 2 CT let t T 2 denote the time it takes for the solution through 2Z to make the transit to Dt We compute immediately using zt ze that tT logz Therefore 1 r JZ Wi logZ J Z 1 1 For simplicity we will regard yy as a map from the 69 z cylinder to the 7101 plane Note that a vertical line given by 6 6 in Ct is mapped by Wy to the spiral ay fH 6 log V which spirals down to the point r 0 in D since log o0 as a 0 To define the second piece of the system we assume that the branch of the unstable curve leaving the origin through Dt is a homoclinic solu tion That is eventually returns to the stable plane See Figure 161 We assume that first meets the cylinder C at the point r10 0z0 More precisely we assume that there is a time such that 06 1 100 in r6and zcoordinates ct Figure 161 Homoclinic orbit 364 Chapter 16 Homoclinic Phenomena Therefore we may define a second map wz by following solutions beginning near r 0 in D until they reach C We will assume that 2 is in fact defined on all of D In Cartesian coordinates on Dt we assume that 2 takes x y D to 6121 C via the rule x A y2 mG 2 ca In polar coordinates yy is given by 6 rsin 2 Z rcos2 Of course this is a major assumption since writing down such a map for a particular nonlinear system would be virtually impossible Now the composition yf yy defines a Poincaré map on C The map wy is defined on C and takes values in Dt and then y takes values in C We have Ct C where 4 fasin é log 2 Zi 5 cos 6 logZ J See Figure 162 As in the Lorenz system we have now reduced the study of the flow of this threedimensional system to the study of a planar discrete dynamical system As we will see in the next section this type of mapping has incredibly rich dynamics that may be partially analyzed using symbolic dynamics For a oF Dt a AN Figure 162 Map yo Dt C 161 The Shilnikov System 365 little taste of what is to come we content ourselves here with just finding the fixed points of To do this we need to solve 1 0 5 V2 sin 60 log20 1 n 7 V2 008 0 logZ These equations look pretty formidable However if we square both equations and add them we find 0 72 0 3 so that 1 2 09 74 H 425 2 which is well defined provided that 0 z 14 Substituting this expression into the preceding second equation we find that 1 2 cos 5 425 log z 2 Now the term 2Z 42 tends to zero as z 0 but logz oo Therefore the graph of the left side of this equation oscillates infinitely many times between 1 as z 0 Thus there must be infinitely many places where this graph meets that of 2zo and so there are infinitely many solutions of this equation This in turn yields infinitely many fixed points for Each of these fixed points then corresponds to a periodic solution of the system that starts in Ct winds a number of times around the zaxis near the origin and then travels around close to the homoclinic orbit until closing up when it returns to CT See Figure 163 We now describe the geometry of this map in the next section we use these ideas to investigate the dynamics of a simplified version of it First note that the circles z a in Ct are mapped by y to circles r a centered at r 0 in Dt since wv A fn Ja Na O 0o loga Then w maps these circles to circles of radius a2 centered at 0 z 0 in C To be precise these are circles in the zplane in the cylinder these 366 Chapter 16 Homoclinic Phenomena c J SE SS Figure 163 A periodic solution y near the homoclinic solution circles are bent In particular we see that onehalf of the domain C is mapped into the lower part of the cylinder C and therefore no longer comes into play Let H denote the halfdisk CT Nz 0 H has center at 6 z 0 and radius 12 The preimage of H in C consists of all points 9 20 with images that satisfy z 0 so that we must have 1 ZyA 5 v0 Cos 4 logZ 0 It follows that the preimage of H is given by H 0520 12 00 log2 12 where 0 z 1 This is a region bounded by the two curves 4 logzo 2 2 each of which spirals downward in C toward the circle z 0 This follows since as z 0 we must have 0 oo More generally consider the curves 4 given by 6 logzo for 12 a 72 These curves fill the preimage H and each spirals around C just as the boundary curves do Now we have J sina Plq Ncosa 161 The Shilnikov System 367 Dt a 1H Figure 164 Halfdisk H and its preimage in C so maps each fy to a ray that emanates from 06 z0 in Ct and is parametrized by zo In particular maps each of the boundary curves linj2tozO0inC Since the curves 42 spiral down toward the circle z 0 in C it follows that H meets H in infinitely many strips that are nearly horizontal close to z 0 See Figure 164 We denote these strips by Hy for k sufficiently large More precisely let Hy denote the component of HH for which we have 2k 69 2km 1 wn 29 2 The top boundary of Hy is given by a portion of the spiral 72 and the bottom boundary by a piece of 2 Using the fact that 1 1 5 90 log V2 5 we find that if 69 2 Hy then 4k 1 1 m 26 2logz 1 20 4k1x 1 from which we conclude that exp4k 1 1 a exp4k 1a 1 Now consider the image of Hy under The upper and lower boundaries of Hx are mapped to z0 The curves 4 Hx are mapped to arcs in rays 368 Chapter 16 Homoclinic Phenomena LESS v H PT Figure 165 The image of Hy isa horseshoe that crosses H twice incr emanating from 6 z 0 These rays are given as before by z sina 27 om In particular the curve 9 is mapped to the vertical line 6 0 z z2 Using the preceding estimate of the size of z in Hx one checks easily that the image of 9 lies completely above Hy when k 2 Therefore the image of H is a horseshoeshaped region that crosses Hy twice as shown in Figure 165 In particular if k is large the curves 1 H meet the horsehoe H in nearly horizontal subarcs Such a map is called a horseshoe map in the next section we discuss the prototype of such a function 162 The Horseshoe Map Symbolic dynamics which play such a crucial role in our understanding of the onedimensional logistic map can also be used to study higherdimensional phenomena In this section we describe an important example in R called the horseshoe map 43 We will see that this map has much in common with the Poincaré map described in the previous section To define the horseshoe map we consider a region D consisting of three components a central square S with sides of length 1 together with two semicircles D and Dy at the top and bottom D is shaped like a stadium The horseshoe map F takes D inside itself according to the following pre scription First F linearly contracts S in the horizontal direction by a factor 5 12 and expands it in the vertical direction by a factor of 16 so that S is long and thin then F curls S back inside D in a horseshoeshaped figure as shown in Figure 166 We stipulate that F maps S linearly onto the two vertical legs of the horseshoe We assume that the semicircular regions D and D2 are mapped inside D as shown We also assume that there is a fixed point in D that attracts all other Hirsch Ch169780123820105 201222 1253 Page 369 9 162 The Horseshoe Map 369 D1 D2 S F FD1 FD2 FS Figure 166 First iterate of the horseshoe map V0 V1 H1 H0 Figure 167 Rectangles H0 and H1 and their images V0 and V1 orbits in D1 Note that FD D and that F is one to one However since F is not onto the inverse of F is not globally defined The remainder of this section is devoted to the study of the dynamics of F in D Note first that the preimage of S consists of two horizontal rectangles H0 and H1 that are mapped linearly onto the two vertical components V0 and V1 of FS S The width of V0 and V1 is therefore δ as is the height of H0 and H1 See Figure 167 By linearity of F H0 V0 and F H1 V1 we know that F takes horizontal and vertical lines in Hj to horizontal and vertical lines in Vj for j 12 As a consequence if both h and Fh are horizontal line segments in S then the length of Fh is δ times the length of h Similarly if v 370 Chapter 16 Homoclinic Phenomena is a vertical line segment in S with an image that also lies in S then the length of Fv is 16 times the length of v We now describe the forward orbit of each point X D Recall that the for ward orbit of X is given by FXn 0 By assumption F has a unique fixed point Xp in D and limyoFX Xp for all X D Also since FD2 C Dj all forward orbits in D behave likewise Similarly if X S but FEX S for some k 0 then we must have that FEX D UD so that FX Xo as n oo as well Consequently we understand the forward orbits of any X D with an orbit that enters Dj so it suffices to consider the set of points with forward orbits that never enter D and so lie completely in S Let Ay X SFX S for n012 We claim that A has properties similar to the corresponding