Differential equations

Hirsch, Morris W., 1933- Differential Equations, dynamical systems, and linear algebra... Preface x CHAPTER 1 First-Order Equations 1 1.1 The Simplest Example 1 1.2 The Logistic Populati

DIFFERENT IA L EQUA T I ONS,

DY NA MI C A L SY ST EMS, A ND

A N I NT RODUC T I ON

T O C HA OS

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Includes bibliographical references and index.

ISBN 0-12-349703-5 (alk paper)

1 Differential equations 2 Algebras, Linear 3 Chaotic behavior in systems I Smale, Stephen, 1930- II Devaney, Robert L., 1948- III Hirsch, Morris W., 1933-

Differential Equations, dynamical systems, and linear algebra IV Title.

QA372.H67 2003

515’.35 dc22

2003058255 PRINTED IN THE UNITED STATES OF AMERICA

03 04 05 06 07 08 9 8 7 6 5 4 3 2 1

Preface x

CHAPTER 1 First-Order Equations 1

1.1 The Simplest Example 1 1.2 The Logistic Population Model 4 1.3 Constant Harvesting and Bifurcations 7 1.4 Periodic Harvesting and Periodic

Solutions 9 1.5 Computing the Poincaré Map 12 1.6 Exploration: A Two-Parameter Family 15

CHAPTER 2 Planar Linear Systems 21

2.1 Second-Order Differential Equations 23 2.2 Planar Systems 24

2.3 Preliminaries from Algebra 26 2.4 Planar Linear Systems 29 2.5 Eigenvalues and Eigenvectors 30 2.6 Solving Linear Systems 33 2.7 The Linearity Principle 36

CHAPTER 3 Phase Portraits for Planar Systems 39

3.1 Real Distinct Eigenvalues 39 3.2 Complex Eigenvalues 44

v

3.3 Repeated Eigenvalues 47 3.4 Changing Coordinates 49

CHAPTER 4 Classification of Planar Systems 61

4.1 The Trace-Determinant Plane 61 4.2 Dynamical Classification 64 4.3 Exploration: A 3D Parameter Space 71

CHAPTER 5 Higher Dimensional Linear Algebra 75

5.1 Preliminaries from Linear Algebra 75 5.2 Eigenvalues and Eigenvectors 83 5.3 Complex Eigenvalues 86

5.4 Bases and Subspaces 89 5.5 Repeated Eigenvalues 95 5.6 Genericity 101

CHAPTER 6 Higher Dimensional Linear Systems 107

6.1 Distinct Eigenvalues 107 6.2 Harmonic Oscillators 114 6.3 Repeated Eigenvalues 119 6.4 The Exponential of a Matrix 123 6.5 Nonautonomous Linear Systems 130

CHAPTER 7 Nonlinear Systems 139

7.1 Dynamical Systems 140 7.2 The Existence and Uniqueness Theorem 142

7.3 Continuous Dependence of Solutions 147 7.4 The Variational Equation 149

7.5 Exploration: Numerical Methods 153

CHAPTER 8 Equilibria in Nonlinear Systems 159

8.1 Some Illustrative Examples 159 8.2 Nonlinear Sinks and Sources 165 8.3 Saddles 168

8.4 Stability 174 8.5 Bifurcations 176 8.6 Exploration: Complex Vector Fields 182

Contents vii

CHAPTER 9 Global Nonlinear Techniques 189

9.1 Nullclines 189 9.2 Stability of Equilibria 194 9.3 Gradient Systems 203 9.4 Hamiltonian Systems 207 9.5 Exploration: The Pendulum with Constant Forcing 210

CHAPTER 10 Closed Orbits and Limit Sets 215

10.1 Limit Sets 215 10.2 Local Sections and Flow Boxes 218 10.3 The Poincaré Map 220

10.4 Monotone Sequences in Planar Dynamical Systems 222

10.5 The Poincaré-Bendixson Theorem 225 10.6 Applications of Poincaré-Bendixson 227 10.7 Exploration: Chemical Reactions That Oscillate 230

CHAPTER 11 Applications in Biology 235

11.1 Infectious Diseases 235 11.2 Predator/Prey Systems 239 11.3 Competitive Species 246 11.4 Exploration: Competition and Harvesting 252