set for the one dimensional logistic map described in Chapter 15 If X A then FX S and so we must have that either X Hy or X Hy for all other points in S are mapped into D or D3 Since FX Sas well we must also have FX Hy U Hj so that X FHpy UH Here FW denotes the preimage of a set W lying in D In general since FX S we have X FHo U H Thus we may write CO A FHy UH n0 Now if H is any horizontal strip connecting the left and right boundaries of S with height h then FH consists of a pair of narrower horizontal strips of height 5h one in each of Hp and Hj The images under F of these narrower strips are given by HM Vp and HN Vj In particular if H Hj F7H isa pair of horizontal strips each of height 5 with one in Ho and the other in Hj Similarly FFH FH consists of four horizontal strips each of height 6 and FH consists of 2 horizontal strips of width 5 Thus the same procedure we used in Section 155 shows that A is a Cantor set of line segments each extending horizontally across S The main difference between the horseshoe and the logistic map is that in the horseshoe case there is a single backward orbit rather than infinitely many such orbits The backward orbit of X S is FXn 12 provided FX is defined and in D If FX is not defined then the backward orbit of X terminates Let A denote the set of points with a backward orbit defined for all n and that lies entirely in S If X A then we have FX S for all n 1 which implies that X FS for all n 1 As before this forces 162 The Horseshoe Map 371 X FHo U for all n 1 Therefore we may also write Co A FHo U H n1 On the other hand if X S and FX S then we must have X FSNS so that X Vo or X Vj Similarly if FX S as well then X FS NS which consists of four narrower vertical strips two in Vo and two in V In Figure 168 we show how the image of D under F Arguing entirely analogously as earlier it is easy to check that A consists of a Cantor set of vertical lines Let A Ay a AL be the intersection of these two sets Any point in A has its entire orbit both the backward and forward orbit in S To introduce symbolic dynamics into this picture we will assign a doubly infinite sequence of 0s and 1s to each point in A If X A then from the TZ A Figure 168 Second iterate of the horseshoe map 372 Chapter 16 Homoclinic Phenomena preceding we have Co Xe FHoUH nOCOo Thus we associate with X the itinerary SX S2S15S0S1 52 ds where s 0 or 1 and s k if and only if FiX Hy This then provides us with the symbolic dynamics on A Let Xz denote the set of all doubly infinite sequences of 0s and 1s 2 s s25159512sj 0 or 1 We impose a distance function on X by defining si til 1 ds t Xu sr 1CO as in Section 155 Thus two sequences in XZ are close if they agree in all k spots where k n for some large n Define the twosided shift map o by OS281595152 S2S159SS52 That is o simply shifts each sequence in X2 one unit to the left equivalently o shifts the decimal point one unit to the right Unlike our previous one sided shift map this map has an inverse Clearly shifting one unit to the right gives this inverse It is easy to check that o is a homeomorphism on Y see Exercise 2 at the end of this chapter The shift map is now the model for the restriction of F to A Indeed the itinerary map S gives a conjugacy between F on A and o on Xp For if XeA and SX s2s1595152 then we have X H FX H FlxXe H and so forth But then we have FX H FFX H X F7FXe H and so forth This tells us that the itinerary of FX is 159S1 So that SFx 5159S 52 a SX which is the conjugacy equation We leave the proof of the fact that S is a homeomorphism to the reader see Exercise 3 at the end of this chapter 162 The Horseshoe Map 373 All of the properties that held for the old onesided shift from the previous chapter hold for the twosided shift o as well For example there are precisely 2 periodic points of period n for o and there is a dense orbit for 0 Moreover F is chaotic on A see Exercises 4 and 5 at the end of this chapter But there are new phenomena present as well We say that two points X and X are forward asymptotic if FX FX2 D for all n 0 and lim FX FX2 0 nao X and X2 are backward asymptotic if their backward orbits are defined for all nand the preceding limit is zero as n oo Intuitively two points in D are forward asymptotic if their orbits approach each other as n oo Note that any point that leaves S under forward iteration of F is forward asymptotic to the fixed point Xp D Also if X and X lie on the same horizontal line in Ax then X and X2 are forward asymptotic If X and X lie on the same vertical line in A then they are backward asymptotic We define the stable set of X to be WX ZFZ FX 0 as n oo The unstable set of X is given by WX ZFX FZ 0as n oo Equivalently a point Z lies in WX if X and Z are forward asymptotic As before any point in S where the orbit leaves S under forward iteration of the horseshoe map lies in the stable set of the fixed point in D The stable and unstable sets of points in A are more complicated For exam ple consider the fixed point X which lies in Hp and therefore has the itinerary 00000 Any point that lies on the horizontal segment through X lies in WX But there are many other points in this stable set Suppose the point Y eventually maps into Then there is an integer n such that FY X 1 Thus JFEY X 6 and it follows that Y WX Thus the union of horizontal intervals given by Fé for k 123 all lie in WX The reader may easily check that there are 2 such intervals Since FD C D the unstable set of the fixed point X assumes a somewhat different form The vertical line segment through X in D clearly lies in WX As before all of the forward images of also lie in D One may easily check that F k isa snakelike curve in D that cuts vertically across 374 Chapter 16 Homoclinic Phenomena x Xo 4 Figure 169 Unstable set for X in D S exactly 2 times See Figure 169 The union of these forward images is then a very complicated curve that passes through S infinitely often The closure of this curve in fact contains all points in A as well as all of their unstable curves see Exercise 12 at the end of this chapter The stable and unstable sets in A are easy to describe on the shift level Let s 5 55 159 5755 Do Clearly if t is a sequence with entries that agree with those of s to the right of some entry then t Ws The converse of this is also true as is shown in Exercise 6 at the end of this chapter A natural question that arises is the relationship between the set A for the onedimensional logistic map and the corresponding A for the horseshoe map Intuitively it may appear that the A for the horseshoe has many more points However both As are actually homeomorphic This is best seen on the shift level Let denote the set of onesided sequences of 0s and 1s and the set of twosided such sequences Define a map D YD by PD 5951 52 oe 5 351S9S254 163 The Double Scroll Attractor 375 It is easy to check that is a homeomorphism between LZ and Lp see exercise 11 at the end of this chapter Finally to return to the subject of Section 161 note that the return map investigated in that section consists of infinitely many pieces that resemble the horseshoe map of this section Of course the horseshoe map here was effectively linear in the region where the map was chaotic so the results in this section do not go over immediately to prove that the return maps near the homoclinic orbit have similar properties This can be done however the techniques for doing so involving a generalized notion of hyperbolicity are beyond the scope of this book See Devaney 13 or Robinson 37 for details 163 The Double Scroll Attractor In this section we continue the study of behavior near homoclinic solutions in a threedimensional system We return to the system described in Section 161 only now we assume that the vector field is skewsymmetric about the origin In particular this means that both branches of the unstable curve at the origin now yield homoclinic solutions as shown in Figure 1610 We assume that c meets the cylinder C given by r 1 z 1 at the point 9 0 z0 so that meets the cylinder at the diametrically opposite point 6 z z0 As in Section 161 we have a Poincaré map defined on the cylinder C This time however we cannot disregard solutions that reach C in the region z 0 now these solutions follow the second homoclinic solution until they reintersect C Thus is defined on all of C z 0 As before the Poincaré map defined in the top half of the cylinder C is given by 0 8 iBsin toea 20 41 5 cos 0 logZ0 Invoking the symmetry a computation shows that on C is given by oo 2 5vasin 60 log2 ZI 1 x cos 0 logz J where