CHAPTER 12 Applications in Circuit Theory 257

12.1 An RLC Circuit 257 12.2 The Lienard Equation 261 12.3 The van der Pol Equation 262 12.4 A Hopf Bifurcation 270 12.5 Exploration: Neurodynamics 272

CHAPTER 13 Applications in Mechanics 277

13.1 Newton’s Second Law 277 13.2 Conservative Systems 280 13.3 Central Force Fields 281 13.4 The Newtonian Central Force System 285

13.5 Kepler’s First Law 289 13.6 The Two-Body Problem 292 13.7 Blowing Up the Singularity 293 13.8 Exploration: Other Central Force Problems 297

13.9 Exploration: Classical Limits of Quantum Mechanical Systems 298

CHAPTER 14 The Lorenz System 303

14.1 Introduction to the Lorenz System 304 14.2 Elementary Properties of the Lorenz System 306

14.3 The Lorenz Attractor 310 14.4 A Model for the Lorenz Attractor 314 14.5 The Chaotic Attractor 319

14.6 Exploration: The Rössler Attractor 324

CHAPTER 15 Discrete Dynamical Systems 327

15.1 Introduction to Discrete Dynamical Systems 327

15.2 Bifurcations 332 15.3 The Discrete Logistic Model 335 15.4 Chaos 337

15.5 Symbolic Dynamics 342 15.6 The Shift Map 347 15.7 The Cantor Middle-Thirds Set 349 15.8 Exploration: Cubic Chaos 352 15.9 Exploration: The Orbit Diagram 353

CHAPTER 16 Homoclinic Phenomena 359

16.1 The Shil’nikov System 359 16.2 The Horseshoe Map 366 16.3 The Double Scroll Attractor 372 16.4 Homoclinic Bifurcations 375 16.5 Exploration: The Chua Circuit 379

17.1 The Existence and Uniqueness Theorem 383

17.2 Proof of Existence and Uniqueness 385

Contents ix

17.3 Continuous Dependence on Initial Conditions 392

17.4 Extending Solutions 395 17.5 Nonautonomous Systems 398 17.6 Differentiability of the Flow 400

Bibliography 407

Index 411

In the 30 years since the publication of the first edition of this book, muchhas changed in the field of mathematics known as dynamical systems In theearly 1970s, we had very little access to high-speed computers and computer

graphics The word chaos had never been used in a mathematical setting, and

most of the interest in the theory of differential equations and dynamicalsystems was confined to a relatively small group of mathematicians

Things have changed dramatically in the ensuing 3 decades Computers areeverywhere, 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 such cated dynamical systems as the horseshoe map, homoclinic tangles, and theLorenz system, and their mathematical analyses, convinced scientists that sim-ple stable motions such as equilibria or periodic solutions were not always themost important behavior of solutions of differential equations The beautyand relative accessibility of these chaotic phenomena motivated scientists andengineers in many disciplines to look more carefully at the important differen-tial equations in their own fields In many cases, they found chaotic behavior inthese systems as well Now dynamical systems phenomena appear in virtuallyevery area of science, from the oscillating Belousov-Zhabotinsky reaction inchemistry to the chaotic Chua circuit in electrical engineering, from compli-cated motions in celestial mechanics to the bifurcations arising in ecologicalsystems

compli-As a consequence, the audience for a text on differential equations anddynamical systems is considerably larger and more diverse than it was in

x

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 theLorenz attractor, the Shil’nikov system, and the double scroll attractor

3 Many new applications are included; previous applications have beenupdated

4 There are now several chapters dealing with discrete dynamical systems

5 We deal primarily with systems that are C∞, thereby simplifying many ofthe hypotheses of theorems

The book consists of three main parts The first part deals with linear systems

of differential equations together with some first-order nonlinear equations.The second part of the book is the main part of the text: Here we concentrate onnonlinear systems, primarily two dimensional, as well as applications of thesesystems in a wide variety of fields The third part deals with higher dimensionalsystems Here we emphasize the types of chaotic behavior that do not occur inplanar systems, as well as the principal means of studying such behavior, thereduction to a discrete dynamical system