z 0 and 6 is arbitrary Thus C is the disk of radius 12 centered at 9 0 z0 while C is a similar disk centered at 9 2z0 The centers of these disks do not lie in the image as these are the points where enters C See Figure 1611 376 Chapter 16 Homoclinic Phenomena cake Figure 1610 Homoclinic orbits c 1 Zz ct C C a 1 Figure 1611 C C where we show the cylinder Cas a strip Now let X C Either the solution through X lies on the stable surface of the origin or X is defined so that the solution through X returns to C at some later time As a consequence each point X C has the property that 1 Either the solution through X crosses C infinitely many times as tf oo so that X is defined for all n 0 or 2 The solution through X eventually meets z 0 and thus lies on the stable surface through the origin In backward time the situation is different Only those points that lie in C have solutions that return to C strictly speaking we have not defined 164 Homoclinic Bifurcations 377 the backward solution of points in C C but we think of these solu tions as being defined in R and eventually meeting C after which time these solutions continually revist C As in the case of the Lorenz attractor we let Co AoO n0 where C denotes the closure of the set C Then we set A U oa U00 teR Note that C C is just the two intersection points of the homoclinic solutions with C Therefore we only need to add the origin to A to ensure that A is a closed set The proof of the following result is similar in spirit to the corresponding result for the Lorenz attractor in Section 144 Proposition The set A has the following properties 1 A is closed and invariant 2 IfP EC thenwP CA 3 NteRpiC A g Thus A has all of the properties of an attractor except the transitivity property Nonetheless A is traditionally called a double scroll attractor We cannot compute the divergence of the double scroll vector field as we did in the Lorenz case for the simple reason that we have not written down the formula for this system However we do have an expression for the Poincaré map A straightforward computation shows that det D 18 That is the Poincaré map shrinks areas by a factor of 18 at each iteration Thus ANMn0PC has area 0 in C and we have the following proposition Proposition The volume of the double scroll attractor A is zero L 164 Homoclinic Bifurcations In higher dimensions bifurcations associated with homoclinic orbits may lead to horribly or wonderfully depending on your point of view complicated 378 Chapter 16 Homoclinic Phenomena B a n Fx0 FxB Fa FoB a b Figure 1612 Images FR for O anda1 behavior In this section we give a brief indication of some of the ramifications of this type of bifurcation We deal here with a specific perturbation of the double scroll vector field that breaks both of the homoclinic connections The full picture of this bifurcation involves understanding the unfolding of infinitely many horseshoe maps By this we mean the following Consider a family of maps F defined on a rectangle R with parameter A 01 The image of F R is a horseshoe as shown in Figure 1612 When A 0 FR lies below R As A increases FR rises monotonically When A 1 FR crosses R twice and we assume that F is the horseshoe map described in Section 162 Clearly Fo has no periodic points whatsoever in R but by the time A has reached 1 infinitely many periodic points have been born and other chaotic behavior has appeared The family F has undergone infinitely many bifur cations en route to the horseshoe map How these bifurcations occur is the subject of much contemporary research in mathematics The situation here is significantly more complex than the bifurcations that occur for the onedimensional logistic family fx Ax1 x with 0A 4 The bifurcation structure of the logistic family has recently been completely determined the planar case is far from being resolved We now introduce a parameter into the double scroll system When 0 the system will be the double scroll system considered in the previous section When 40 we change the system by simply translating M C and the corre sponding transit map in the zdirection by More precisely we assume that the system remains unchanged in the cylindrical region r 1 z 1 but we change the transit map defined on the upper disk Dt by adding 0 to the image That is the new Poincaré map is given on C by 1 zsin 6 logz O29z 5zcos 6 logz 164 Homoclinic Bifurcations 379 is defined similarly using the skew symmetry of the system We further assume that is chosen small enough é 12 so that oC CC When 0 intersects C in the upper cylindrical region Ct and then after passing close to the origin winds around itself before reintersecting C a second time When 0 now meets C in C and then takes a very different route back to C this time winding around Recall that j has infinitely many fixed points in C This changes dramatically when 4 0 Proposition The maps each have only finitely many fixed points in C whene 0 Proof To find fixed points of 7 we must solve Z G va sin 0 logz Z Z va cos 0 logz e where 0 As in Section 161 we must therefore have Zz Z g2 1 z so that 1 Vz4z 2 In particular we must have zAz 0 or equivalently 2 4z L This inequality holds provided z lies in the interval J defined by 1 1 1 1 gti gyi tle szs 5 tet evi t loc 380 Chapter 16 Homoclinic Phenomena This puts a further restriction on for to have fixed points namely 116 Note that when 116 we have 1 1 gti gyi tle so that I has length 1 164 and this interval lies to the right of 0 To determine the zvalues of the fixed points we must now solve 1 2z Vz4z641 cos 5V2 z ov We or 51 Aze cos 5 vz 4z logz Zz With a little calculus one may check that the function Aze gz Zz has a single minimum 0 at z and two maxima equal to 1 at the endpoints of I Meanwhile the graph of 1 hz cos 25v zAz62 loxv2 oscillates between 1 only finitely many times in I Thus hz gz at only finitely many zvalues in I These points are the fixed points for O Note that as 0 the interval I tends to 014 and so the number of oscillations of h in I increases without bound Therefore we have this corollary Corollary Given N Z there exists such that if 0 ey then t has at least N fixed points in CT When 0 the unstable curve misses the stable surface in its first pass through C Indeed crosses Ct at 9 0 z This does not mean that there are no homoclinic orbits when 4 0 In fact we have the following proposition Proposition There are infinitely many values of for which are homoclinic solutions that pass twice through C 165 Exploration The Chua Circuit 381 Proof To show this we need to find values of for which 70 lies on the stable surface of the origin Thus we must solve 0 ve cos 0 loge e or 2 cos loge But as in Section 161 the graph of cosloge meets that of 2e infinitely often This completes the proof L For each of the values for which is a homoclinic solution we again have infinitely many fixed points for 0 6 as well as a very different structure for the attractor Clearly a lot is happening as changes We invite the reader who has lasted this long with this book to figure out everything that is happening here Good luck And have fun 165 Exploration The Chua Circuit In this exploration we investigate a nonlinear threedimensional system of differential equations related to an electrical circuit known the Chua circuit These were the first examples of circuit equations to exhibit chaotic behavior Indeed for certain values of the parameters these equations exhibit behav ior similar to the double scroll attractor in Section 163 The original Chua circuit equations possess a piecewise linear nonlinearity Here we investigate a variation of these equations in which the nonlinearity is given by a cubic function For more details on the Chua circuit we refer to Chua Komuro and Matsumoto 11 and Khibnik Roose and Chua 26 The nonlinear Chua circuit system is given by x ay x ysxytz Zz by where a and b are parameters and the function is given by ot x xX x 16 6 Actually the coefficients of this polynomial are usually regarded as parame ters but we will fix them for the sake of definiteness in this exploration Hirsch Ch169780123820105 201222 1253 Page 382 22 382 Chapter 16 Homoclinic Phenomena When a 1091865 and b 14 this system appears to have a pair of symmetric homoclinic orbits as illustrated in Figure 1613 The goal of this exploration is to investigate how this system evolves as the parameter a changes As a consequence we will also fix the parameter b at 14 and then let a vary We caution the explorer that proving any of the chaotic or bifurcation behavior observed next is nearly impossible