Writing a book for a diverse audience whose backgrounds vary greatly poses

a significant challenge We view this book as a text for a second course indifferential equations that is aimed not only at mathematicians, but also atscientists and engineers who are seeking to develop sufficient mathematicalskills to analyze the types of differential equations that arise in their disciplines.Many who come to this book will have strong backgrounds in linear algebraand real analysis, but others will have less exposure to these fields To makethis text accessible to both groups, we begin with a fairly gentle introduction

to low-dimensional systems of differential equations Much of this will be areview for readers with deeper backgrounds in differential equations, so weintersperse some new topics throughout the early part of the book for thesereaders

For example, the first chapter deals with first-order equations We beginthis chapter with a discussion of linear differential equations and the logisticpopulation model, topics that should be familiar to anyone who has a rudimen-tary acquaintance with differential equations Beyond this review, we discussthe logistic model with harvesting, both constant and periodic This allows

us to introduce bifurcations at an early stage as well as to describe Poincarémaps and periodic solutions These are topics that are not usually found inelementary differential equations courses, yet they are accessible to anyone

with a background in multivariable calculus Of course, readers with a limitedbackground 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 two-dimensional linear algebra Chapters

5 and 6 introduce higher dimensional linear systems; however, our sis remains on three- and four-dimensional systems rather than completely

empha-general n-dimensional systems, 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 tion to nonlinear systems Unlike linear systems, nonlinear systems presentsome serious theoretical difficulties such as existence and uniqueness of solu-tions, dependence of solutions on initial conditions and parameters, and thelike 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 illustratewhat these theorems say (and do not say) Proofs of all of these results areincluded in the final chapter of the book

atten-In the first few chapters in the nonlinear part of the book, we introducesuch important techniques as linearization near equilibria, nullcline analysis,stability properties, limit sets, and bifurcation theory In the latter half of thispart, we apply these ideas to a variety of systems that arise in biology, electricalengineering, mechanics, and other fields

Many of the chapters conclude with a section called “Exploration.” Thesesections consist of a series of questions and numerical investigations dealingwith a particular topic or application relevant to the preceding material Ineach Exploration we give a brief introduction to the topic at hand and providereferences for further reading about this subject But we leave it to the reader totackle the behavior of the resulting system using the material presented earlier

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 amajor research project You will not find “answers in the back of the book” forthese 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 behavior

of higher dimensional systems known as chaotic behavior We introduce theseideas via the famous Lorenz system of differential equations As is often thecase in dimensions three and higher, we reduce the problem of comprehendingthe complicated behavior of this differential equation to that of understandingthe dynamics of a discrete dynamical system or iterated function So we thentake a detour into the world of discrete systems, discussing along the way howsymbolic dynamics may be used to describe completely certain chaotic systems

Preface xiii

We then return to nonlinear differential equations to apply these techniques

to other chaotic systems, including those that arise when homoclinic orbitsare present

We maintain a website at math.bu.edu/hsd devoted to issues ing this text Look here for errata, suggestions, and other topics of interest toteachers and students of differential equations We welcome any contributionsfrom readers at this site

regard-It is a special pleasure to thank Bard Ermentrout, John Guckenheimer, TassoKaper, Jerrold Marsden, and Gareth Roberts for their many fine commentsabout an earlier version of this edition Thanks are especially due to DanielLook and Richard Moeckel for a careful reading of the entire manuscript.Many of the phase plane drawings in this book were made using the excellentMathematica package called DynPac: A Dynamical Systems Package for Math-ematica written by Al Clark See www.me.rochester.edu/˜clark/dynpac.html

And, as always, Kier Devaney digested the entire manuscript; all errors that

remain are due to her

I would like to thank the following reviewers:

Bruce Peckham, University of Minnesota

Bard Ermentrout, University of Pittsburgh

Richard Moeckel, University of Minnesota

Jerry Marsden, CalTech

John Guckenheimer, Cornell University

Gareth Roberts, College of Holy Cross

Rick Moeckel, University of Minnesota

Hans Lindblad, University of California San Diego

Tom LoFaro, Gustavus Adolphus College

Daniel M Look, Boston University

xiv

1.1 The Simplest Example

The differential equation familiar to all calculus students

dx

dt = ax

is the simplest differential equation It is also one of the most important

First, what does it mean? Here x = x(t) is an unknown real-valued function

of a real variable t and dx/dt is its derivative (we will also use x or x(t ) for the derivative) Also, a is a parameter; for each value of a we have a