virtually anything you can do in this regard would qualify as an interesting research result 1 As always begin by finding the equilibrium points 2 Determine the types of these equilibria perhaps by using a computer algebra system 3 This system possesses a symmetry describe this symmetry and tell what it implies for solutions 4 Let a vary from 6 to 14 Describe any bifurcations you observe as a varies Be sure to choose pairs of symmetrically located initial conditions in this and other experiments to see the full effect of the bifurcations Pay particular attention to solutions that begin near the origin 5 Are there values of a for which there appears to be an attractor for this system What appears to be happening in this case Can you construct a model 6 Describe the bifurcation that occurs near the following avalues a a 658 b a 73 c a 878 d a 1077 x y z Figure 1613 A pair of homoclinic orbits in the nonlinear Chua system at parameter values a 1091865 and b 14 Exercises 383 EXERCISES 1 Prove that S si ti 1 dO1 Da 1C is a distance function on 2 where Xp is the set of doubly infinite sequences of 0s and Is as described in Section 162 2 Prove that the shift o isa homeomorphism of X 3 Prove that S A Xo gives a conjugacy between o and F 4 Construct a dense orbit for o 5 Prove that periodic points are dense for o 6 Let s Xo Prove that Ws consists of precisely those sequences with entries that agree with those of s to the right of some entry of s 7 Let 0 00000 Xo A sequence s Xp is called homoclinic to 0 if se W0N W0 Describe the entries of a sequence that is homoclinic to 0 Prove that sequences that are homoclinic to 0 are dense in Xp 8 Let 1 11111 Xz A sequence s is a heteroclinic sequence if s W0N W1 Describe the entries of such a heteroclinic sequence Prove that such sequences are dense in 2 9 Generalize the definitions of homoclinic and heteroclinic points to arbitrary periodic points for o and reprove Exercises 7 and 8 in this case 10 Prove that the set of homoclinic points to a given periodic point is countable 11 Let denote the set of onesided sequences of 0s and 1s Define XL Xp by Ds0s1 52 5553517595254 Prove that is a homeomorphism 12 Let X denote the fixed point of F in Hp for the horseshoe map Prove that the closure of WX contains all points in A as well as points on their unstable curves 13 Let R XL Lp be defined by RO S2S 05S152 52S SQ S1S2 Prove that Ro R idand thato o R Roo Conclude thato UoR where U is a map that satisfies UoU id Maps that are their own inverses are called involutions They represent very simple types of Hirsch Ch169780123820105 201222 1253 Page 384 24 384 Chapter 16 Homoclinic Phenomena dynamical systems Thus the shift may be decomposed into a compo sition of two such maps 14 Let s be a sequence that is fixed by R where R is as defined in the previous Exercise Suppose that σ ns is also fixed by R Prove that s is a periodic point of σ of period 2n 15 Rework the previous exercise assuming that σ ns is fixed by U where U is given as in Exercise 13 What is the period of s 16 For the Lorenz system in Chapter 14 investigate numerically the bifur cation that takes place for r between 1392 and 1396 with σ 10 and b 83 Hirsch Ch179780123820105 2012128 426 Page 385 1 17 Existence and Uniqueness Revisited In this chapter we return to the material presented in Chapter 7 this time filling in all of the technical details and proofs that were omitted earlier As a result this chapter is more difficult than the preceding ones it is however central to the rigorous study of ordinary differential equations To compre hend thoroughly many of the proofs in this section the reader should be familiar with such topics from real analysis as uniform continuity uniform convergence of functions and compact sets 171 The Existence and Uniqueness Theorem Consider the autonomous system of differential equations X FX where F Rn Rn In previous chapters we have usually assumed that F is C here we will relax this condition and assume that F is only C1 Recall that this means that F is continuously differentiable That is F and its first partial derivatives exist and are continuous functions on Rn For the first few Differential Equations Dynamical Systems and an Introduction to Chaos DOI 101016B9780123820105000178 c 2013 Elsevier Inc All rights reserved 385 386 Chapter 17 Existence and Uniqueness Revisited sections of this chapter we will deal only with autonomous equations later we will assume that F depends on t as well as X As we know a solution of this system is a differentiable function X J R defined on some interval J C R such that for all t J Xt FXt Geometrically Xt is a curve in R where the tangent vector Xt equals FXt as in previous chapters we think of this vector as being based at Xt so that the map F R R defines a vector field on R An initial condition or initial value for a solution X J R is a specification of the form Xto Xo where fo J and Xp R For simplicity we usually take to 0 A nonlinear differential equation may have several solutions that satisfy a given initial condition For example consider the firstorder nonlinear differential equation x 3x7 In Chapter 7 we saw that the identically zero function uj R R given by uot 0 is a solution satisfying the initial condition u0 0 But ut a is also a solution satisfying this initial condition in addition for any t 0 the function given by ut 0 iftT mw Vtty iftt is also a solution satisfying the initial condition u 0 0 Besides uniqueness there is also the question of existence of solutions When we dealt with linear systems we were able to compute solutions explicitly For nonlinear systems this is often not possible as we have seen Moreover certain initial conditions may not give rise to any solutions For example as we saw in Chapter 7 the differential equation va 1 ifx0 11 ifx0 has no solution that satisfies x0 0 Thus it is clear that to ensure existence and uniqueness of solutions extra conditions must be imposed on the function F The assumption that F is con tinuously differentiable turns out to be sufficient as we shall see In the first example above F is not differentiable at the problematic point x 0 while in the second example F is not continuous at x 0 172 Proof of Existence and Uniqueness 387 The following is the fundamental local theorem of ordinary differential equations The Existence and Uniqueness Theorem Consider the initial value problem x FX X0 Xo where Xo R Suppose that F R R is C Then there exists a unique solution of this initial value problem More precisely there exists a 0 and a unique solution X aa R of this differential equation satisfying the initial condition X0 Xo We will prove this theorem in the next section 172 Proof of Existence and Uniqueness We need to recall some multivariable calculus Let F R R In coordi nates xX on R we write FX fi x15 5 Xy oofnX1 5 Xy Let DFy be the derivative of F at the point X R We may view this derivative in two slightly different ways From one point of view DFx is a linear map defined for each point X IR this linear map assigns to each vector U R the vector FX AU FX DFyU lim LA TRY FRO h0 h where h R Equivalently from the matrix point of view DFx is the nx n Jacobian matrix 0 DFx Ox where each derivative is evaluated at xx Thus the derivative may be viewed as a function that associates different linear maps or matrices to each point in R That is DF R LCR 388 Chapter 17 Existence and Uniqueness Revisited As before the function F is said to be continuously differentiable or C 1 if all of the partial derivatives of the f exist and are continuous We will assume for the remainder of this chapter that F is C For each X R we define the norm DFx of the Jacobian matrix DF by DFx sup DFxU U1 where U R Note that DFx is not necessarily the magnitude of the largest eigenvalue of the Jacobian matrix at X Example Suppose 2 0 DFx i Then indeed DFx 2 and 2 is the largest eigenvalue of DFy However if 1 1 DEx i then 1 1 cosé prsi awe 0 1 Soe o6 Pe 0 1 siné sup cossin6 sin 002n sup V12cossiné sin0 002n 1 whereas is the largest eigenvalue a We do however have DFxV DFxV for any vector V R Indeed if we write V VVV then we have DFxV DFx VIV IV S DFxIV 172 Proof of Existence and Uniqueness 389 since VV has magnitude 1 Moreover the fact that FR R is C implies that the function R LCR that sends X DFy is a continuous function Let O C R be an open set A function F O R is said to be Lipschitz on O if there exists a constant K such that FY FX KYX for all X Y O We call K a Lipschitz constant for F More generally we say that F is locally Lipschitz if each point in O has a neighborhood 0 in O such that the restriction F to is Lipschitz The Lipschitz constant of FO may vary with the neighborhoods 0 Another important notion is that of compactness We say that a set C C R is compact if C is closed and bounded An important fact is that if fC R is continuous and C is compact then first f is bounded on C and second f actually attains its maximum on C See exercise 13 at the end of this chapter Lemma Suppose that the function F O R is C Then F is locally Lipschitz Proof Suppose that F O R is C and let Xo O Let 0 be so small that the closed ball O of radius about Xo is contained in Let K be an upper bound for DFx on O this bound exists because DFx is continuous and Q is compact The set O is convex that is if Y Z O then the straight line segment connecting Y to Z is contained in O This straight line is given by YsUO where UZY and 0s1 Let WsFYsU Using the Chain Rule we find Ws DFysuU Therefore FZ FY v0 1 ys ds 0 1 PFw ds 0 Thus we have 1 FZ FY xi dsKZY 0 390 Chapter 17 Existence and Uniqueness Revisited The following is implicit in the proof of the preceding lemma If O is convex and if DFx K for all X O then K is a Lipschitz constant for F O Suppose that J is an open interval containing zero and X J O satisfies Xt FXt with X0 Xo Integrating we have t Xt Xo FX ds 0 This is the integral form of the differential equation X FX Conversely if X J O satisfies this integral equation then X0 Xp and X satisfies X FX as is seen by differentiation Thus the integral and differential forms of this equation are equivalent as equations for X J O To prove existence of solutions we will use the integral form of the differential equation We now proceed with the proof of existence Here are our assumptions 1 Op is the closed ball of radius p 0 centered at Xo 2 There is a Lipschitz constant K for F on O 3 FX Mon Q 4 Choose a minoM 1K and let J a a We will first define a sequence of functions Up Uj from J to O Then we will prove that these functions converge uniformly to a function satisfying the differential equation Later we will show that there are no other such solutions The lemma that is used to obtain the convergence of the Uj is the following Lemma from analysis Suppose Uz J R k012 is a sequence of continuous functions defined on a closed interval J that satisfy the following Given 0 there is some N 0 such that for every pq N Ut Ut max pt Ut Then there is a continuous function U J R such that max Ux Ut 0 asko te Moreover for any t with t a t t lim Us ds Us ds kco 0 0 172 Proof of Existence and Uniqueness 391 This type of convergence is called uniform convergence of the functions Ux This lemma is proved in elementary analysis books and will not be proved here See Rudin 38 The sequence of functions U is defined recursively using an iteration scheme known as Picard iteration We gave several illustrative examples of this iterative scheme back in Chapter 7 Let Uot Xo For t J define t Uit Xo FUos ds Xo tFXo 0 Since t a and FXo M it follows that Ui t Xo tIFXo aM p so that Ut O for all t J By induction assume that Ut has been defined and that Ut Xo pe for all t J Let t Uti t Xo FOX ds 0 This makes sense since Uxs Op and so the integrand is defined We show that Up41t Xo e so that Ux 4 Op for t J this will imply that the sequence can be continued to Uz42 Up3 and so on This is shown as follows t Uk1 Xol S FUgs ds 0 t M ds 0 Map Next we prove that there is a constant L 0 such that for all k 0 ei 1 Upt aKL 392 Chapter 17 Existence and Uniqueness Revisited Let L be the maximum of Uj tf Upt over a t a By the preceding L aM We have t Uzt Ui t ue FU8 a 0 t xine Ups ds 0 aKL Assuming by induction that for some k 2 we have already proved Uet Ura S aK L for t a we then have t Ukit Uk FUxs FUg1s ds 0 t xf Uxs U1s ds 0 aKaKL aKL Let a aK so that a 1 by assumption Given any 0 we may choose N large enough so that for any r s N we have Co Urt Us SY Ue Ux0 kN Co eal kN eE since the tail of the geometric series may be made as small as we please By the lemma from analysis this shows that the sequence of functions Up U converges uniformly to a continuous function X J R From 172 Proof of Existence and Uniqueness 393 the identity t Ursit Xo f FUR ds 0 and we find by taking limits of both sides that t Xt Xo lim Focsconas k0o 0 t Xo lim FUs ds koo 0 t Xo FEXGds 0 The second equality also follows from the lemma from analysis Therefore X J Oy satisfies the integral form of the differential equation and thus is a solution of the equation itself In particular it follows that X J O is Cl This takes care of the existence part the theorem Now we turn to the uniqueness part Suppose that XY J O are two solutions of the differential equation satisfying X0 Y0 Xo where as before J is the closed interval a a We will show that Xt Yf for all t J Let QmaxXt Yd te This maximum is attained at some point t J Then ty Q Xt Yh oo vspas 0 ty FXs FYsds 0 ty xx0 Ysds 0 akQ Hirsch Ch179780123820105 2012128 426 Page 394 10 394 Chapter 17 Existence and Uniqueness Revisited Since aK 1 this is impossible unless Q 0 Therefore Xt Yt This completes the proof of the theorem To summarize this result we have shown that given any ball Oρ O of radius ρ about X0 on which 1 FX M 2 F has Lipschitz constant K 3 0 a minρM1K there is a unique solution X aa O of the differential equation such that X0 X0 In particular this result holds if F is C1 on O Some remarks are in order First note that two solution curves of X FX cannot cross if F satisfies the hypotheses of the theorem This is an immediate consequence of uniqueness but is worth emphasizing geometrically Sup pose X J O and Y J1 O are two solutions of X FX for which Xt1 Yt2 If t1 t2 we are done immediately by the theorem If t1 t2 then let Y1t Yt2 t1 t Then Y1 is also a solution of the system Since Y1t1 Yt2 Xt1 it follows that Y1 and X agree near t1 by the uniqueness statement of the theorem and thus so do Xt and Yt We emphasize the point that if Yt is a solution then so too is Y1t Yt t1 for any constant t1 In particular if a solution curve X J O of X FX satisfies Xt1 Xt1 w for some t1 and w 0 then that solu tion curve must in fact be a periodic solution in the sense that Xt w Xt for all t 173 Continuous Dependence on Initial Conditions For the Existence and Uniqueness Theorem to be at all interesting in any phys ical or even mathematical sense the result needs to be complemented by the property that the solution Xt depends continuously on the initial condition X0 The next theorem gives a precise statement of this property Theorem Let O Rn be open and suppose F O Rn has Lipschitz con stant K Let Yt and Zt be solutions of X FX that remain in O and are defined on the interval t0t1 Then for all t t0t1 we have Yt Zt Yt0 Zt0expKt t0 173 Continuous Dependence on Initial Conditions 395 Note that this result says that if the solutions Yt and Zt start out close together then they remain close together for tf near f Although these solu tions may separate from each other they do so no faster than exponentially In particular we have the following Corollary Continuous Dependence on Initial Conditions Let t X be the flow of the system X FX where F is C Then is a continuous function of X The proof depends on a famous inequality that we prove first Gronwalls Inequality Let u 0 R be continuous and nonnegative Suppose C 0 and K 0 are such that t ut c Kus ds 0 for allt 0a Then for all t in this interval ut CeX Proof Suppose first that C 0 Let t Ut c Kus ds 0 0 Then ut Ut Differentiating U we find Ut Kut Therefore 7 Ut Kut k Ut Ut Thus d log Ut K lo a 396 Chapter 17 Existence and Uniqueness Revisited so that log Ut log U0 Kt by integration Since U0 C we have by exponentiation Ut Ce and so ut Ce If C 0 we may apply the preceding argument to a sequence of positive c that tends to 0 as i oo This proves Gronwalls Inequality a We turn now to the proof of the theorem Proof Define vt Yt Zt Since t Yt Zt Yt Zt FYs FZs ds to we have t vt vto Kn ds to Now apply Gronwalls Inequality to the function ut vt to get tto ua r vm f Rv ds to t wt f Kuey dr 0 so vt to vt expKt or vt vt expKt f which is just the conclusion of the theorem Hirsch Ch179780123820105 2012128 426 Page 397 13 174 Extending Solutions 397 As we have seen differential equations that arise in applications often depend on parameters For example the harmonic oscillator equations depend on the parameters b the damping constant and k the spring constant circuit equations depend on the resistance capacitance and induc tance and so forth The natural question is how do