1

different differential equation The equation tells us that for every value of t



u(t )e −at

= u(t )e −at + u(t)(−ae −at)

= au(t)e −at − au(t)e −at = 0

Therefore u(t )e −at is a constant k, so u(t ) = ke at This proves our assertion

We have therefore 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 u0of a solution at a single point t0is specified Suppose that a function

x(t ) satisfying the differential equation is also required to satisfy x(t0)= u0

Then we must have ke at0 = u0, so that k = u0e−at0 Thus we have determined

k, and this equation therefore has a unique solution satisfying the specified initial condition x(t0)= u0 For simplicity, we often take t0= 0; then k = u0

There is no loss of generality in taking t0 = 0, for if u(t) is a solution with u(0) = u0, then the function v(t ) = u(t − t0) is a solution with v(t0)= u0

It is common to restate this in the form of an initial value problem:

x= ax, x(0) = u0

A solution x(t ) of an initial value problem must not only solve the differential equation, but it must also take on the prescribed initial value u0at t = 0

Note that there is a special solution of this differential equation when k= 0

This is the constant solution x(t )≡ 0 A constant solution such as 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 a parameter.

If a changes, the equation changes and so do the solutions Can we describe

1.1 The Simplest Example 3

qualitatively the way the solutions change? The sign of a is crucial here:

1 If a > 0, lim t→∞ ke at equals∞ when k > 0, and equals −∞ when k < 0;

solutions tend toward the equilibrium 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 Because the solution x(t ) is a function of time, we may view x(t ) as a particle moving along

the real line At the equilibrium point, the particle remains at rest (indicated

x

t

Figure 1.1 The solution graphs and phase

line for x= ax for a > 0 Each graphrepresents a particular solution.

t x

Figure 1.2 The solution graphs and

phase line for x= ax for a < 0.

by a solid dot), while any other solution moves up or down the x-axis, as

indicated by the arrows in Figure 1.1

The equation x = ax is stable in a certain sense if a = 0 More precisely,

if a is replaced by another constant b whose sign is the same as a, then 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 one-parameter family of equations

x= ax.

1.2 The Logistic Population Model

The differential equation x= ax above can be considered a simplistic model

of population growth when a > 0 The quantity x(t ) measures the population

of some species at time t The assumption that leads to the differential tion is that the rate of growth of the population (namely, dx/dt ) is directly

equa-proportional to the size of the population Of course, this naive assumptionomits many circumstances that govern actual population growth, including,for example, the fact that actual populations cannot increase with bound

To take this restriction into account, we can make the following furtherassumptions about the population model:

1 If the population is small, the growth rate is nearly directly proportional

to the size of the population;

2 but if the population grows too large, the growth rate becomes negative

One differential equation that satisfies these assumptions is the logistic population growth model This differential equation is

x= ax1− x

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− (x/N ) ≈ 1], but if x > N , then x < 0 Thus this simpleequation satisfies the above assumptions We should add here that there aremany other differential equations that correspond to these assumptions; ourchoice is perhaps the simplest

Without loss of generality we will assume that N = 1 That is, we willchoose units so that the carrying capacity is exactly 1 unit of population, and

x(t ) therefore represents the fraction of the ideal population present at time t

1.2 The Logistic Population Model 5

Therefore the logistic equation reduces to

x= f a (x) = ax(1 − x).

This is an example of a first-order, autonomous, nonlinear differential

equa-tion It is first order since only the first derivative of x appears in the equaequa-tion.