solutions of these equations depend on these parameters As in the previous case solutions depend continuously on these parameters provided that the system depends on the parameters in a continuously differ entiable fashion We can see this easily by using a special little trick Suppose the system X FaX depends on the parameter a in a C1 fashion Lets consider an artificially augmented system of differential equations given by x 1 f1x1xna x n fnx1xna a 0 This is now an autonomous system of n 1 differential equations Although this expansion of the system may seem trivial we may now invoke the previous result about continuous dependence of solutions on initial conditions to verify that solutions of the original system depend continuously on a as well Theorem Continuous Dependence on Parameters Let X FaX be a system of differential equations for which Fa is continuously differentiable in both X and a Then the flow of this system depends continuously on a as well as X 174 Extending Solutions Suppose we have two solutions Yt Zt of the differential equation X FX where F is C1 Suppose also that Yt and Zt satisfy Yt0 Zt0 and that both solutions are defined on an interval J about t0 Now the Existence and Uniqueness Theorem guarantees that Yt Zt for all t in an interval about t0 that may a priori be smaller than J However this is not the case To see this suppose that J is the largest interval on which Yt Zt If J J there is an endpoint t1 of J and t1 J By continuity we have 398 Chapter 17 Existence and Uniqueness Revisited Yt Zh Now the uniqueness part of the theorem guarantees that in fact Yt and Zt agree on an interval containing t This contradicts the assertion that J is the largest interval on which the two solutions agree Thus we can always assume that we have a unique solution defined on a maximal time domain There is however no guarantee that a solution Xt can be defined for all time For example the differential equation x 14x has as solutions the functions xt tant c for any constant c Such a function cannot be extended over an interval larger than To 1 Iv ctc 2 2 since xt tooast c72 Next we investigate what happens to a solution as the limits of its domain are approached We state the result only for the righthand limit the other case is similar Theorem Let O CR be open and let F O R be C Let Yt be a solution of X FX defined on a maximal open interval J aB CR with B oo Then given any compact setC C O there is some to a 8 with Yto C This theorem says that if a solution Yt cannot be extended to a larger time interval then this solution leaves any compact set in O This implies that as t B either Yt accumulates on the boundary of O or else a subsequence Yt tends to oo or both Proof Suppose Yt CC for all t a 6 Since F is continuous and C is compact there exists M 0 such that FX M for all X EC Let y af We claim that Y extends to a continuous function Y y 8 C To see this it suffices to prove that Y is uniformly continuous on J For to t J we have ty Yt Yq ro a 0 ty FYs ds to 4 M 174 Extending Solutions 399 This proves uniform continuity on J Thus we may define YB lim Yt tB We next claim that the extended curve Y y 6 R is differentiable at Bf and is a solution of the differential equation We have t vip Yy lim f YG a t B Y t Yy lim row ds t B Y B Yy FYs ds Y where we have used uniform continuity of FYs Therefore t YOYY FYs ds Y for all t between y and f Thus Y is differentiable at 6 and in fact YB FYB Therefore Y is a solution on yf Since there must then be a solution on an interval 85 for some 6 6 we can extend Y to the interval a6 Thus 8B could not have been a maximal domain of a solution This completes the proof of the theorem This important fact follows immediately from the preceding theorem Corollary Let C be a compact subset of the open set O CR and let F OR be Cl Let Yo C and suppose that every solution curve of the form Y 08 O with Y0 Yo lies entirely in C Then there is a solu tion Y 000 O satisfying Y0 Yo and Yt C for all t 0 so this solution is defined for all forward time Given these results we can now give a slightly stronger theorem on the con tinuity of solutions in terms of initial conditions than the result discussed in Section 173 In that section we assumed that both solutions were defined on the same interval In the next theorem we drop this requirement The theorem shows that solutions starting at nearby points are defined on the same closed interval and also remain close to each other on this interval Hirsch Ch179780123820105 2012128 426 Page 400 16 400 Chapter 17 Existence and Uniqueness Revisited Theorem Let F O Rn be C1 Let Yt be a solution of X FX that is defined on the closed interval t0t1 with Yt0 Y0 There is a neighborhood U Rn of Y0 and a constant K such that if Z0 U then there is a unique solution Zt also defined on t0t1 with Zt0 Z0 Moreover Z satisfies Yt Zt Y0 Z0expKt t0 for all t t0t1 For the proof of the preceding theorem will need the following lemma Lemma If F O Rn is locally Lipschitz and C O is a compact set then FC is Lipschitz Proof Suppose not Then for every k 0 no matter how large we can find X and Y in C with FX FY kX Y In particular we can find XnYn such that FXn FYn nXn Yn for n 12 Since C is compact we can choose convergent subsequences of the Xn and Yn Relabeling we may assume Xn X and Yn Y with X and Y in C Note that we must have X Y since for all n X Y lim nXn Yn lim nn1FXn FYn lim nn12M where M is the maximum value of FX on C There is a neighborhood O0 of X on which FO0 has Lipschitz constant K Also there is an n0 such that Xn O0 if n n0 Therefore for n n0 FXn FYn KXn Yn which contradicts the assertion just made for n n0 This proves the lemma The proof of the theorem now goes as follows Proof By compactness of t0t1 there exists ϵ 0 such that X O if X Yt ϵ for some t t0t1 The set of all such points is a compact subset C of O The C1 map F is locally Lipschitz as we saw in Section 172 By the lemma it follows that FC has a Lipschitz constant K Hirsch Ch179780123820105 2012128 426 Page 401 17 175 Nonautonomous Systems 401 Let δ 0 be so small that δ ϵ and δ expKt1 t0 ϵ We claim that if Z0 Y0 δ then there is a unique solution through Z0 defined on all of t0t1 First of all Z0 O since Z0 Yt0 ϵ so there is a solution Zt through Z0 on a maximal interval t0β We claim that β t1 because if we suppose β t1 then by Gronwalls Inequality for all t t0β we have Zt Yt Z0 Y0expK t t0 δ expK t t0 ϵ Thus Zt lies in the compact set C By the preceding results t0β could not be a maximal solution domain Therefore Zt is defined on t0t1 The uniqueness of Zt then follows immediately This completes the proof 175 Nonautonomous Systems We turn our attention briefly in this section to nonautonomous differ ential equations Even though our main emphasis in this book has been on autonomous equations the theory of nonautonomous linear equa tions is needed as a technical device for establishing the differentiability of autonomous flows Let O R Rn be an open set and let F O Rn be a function that is C1 in X but perhaps only continuous in t Let t0X0 O Consider the nonautonomous differential equation Xt FtX Xt0 X0 As usual a solution of this system is a differentiable curve Xt in Rn defined for t in some interval J having the following properties 1 t0 J and Xt0 X0 2 tXt O and Xt FtXt for all t J The fundamental local theorem for nonautonomous equations is as follows Theorem Let O R Rn be open and F O Rn a function that is C1 in X and continuous in t If t0X0 O there is an open interval J con taining t and a unique solution of X FtX defined on J and satisfying Xt0 X0 402 Chapter 17 Existence and Uniqueness Revisited The proof is the same as that of the fundamental theorem for autonomous equations Section 172 the extra variable t being inserted where appropriate An important corollary of this result is the following Corollary Let At be a continuous family of n x n matrices Let tXo J x R Then the initial value problem x AtX Xto Xo has a unique solution on all of J We call the function Ft X Lipschitz in X if there is a constant K 0 such that FtX1 Ft X2 KX1 XQ for all t X and tf X2 in O Locally Lipschitz in X is defined analogously As in the autonomous case solutions of nonautonomous equations are con tinuous with respect to initial conditions if Ft X is locally Lipschitz in X We leave the precise formulation and proof of this fact to the reader A different kind of continuity is continuity of solutions as functions of the data Ft X That is if F O R and G O R are both C in X and F G is uniformly small we expect solutions to X