It is autonomous since the right-hand side of the equation depends on x alone, not on time t And it is nonlinear since f a (x) is a nonlinear function of x The previous example, x = ax, is a first-order, autonomous, linear differential



a dt

The method of partial fractions allows us to rewrite the left integral as

 1

So this solution is valid for any initial population x(0) When x(0) = 1,

we have an equilibrium solution, since x(t ) reduces to x(t ) ≡ 1 Similarly,

x(t )≡ 0 is also 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 to this equation at thisstage; we will return to this issue when we discuss the existence and uniquenessproblem 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-hand side of the differential equation determines the slope of the graph of any solution at each time t Hence we may plot little slope lines in the tx–plane as in Figure 1.3, with the slope of the line at (t , x) given by the quantity ax(1 − x) Our solutions must therefore

have graphs that are everywhere tangent to this slope field From these graphs,

we see immediately that, in agreement with our assumptions, all solutions

for which x(0) > 0 tend to the ideal population x(t ) ≡ 1 For x(0) < 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

fa (x) = ax(1 − x) This graph, displayed in Figure 1.4, crosses the x-axis at the two points x = 0 and x = 1, so these represent our equilibrium points When 0 < x < 1, we have f (x) > 0 Hence slopes are positive at any (t , x) with 0 < x < 1, and so solutions must increase in this region When x < 0 or

x > 1, we have f (x) < 0 and so solutions must decrease, as we see in both the

solution graphs and the phase lines in Figure 1.3

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 and

so 0 is a source Similarly, 1 is a sink

We may also determine this information analytically We have f a(x) =

a − 2ax so that f

a(0) = a > 0 and f

a(1) = −a < 0 Since f

a (0) > 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 Hence solutions must tend away from x = 0 In similar fashion, f

a (1) < 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

1.3 Constant Harvesting and Bifurcations 7

0.8

Figure 1.4 The graph of the

function f (x) = ax(1 – x) with

Figure 1.5 The slope field, solution graphs, and phase line for x= x – x3

Example As a further illustration of these qualitative ideas, consider the

differential equation

x = g(x) = x − x3

There are three equilibrium points, at x = 0, ±1 Since g(x) = 1 − 3x2, we

have g(0)= 1, so the equilibrium point 0 is a source Also, g(±1) = −2, sothe 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 canimmediately display the phase line, which is shown in Figure 1.5 

1.3 Constant Harvesting and

Bifurcations

Now let’s modify the logistic model to take into account harvesting of the ulation Suppose that the population obeys the logistic assumptions with the

Rather than solving this equation explicitly (which can be done — seeExercise 6 at the end of this chapter), we use the graphs of the functions

fh (x) = x(1 − x) − h

to “read off ” the qualitative behavior of solutions In Figure 1.6 we display

the graph of f h in three different cases: 0 < h < 1/4, h = 1/4, and h > 1/4.

It is straightforward to check that f h has two roots when 0 ≤ h < 1/4, one root when h = 1/4, and no roots if h > 1/4, as illustrated in the graphs As a consequence, the differential equation has two equilibrium points x  and x r

with 0≤ x  < xr when 0 < h < 1/4 It is also easy to check that f h(x  ) > 0, so that x  is a source, and f h(x r ) < 0 so that x ris a sink

As h passes through h= 1/4, we encounter another example of a bifurcation

The two equilibria x  and x r coalesce as h increases through 1/4 and then disappear when h > 1/4 Moreover, when h > 1/4, we have f h (x) < 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 this diagram we plot the parameter h horizontally Over each h-value 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 1/4 (see

Figure 1.7)

Ecologically, this bifurcation corresponds to a disaster for the species understudy For rates of harvesting 1/4 or lower, the population persists, provided

1.4 Periodic Harvesting and Periodic Solutions 9

the initial population is sufficiently large (x(0) ≥ x ) But a very small change

in the rate of harvesting when h = 1/4 leads to a major change in the fate of

the population: At any rate of harvesting h > 1/4, the species becomes extinct.

This phenomenon highlights the importance of detecting bifurcations infamilies of differential equations, a procedure that we will encounter manytimes in later chapters We should also mention that, despite the simplicity ofthis population model, the prediction that small changes in harvesting ratescan lead to disastrous changes in population has been observed many times inreal situations on earth

Example As another example of a bifurcation, consider the family of

differential equations

x = g a (x) = x2− ax = x(x − a) which depends on a parameter a The equilibrium points are given by x = 0

and x = a We compute g

a(0)= −a, so 0 is a sink if a > 0 and a source if

a < 0 Similarly, g a(a) = 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

x ⫽a

a

Figure 1.8 The bifurcation

diagram for x= x 2 – ax.