FtX and Y Gt Y having the same initial values to be close This is true in fact we have the following more precise result Theorem Let O CR xR be an open set containing 0 Xo and suppose that FGO R are C in X and continuous in t Suppose also that for all tXO FtX Gt X Let K be a Lipschitz constant in X for Ft X If Xt and Yt are solutions of the equations X Ft X and Y Gt Y respectively on some interval J and X0 Xp Y0 then IX YOlS expKtl 1 forallt J 175 Nonautonomous Systems 403 Proof For t J we have t Xt Yt oo Ys ds 0 t oxen Gs Ys ds 0 Thus t Xt Y FsXs Fs Ys ds 0 t f Fs Ys Gs Ys ds 0 t t xixe Ys as eds 0 0 Let ut Xt Y4 Then t ut xf ws ds 0 so that t ui KR K ui ds 0 It follows from Gronwalls Inequality that t SexpKit u Kk exp which yields the theorem Hirsch Ch179780123820105 2012128 426 Page 404 20 404 Chapter 17 Existence and Uniqueness Revisited 176 Differentiability of the Flow Now we return to the case of an autonomous differential equation X FX where F is assumed to be C1 Our aim is to show that the flow φtX φtX determined by this equation is a C1 function of the two variables and to identify φX We know of course that φ is continuously differentiable in the variable t so it suffices to prove differentiability in X Toward that end let Xt be a particular solution of the system defined for t in a closed interval J about 0 Suppose X0 X0 For each t J let At DFXt That is At denotes the Jacobian matrix of F at the point Xt Since F is C1 At is continuous We define the nonautonomous linear equation U AtU This equation is known as the variational equation along the solution Xt From the previous section we know that the variational equation has a solution on all of J for every initial condition U0 U0 Also as in the autonomous case solutions of this system satisfy the Linearity Principle The significance of this equation is that if U0 is small then the function t Xt Ut is a good approximation to the solution Xt of the original autonomous equation with initial value X0 X0 U0 To make this precise suppose that Utξ is the solution of the variational equation that satisfies U0ξ ξ where ξ Rn If ξ and X0 ξ belong to O let Ytξ be the solution of the autonomous equation X FX that satisfies Y0 X0 ξ Proposition Let J be the closed interval containing 0 on which Xt is defined Then lim ξ0 Ytξ Xt Utξ ξ converges to 0 uniformly for t J This means that for every ϵ 0 there exists δ 0 such that if ξ δ then Ytξ Xt Utξ ϵξ Hirsch Ch179780123820105 2012128 426 Page 405 21 176 Differentiability of the Flow 405 for all t J Thus as ξ 0 the curve t Xt Utξ is a better and better approximation to Ytξ In many applications Xt Utξ is used in place of Ytξ this is convenient because Utξ is linear in ξ We will prove the proposition momentarily but first we use this result to prove the following theorem Theorem Smoothness of Flows The flow φtX of the autonomous system X FX is a C1 function that is φt and φX exist and are continuous in t and X Proof Of course φtXt is just FφtX which is continuous To compute φX we have for small ξ φtX0 ξ φtX0 Ytξ Xt The proposition now implies that φtX0X is the linear map ξ Utξ The continuity of φX is then a consequence of the continuity in initial conditions and data of solutions for the variational equation Denoting the flow again by φtX we note that for each t the derivative DφtX of the map φt at X O is the same as φtXX We call this the space derivative of the flow as opposed to the time derivative φtXt The proof of the preceding theorem actually shows that DφtX is the solu tion of an initial value problem in the space of linear maps on Rn For each X0 O the space derivative of the flow satisfies the differential equation d dt DφtX0 DFφtX0DφtX0 with the initial condition Dφ0X0 I Here we may regard X0 as a parameter An important special case is that of an equilibrium solution X so that φt X X Putting DF X A we get the differential equation d dt Dφt X ADφt X with Dφ0 X I The solution of this equation is Dφt X exptA This means that in a neighborhood of an equilibrium point the flow is approximately linear 406 Chapter 17 Existence and Uniqueness Revisited We now prove the proposition The integral equations satisfied by Xt Yt and Uté are t xi Xo FOX9 as 0 t Yt Xo f FOE as 0 t Uté DF Use as 0 From these we get t Yt Xt U FYs FXs DFxsUs ds 0 The Taylor approximation of F at a point Z says FY FZ DFZY Z RZY Z where RZY lim RY 2 0 YsZ YZ uniformly in Y for Y ina given compact set We apply this to Y Ys Z Xs From the linearity of DFx we get t Yt Xt UG DFxs Ys Xs Us ds 0 t f RXs Ys Xs ds 0 Denote the left side of this expression by gt and set N maxDFxs J Exercises 407 Then we have t t gt n 20s a RXs Ys Xs ds 0 0 Fix 0 and pick 59 0 so small that RXs Ys Xs Ys X if Ys Xs 69 ands J From Section 173 there are constants K 0 and 6 0 such that Ys8 X léleS 8 if E 6 andseJ Assume now that 5 From the preceding we find for t J t t gt vs as elgleas 0 0 so that t gt nfs ds Celé 0 for some constant C depending only on K and the length of J Applying Gronwalls Inequality we obtain gt Cee g if te J and 6 Recall that 5 depends on Since is any positive number this shows that gt 0 uniformly in t J which proves the proposition EXERCISES 1 Write out the first few terms of the Picard iteration scheme for each of the following initial value problems Where possible use any method to find explicit solutions Discuss the domain of the solution 408 Chapter 17 Existence and Uniqueness Revisited a x x2x01 b x x43x0 0 c x x43x0 1 d x cosxx0 0 e x 12xx11 2 Let A be an n x n matrix Show that the Picard method for solving X AX X0 Xo gives the solution exptAXpo 3 Derive the Taylor series for cost by applying the Picard method to the firstorder system corresponding to the secondorder initial value problem x x x01 x00 4 For each of the following functions find a Lipschitz constant on the region indicated or prove there is none a fx x00o x 00 b fx x1x1 c fx 1x31x 0 d fy x2yy On R xy 7 2 e fxy lety y 4 5 Consider the differential equation xl3 How many different solutions satisfy x0 0 6 What can be said about solutions of the differential equation x xt 7 Define f R R by fx 1 if x 1 fx 2 if x 1 What can be said about solutions of x fx satisfying x0 1 where the right side of the differential equation is discontinuous What happens if you have instead fx Oifx 1 8 Let At be a continuous family of m x n matrices and let Pt be the matrix solution to the initial value problem P AtP P0 Po Show that t det Pt det Po exp Tr As 0 9 Suppose F is a gradient vector field Show that DFx is the magnitude of the largest eigenvalue of DFy Hint DFx is a symmetric matrix Exercises 409 10 Show that there is no solution to the secondorder twopoint boundary value problem x x x0 0 x7 1 11 What happens if you replace the differential equation in the previous Exercise by x kx with k 0 12 Prove the following general fact see also Section 173 If C0 and uv 08 R are continuous and nonnegative and t ut c usvs ds 0 for all t 0 B then ut Ce where t VOH vs ds 0 13 Suppose C C R is compact and f C R is continuous Prove that f is bounded on C and that f attains its maximum value at some point in C 14 In a lengthy essay not to exceed 50 pages describe the behavior of all solutions of the system X 0 where X R Ah yes Another free and final gift from the Math Department Hirsch Bibliography9780123820105 201212 1027 Page 411 1 Bibliography 1 Abraham R and Marsden J Foundations of Mechanics Reading MA Benjamin Cummings 1978 2 Abraham R and Shaw C Dynamics The Geometry of Behavior Redwood City CA AddisonWesley 1992 3 Alligood K Sauer T and Yorke J Chaos An Introduction to Dynamical Systems New York SpringerVerlag 1997 4 Arnold V I Ordinary Differential Equ0ations Cambridge MIT Press 1973 5 Arnold V I Mathematical Methods of Classical Mechanics New York SpringerVerlag 1978 6 Afraimovich V S and Shilnikov L P Strange attractors and quasi attractors In Nonlinear Dynamics and Turbulence 1 Boston Pitman 1983 7 Arrowsmith D and Place C An Introduction to Dynamical Systems Cambridge UK Cambridge University Press 1990 8 Banks J Brooks J Cairns G Davis G and Stacey P On Devaneys definition of chaos Amer Math Monthly 99 1992 332 9 Blanchard P Devaney R L and Hall G R Differential Equations Pacific Grove CA BrooksCole 2002 10 Birman J S and Williams R F Knotted 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Global Theory of Dynamical Systems Bristol Institute Physics 2001 277 20 Gutzwiller M The anisotropic Kepler problem in two dimensions J Math Phys 14 1973 139 21 Guckenheimer J and Williams R F Structural stability of Lorenz