populations of many species of fish are harvested at a higher rate in warmerseasons than in colder months So we assume that the population is harvested

at a periodic rate One such model is then

x = f (t, x) = ax(1 − x) − h(1 + sin(2πt)) 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 x(t ) of this equation must satisfy x(t ) =

f (t , x(t )) for all t Also, this differential equation is no longer separable, so we

cannot generate an analytic formula for its solution using the usual methodsfrom calculus Thus we are forced to take a more qualitative approach

To describe the fate of the population in this case, we first note that the hand side of the differential equation is periodic with period 1 in the time

right-variable That is, f (t + 1, x) = f (t, x) This fact simplifies somewhat the

problem of finding solutions Suppose that we know the solution of all initialvalue problems, not for all times, but only for 0 ≤ t ≤ 1 Then in fact we know the solutions for all time For example, suppose x1(t ) is the solution that

is defined for 0 ≤ t ≤ 1 and satisfies x1(0) = x0 Suppose that x2(t ) is the solution that satisfies x2(0) = x1(1) Then we may extend the solution x1by

defining x1(t + 1) = x2(t ) for 0 ≤ t ≤ 1 The extended function is a solution

since we have

x1(t + 1) = x2(t ) = f (t, x2(t ))

= f (t + 1, x1(t+ 1))

1.4 Periodic Harvesting and Periodic Solutions 11

Figure 1.9 The slope field for f (x) =

x (1 – x) – h (1 + sin (2π t)).

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 knowthe behavior of solutions for all time

Secondly, suppose that we know the value at time t = 1 of the solution

satisfying any initial condition x(0) = x0 Then, to each such initial condition

x0 , we can associate the value x(1) of the solution x(t ) that satisfies x(0) = x0

This gives us a function p(x0)= x(1) If we compose this function with itself,

we derive the value of the solution through x0at time 2; that is, p(p(x0))=

x(2) 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 Poincaré map for this differential equation Having

such a function allows us to move from the realm of continuous dynamicalsystems (differential equations) to the often easier-to-understand realm ofdiscrete dynamical systems (iterated functions) For example, suppose that

we know that p(x0) = x0for some initial condition x0 That is, x0is a fixed point for the function p Then from our previous observations, we know that x(n) = x0for each integer n Moreover, for each time t with 0 < t < 1, we also have x(t ) = x(t + 1) and hence x(t + n) = x(t) for each integer n That is, the solution satisfying the initial condition x(0) = x0is a periodic

function of t with period 1 Such solutions are called periodic solutions of the

differential equation In Figure 1.10, we have displayed several solutions of thelogistic equation with periodic harvesting Note that the solution satisfying

the initial condition x(0) = x0 is a periodic solution, and we have x0 =

p(x0) = p(p(x0)) Similarly, the solution satisfying the initial condition x(0) = ˆx0also appears to be a periodic solution, so we should have p( ˆx0)= ˆx0.Unfortunately, it is usually the case that computing a Poincaré map for adifferential equation is impossible, but for the logistic equation with periodicharvesting we get lucky

1.5 Computing the Poincaré Map

Before computing the Poincaré map for this equation, we introduce someimportant terminology To emphasize the dependence of a solution on the

initial value x0, we will denote the corresponding solution by φ(t , x0) This

function φ:R× R → R is called the flow associated to the differential equation If we hold the variable x0fixed, then the function

t → φ(t, x0)

is just an alternative expression for the solution of the differential equation

satisfying the initial condition x0 Sometimes we write this function as φ t (x0)

Example For our first example, x= ax, the flow is given by

φ (t , x0)= x0eat.For the logistic equation (without harvesting), the flow is

φ (t , x0)= x(0)e at

1− x(0) + x(0)e at.Now we return to the logistic differential equation with periodic harvesting

x = f (t, x) = ax(1 − x) − h(1 + sin(2πt)).

1.5 Computing the Poincaré Map 13

The solution satisfying the initial condition x(0) = x0is given by t → φ(t, x0).While we do not have a formula for this expression, we do know that, by thefundamental theorem of calculus, this solution satisfies

φ (t , x0)= x0+

 t0

Again, we do not know φ(t , x0) explicitly, but this equation does tell us that

z(t ) solves the differential equation

z(t )= ∂ f

∂ x0 (t , φ(t , x0)) z(t ) with z(0) = 1 Consequently, via separation of variables, we may computethat the solution of this equation is

z(t )= exp

 t0

∂f

∂x0(s, φ(s, x0)) ds

and so we find

∂φ

∂x0 (1, x0)= exp

 10

∂f

∂x0 (s, φ(s, x0)) ds.

Since p(x0) = φ(1, x0), we have determined the derivative p(x0) of the

Poincaré map Note that p(x0) > 0 Therefore p is an increasing function.

Differentiating once more, we find

p(x0)= p(x0)

 1 0

of x for which p(x) = 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 x(t +1) = x(t) for all t Another way

to say this is that the flow φ(t , x0) is a periodic function in t with period 1 when the initial condition x0is one of the fixed points We saw these two solutions

in the particular case when h = 0 8 in Figure 1.10 In Figure 1.11, we againsee two solutions that appear to be periodic Note that one of these solutionsappears to attract all nearby solutions, while the other appears to repel them

We will return to these concepts often and make them more precise later inthe book

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 1.11 Differentiating f with respect to h, we find

∂ f

∂ h (t , x0)= −(1 + sin 2πt) Hence ∂f /∂h < 0 (except when t = 3/4) This implies that the slopes of the

slope field lines at each point (t , x0) decrease as h increases As a consequence, the values of the Poincaré map also decrease as h increases Hence there is a unique value h∗for which the Poincaré map has exactly one fixed point For

h > h∗, there are no fixed points for p and so p(x0) < x0for all initial values

It then follows that the population again dies out 

1.6 Exploration: A Two-Parameter Family 15

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 two-dimensional parameter plane (the ab–plane) for this differential

equation Feel free to use a computer to experiment with this differentialequation at first, but then try to verify your observations rigorously

1 First fix a = 1 Use the graph of f 1,bto construct the bifurcation diagram

for this family of differential equations depending on b.

2 Repeat the previous step 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 ab–plane, 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 f a,b (x), 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 and/or b

vary

8 Consider the differential equation x= x − x3− b sin(2πt) where |b| is

small What can you say about solutions of this equation? Are there anyperiodic 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

solutions and determine if they are sinks, sources, or neither Also, sketchthe phase line

3 Each of the following families of differential equations depends on a

parameter a Sketch the corresponding bifurcation diagrams.

f (x)

Figure 1.12 The graph of the

function f.

Exercises 17

(a) Sketch the phase line corresponding to the differential equation x=

f (x).

(b) Let g a (x) = f (x)+a Sketch the bifurcation diagram corresponding

to the family of differential equations x= g a (x).

(c) Describe the different bifurcations that occur in this family

5 Consider the family of differential equations

x= ax + sin x 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 differentialequations

6 Find the general solution of the logistic differential equation with

constant harvesting

x= x(1 − x) − h for all values of the parameter h > 0.

7 Consider the nonautonomous differential equation

8 Consider a first-order linear equation of the form x = ax + f (t) where

a ∈R Let y(t ) be any solution of this equation Prove that the general solution is y(t ) + c exp(at) where c ∈Ris arbitrary

9 Consider a first-order, linear, nonautonomous equation of the form

x(t ) = a(t)x.

(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 + cos t.

(a) Find the general solution of this equation

(b) Prove that there is a unique periodic solution for this equation

(c) Compute the Poincaré map p : {t = 0} → {t = 2π} for this

equa-tion and use this to verify again that there is a unique periodicsolution

11 First-order differential equations need not have solutions that are defined

for all times

(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 x(0) = 0 is defined only for −1 < t < 1.

12 First-order differential equations need not have unique solutions

satis-fying a given initial condition

(a) Prove that there are infinitely many different solutions of the

differential equations x = x1/3satisfying x(0)= 0

(b) Discuss the corresponding situation that occurs for x = x/t, x(0) = x0

(c) Discuss the situation that occurs for x = x/t2, x(0)= 0

13 Let x = f (x) be an autonomous first-order differential equation with

an equilibrium point at x0

(a) Suppose f(x0) = 0 What can you say about the behavior of

solutions near x0? Give examples

(b) Suppose f(x0)= 0 and f(x0)= 0 What can you now say?

p(s) ds= 0

15 Consider the differential equation x = f (t, x) where f (t, x) is ously differentiable in t and x Suppose that

continu-f (t + T, x) = f (t, x)

Exercises 19

for all t Suppose there are constants p, q such that

f (t , p) > 0, f (t , q) < 0 for all t Prove that there is a periodic solution x(t ) for this equation with

p < x(0) < q.