attractors Publ Math IHES 50 1979 59 22 Guckenheimer J and Holmes P Nonlinear Oscillations Dynamical Systems and Bifurcations of Vector Fields New York SpringerVerlag 1983 23 Hodgkin A L and Huxley A F A quantitative description of membrane current and its application to conduction and excitation in nerves J Physiol 117 1952 500 24 Katok A and Hasselblatt B Introduction to the Modern Theory of Dynamical Systems Cambridge UK Cambridge University Press 1995 25 Kraft R Chaos Cantor sets and hyperbolicity for the logistic maps Amer Math Monthly 106 1999 400 26 Khibnik A Roose D and Chua L On periodic orbits and homoclinic bifurcations in Chuas circuit with a smooth nonlinearity Intl J Bifurcation and Chaos 3 1993 363 27 Lengyel I Rabai G and Epstein I Experimental and modeling study of oscillations in the chlorine dioxideiodinemalonic acid reaction J Amer Chem Soc 112 1990 9104 28 Liapunov A M The General Problem of Stability of Motion London Taylor Francis 1992 29 Lorenz E Deterministic nonperiodic flow J Atmos Sci 20 1963 130 30 Marsden J E and McCracken M The Hopf Bifurcation and Its Applications New York SpringerVerlag 1976 31 May R M Theoretical Ecology Principles and Applications Oxford Blackwell 1981 32 McGehee R Triple collision in the collinear three body problem Inventiones Math 27 1974 191 33 Moeckel R Chaotic dynamics near triple collision Arch Rational Mech Anal 107 1989 37 34 Murray J D Mathematical Biology Berlin SpringerVerlag 1993 35 Nagumo J S Arimoto S and Yoshizawa S An active pulse transmission line stimulating nerve axon Proc IRE 50 1962 2061 36 Rossler O E An equation for continuous chaos Phys Lett A 57 1976 397 37 Robinson C Dynamical Systems Stability Symbolic Dynamics and Chaos Boca Raton FL CRC Press 1995 Hirsch Bibliography9780123820105 201212 1027 Page 413 3 Bibliography 413 38 Rudin W Principles of Mathematical Analysis New York McGrawHill 1976 39 Schneider G and Wayne C E Kawahara dynamics in dispersive media Physica D 152 2001 384 40 Shilnikov L P A case of the existence of a countable set of periodic motions Sov Math Dokl 6 1965 163 41 Shilnikov L P Chuas circuit rigorous results and future problems Int J Bifurcation and Chaos 4 1994 489 42 Siegel C and Moser J Lectures on Celestial Mechanics Berlin Springer Verlag 1971 43 Smale S Diffeomorphisms with many periodic points In Differential and Combinatorial Topology Princeton Princeton University Press 1965 63 44 Sparrow C The Lorenz Equations Bifurcations Chaos and Strange Attractors New York SpringerVerlag 1982 45 Strogatz S Nonlinear Dynamics and Chaos Reading MA AddisonWesley 1994 46 Tucker W The Lorenz attractor exists C R Acad Sci Paris Ser I Math 328 1999 1197 47 Winfree A T The prehistory of the BelousovZhabotinsky reaction J Chem Educ 61 1984 661 Hirsch Index9780123820105 2012110 1157 Page 415 1 Index C1 function 141 Ck function 141 α limit set 201 213 ω limit set 201 213 A angular momentum 284 anisotropic Kepler problem 299 asymptotic backward 373 forward 373 attractor 312 double scroll 375 Lorenz 307 312 autonomous 5 22 B basic region 188 246 basin of attraction 192 basis 28 88 BelousovZhabotinsky reaction 228 bifurcation 4 175 334 exchange 336 heteroclinic 190 homoclinic 377 Hopf 182 270 period doubling 336 pitchfork 178 saddlenode 176 179 334 tangent 334 bifurcation diagram 8 63 178 BZreaction 228 C canonical form 49 Cantor middlethirds set 351 Cantor set 344 capacitance 260 capacitor 258 carrying capacity 4 CauchyRiemann equations 183 center 46 central force field 282 chaos 156 326 340 characteristic 259 characteristic equation 32 characteristic polynomial 32 chemical reactions 228 Chua circuit 381 closed orbit 213 coefficient matrix 29 collision surface 296 collisionejection orbit 288 compact set 389 competitive species 244 complex eigenvalue 44 complex vector field 182 415 Hirsch Index9780123820105 2012110 1157 Page 416 2 416 Index configuration space 278 conjugacy 65 342 conjugacy equation 342 conservative system 280 constant coefficient 25 constant of the motion 207 281 continuously differentiable 141 coordinate change 49 critical point 203 cross product 279 current 258 current state 258 cycle 330 D damping constant 26 dense 324 dense set 100 determinant 27 diffeomorphism 66 difference equation 338 differential equation 1 dimension 90 direction field 24 discrete dynamical system 329 discrete logistic model 338 distance function 346 dot product 279 double scroll attractor 377 dynamical system 140 141 continuous 141 discrete 141 smooth 141 E eccentricity 293 eigenvalue 30 82 complex 44 real 39 repeated 47 zero 44 eigenvector 30 82 elementary matrix 78 elementary row operation 77 energy 281 energy surface 286 equilibrium point 2 22 equilibrium solution 2 Eulers method 154 Existence and Uniqueness Theorem 142 144 387 F faces authors 63 masks 306 Faradays law 260 first integral 207 281 first order 5 FitzhughNagumo 272 fixed point 11 330 attracting 331 indifferent 331 neutral 331 repelling 331 flow 12 64 flow box 216 force field 277 free gift 155 409 G general solution 2 35 generic property 102 glider 301 gradient 279 gradient system 202 graphical iteration 331 Gronwalls inequality 395 H Hamiltonian function 281 Hamiltonian system 206 281 harmonic oscillator 25 206 forced 23 harvesting constant 7 periodic 10 heteroclinic 191 383 HodgkinHuxley 272 homeomorphism 65 homoclinic 208 224 361 383 homogeneous 25 Hopf bifurcation 182 horseshoe map 368 hyperbolic 66 166 I improved Eulers method 154 inductance 260 inductor 257 infectious diseases 233 initial condition 2 142 386 Hirsch Index9780123820105 2012110 1157 Page 417 3 Index 417 initial value 142 386 initial value problem 2 142 invariant 180 199 positively 199 inverse 77 inverse matrix 50 inverse square law 285 invertible 50 77 itinerary 346 372 K KCL 258 Keplers first law 293 kernel 90 kinetic energy 281 Kirchoff voltage law 259 Kirchoffs current law 258 knot 328 KVL 259 L Lasalle Invariance Principle 199 level surface 203 Liapunov function 193 Liapunov Stability Theorem 193 Lienard equation 261 limit cycle 225 limit set 213 linear combination 28 linear map 50 90 linear transformation 50 90 linearity principle 36 linearization 159 linearized system 152 166 linearly dependent 27 74 linearly independent 27 74 Liouvilles theorem 311 Lipschitz 389 constant 389 locally 389 local section 216 logistic differential equation 4 logistic map 337 338 Lorenz attractor 312 Lorenz system 305 M mechanical system 278 metric 346 minimal period 330 momentum vector 281 N neat picture 306 neurodynamics 272 Newtons equation 23 Newtons law of gravitation 285 Newtons second law 277 Newtonian central force system 285 nonautonomous 10 nonlinear 5 nullcline 187 numerical methods 153 O Ohms law 259 generalized 259 open set 100 orbit 330 backward 322 370 forward 322 370 orbit diagram 355 P pendulum 194 forced 209 ideal 194 207 periodic point 324 330 periodic solution 11 213 phase line 3 phase plane 40 phase portrait 40 phase space 278 physical state 259 Picard iteration 144 391 pitchfork bifurcation 178 Poincare map 11 219 264 potential energy 280 predatorprey system 237 R range 90 real eigenvalues 39 regular point 203 regular value 203 relaxation oscillation 275 repeated eigenvlaues 47 resistor 257 passive 262 RLC circuit 23 257 row echelon form 77 RungeKutta method 154 Hirsch Index9780123820105 2012110 1157 Page 418 4 418 Index S saddle 40 168 saddle connection 191 saddlenode bifurcation 176 second order 23 seed 330 semiconjugacy 343 sensitive dependence 307 324 340 sensitive dependence on initial conditions 156 sensitivity constant 340 shift map 349 372 Shilnikov system 361 sink 3 43 167 331 spiral 47 SIR model 233 SIRS model 235 slope field 5 source 3 331 spiral 47 span 75 88 spiral sink 47 spiral source 47 spring constant 26 stable 3 174 asymptotically 174 stable curve 168 local 169 stable curve theorem 169 stable line 40 stable set 373 standard basis 28 74 state 258 state space 259 278 straight line solution 33 subspace 75 88 symbolic dynamics 344 symmetric matrix 205 system of differential equations 21 T tangent plane 286 tangent space 286 threshold level 236 time t map 64 total energy 281 trace 62 tracedeterminant plane 61 transitive 325 340 transverse line 216 two body problem 293 U uniform convergence 391 unstable 175 unstable curve 168 local 169 unstable curve theorem 169 unstable line 40 unstable set 373 V van der Pol equation 261 263 variational equation 151 404 vector field 24 voltage 258 voltage state 259 VolterraLotka system 238 W web diagram 332 Z zero velocity curve 287 zombies 251