16 Consider the differential equation x = x2− 1 − cos(t) What can be

said about the existence of periodic solutions for this equation?

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

x1 = f1(t , x1, x2, , x n)

x2 = f2(t , x1, x2, , x n)

x n = f n (t , x1, x2, , x n)

Here the functions f j are real-valued functions of the n +1 variables x1, x2, ,

xn , and t Unless otherwise specified, we will always assume that the f j are C∞functions This means that the partial derivatives of all orders of the f j existand are continuous

To simplify notation, we will use vector notation:

Our system may then be written more concisely as

A solution of this system is then a function of the form X (t ) =

(x1(t ), , x n (t )) that satisfies the equation, so that

X(t ) = F(t, X(t)) where X(t ) = (x

1(t ), , x n(t )) Of course, at this stage, we have no guarantee

that there is such a solution, but we will begin to discuss this complicatedquestion in Section 2.7

The system of equations is called autonomous if none of the f j depends on

t , so the system becomes X = F(X) For most of the rest of this book we will

be concerned with autonomous systems

In analogy with first-order differential equations, a vector X0 for which

F (X0)= 0 is called an equilibrium point for the system An equilibrium point corresponds to a constant solution X (t ) ≡ X0of the system as before.Just to set some notation once and for all, we will always denote real variables

by lowercase letters such as x, y, x1, x2, t , and so forth Real-valued functions will also be written in lowercase such as f (x, y) or f1(x1, , x n , t ) We will reserve capital letters for vectors such as X = (x1, , x n), or for vector-valuedfunctions such as

We will denote n-dimensional Euclidean space byRn, so thatRnconsists of

all vectors of the form X = (x1, , x n)

2.1 Second-Order Differential Equations 23

2.1 Second-Order Differential Equations

Many of the most important differential equations encountered in scienceand engineering are second-order differential equations These are differentialequations of the form

defined by simply introducing a second variable y = x.

For example, consider a second-order constant coefficient equation of theform

2.2 Planar Systems

For the remainder of this chapter we will deal with autonomous systems in

R2, which we will write in the form

x = f (x, y)

y = g(x, y)

thus eliminating the annoying subscripts on the functions and variables As

above, we often use the abbreviated notation X = F(X) where X = (x, y) and F (X ) = F(x, y) = (f (x, y), g(x, y)).

In analogy with the slope fields of Chapter 1, we regard the right-hand side

of this equation as defining a vector field onR2 That is, we think of F (x, y)

as representing a vector whose x- and y-components are f (x, y) and g (x, y), respectively We visualize this vector as being based at the point (x, y) For

example, the vector field associated to the system

x= y

y= −x

is displayed in Figure 2.1 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 parameterized curve

in the plane of the form (x(t ), y(t )) such that, for each t , the tangent vector at the point (x(t ), y(t )) is F (x(t ), y(t )) That is, the solution curve (x(t ), y(t )) winds its way through the plane always tangent to the given vector F (x(t ), y(t )) based at (x(t ), y(t )).

Figure 2.1 The vector field, direction field, and

several solutions for the system x= y, y= –x.

2.2 Planar Systems 25

Example The curve



x(t ) y(t )

as required by the differential equation These curves define circles of radius

|a| in the plane that are traversed in the clockwise direction as t increases When a = 0, the solutions are the constant functions x(t) ≡ 0 ≡ y(t). 

Note that this example is equivalent to the second-order differential

equa-tion x = −x by simply introducing the second variable y = x This is an

example of a linear second-order differential equation, which, in more general

form, can be written

An even more special case is the homogeneous equation in which f (t )≡ 0

Example One of the simplest yet most important second-order, linear,

con-stant coefficient differential equations is the equation for a harmonic oscillator.

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

hori-zontal track We let x denote the displacement of the mass from its natural