NONLINEAR

    DYNAMICS AND

    CHAOS

    Strogatz-CROPPED2.pdf 1 5232014 8:40:05 AMNONLINEAR

    DYNAMICS AND

    CHAOS

    With Applications to

    Physics, Biology, Chemis t r y,and Engineering

    Steven H. Strogatz

    Strogatz-CROPPED2.pdf 3 5232014 8:40:05 AM

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    CONTENTS

    Preface to the Second Edition ix

    Preface to the First Edition xi

    1 Overview 1

    1.0 Chaos, Fractals, and Dynamics 1

    1.1 Capsule History of Dynamics 2

    1.2 The Importance of Being Nonlinear 4

    1.3 A Dynamical View of the World 9

    Part I One-Dimensional Flows

    2 Flows on the Line 15

    2.0 Introduction 15

    2.1 A Geometric Way of Thinking 16

    2.2 Fixed Points and Stability 18

    2.3 Population Growth 21

    2.4 Linear Stability Analysis 24

    2.5 Existence and Uniqueness 26

    2.6 Impossibility of Oscillations 28

    2.7 Potentials 30

    2.8 Solving Equations on the Computer 32

    Exercises for Chapter 2 36

    3 Bifurcations 45

    3.0 Introduction 45

    3.1 Saddle-Node Bifurcation 46

    3.2 Transcritical Bifurcation 51

    3.3 Laser Threshold 54

    3.4 Pitchfork Bifurcation 56

    Strogatz-CROPPED2.pdf 5 5232014 8:40:05 AMVI

    3.5 Overdamped Bead on a Rotating Hoop 62

    3.6 Imperfect Bifurcations and Catastrophes 70

    3.7 Insect Outbreak 74

    Exercises for Chapter 3 80

    4 Flows on the Circle 95

    4.0 Introduction 95

    4.1 Examples and Deinitions 95

    4.2 Uniform Oscillator 97

    4.3 Nonuniform Oscillator 98

    4.4 Overdamped Pendulum 103

    4.5 Firelies 105

    4.6 Superconducting Josephson Junctions 109

    Exercises for Chapter 4 115

    Part II Two-Dimensional Flows

    5 Linear Systems 125

    5.0 Introduction 125

    5.1 Deinitions and Examples 125

    5.2 Classiication of Linear Systems 131

    5.3 Love Affairs 139

    Exercises for Chapter 5 142

    6 Phase Plane 146

    6.0 Introduction 146

    6.1 Phase Portraits 146

    6.2 Existence, Uniqueness, and Topological

    Consequences 149

    6.3 Fixed Points and Linearization 151

    6.4 Rabbits versus Sheep 156

    6.5 Conservative Systems 160

    6.6 Reversible Systems 164

    6.7 Pendulum 168

    6.8 Index Theory 174

    Exercises for Chapter 6 181

    7 Limit Cycles 198

    7.0 Introduction 198

    7.1 Examples 199

    7.2 Ruling Out Closed Orbits 201

    7.3 Poincaré?Bendixson Theorem 205

    7.4 Liénard Systems 212

    7.5 Relaxation Oscillations 213

    Strogatz-CROPPED2.pdf 6 5232014 8:40:05 AMVII

    7.6 Weakly Nonlinear Oscillators 217

    Exercises for Chapter 7 230

    8 Bifurcations Revisited 244

    8.0 Introduction 244

    8.1 Saddle-Node, Transcritical, and Pitchfork

    Bifurcations 244

    8.2 Hopf Bifurcations 251

    8.3 Oscillating Chemical Reactions 257

    8.4 Global Bifurcations of Cycles 264

    8.5 Hysteresis in the Driven Pendulum and Josephson

    Junction 268

    8.6 Coupled Oscillators and Quasiperiodicity 276

    8.7 Poincaré Maps 281

    Exercises for Chapter 8 287

    Part III Chaos

    9 Lorenz Equations 309

    9.0 Introduction 309

    9.1 A Chaotic Waterwheel 310

    9.2 Simple Properties of the Lorenz Equations 319

    9.3 Chaos on a Strange Attractor 325

    9.4 Lorenz Map 333

    9.5 Exploring Parameter Space 337

    9.6 Using Chaos to Send Secret Messages 342

    Exercises for Chapter 9 348

    10 One-Dimensional Maps 355

    10.0 Introduction 355

    10.1 Fixed Points and Cobwebs 356

    10.2 Logistic Map: Numerics 360

    10.3 Logistic Map: Analysis 364

    10.4 Periodic Windows 368

    10.5 Liapunov Exponent 373

    10.6 Universality and Experiments 376

    10.7 Renormalization 386

    Exercises for Chapter 10 394

    11 Fractals 405

    11.0 Introduct ion 405

    11.1 Countable and Uncountable Sets 406

    11.2 Cantor Set 408

    11.3 Dimension of Self-Similar Fractals 411

    Strogatz-CROPPED2.pdf 7 5232014 8:40:05 AMVIII

    11.4 Box Dimension 416

    11.5 Pointwise and Correlation Dimensions 418

    Exercises for Chapter 11 423

    12 Strange Attractors 429

    12.0 Introduction 429

    12.1 The Simplest Examples 429

    12.2 Hénon Map 435

    12.3 R?ssler System 440

    12.4 Chemical Chaos and Attractor Reconstruction 443

    12.5 Forced Double-Well Oscillator 447

    Exercises for Chapter 12 454

    Answers to Selected Exercises 460

    References 470

    Author Index 483

    Subject Index 487

    Strogatz-CROPPED2.pdf 8 5232014 8:40:05 AMIX PREFACE TO THE SECOND EDITION

    PREFACE TO THE SECOND

    EDITION

    Welcome to this second edition of Nonlinear Dynamics and Chaos, now avail-

    able in e-book format as well as traditional print.

    In the twenty years since this book irst appeared, the ideas and techniques

    of nonlinear dynamics and chaos have found application in such exciting new

    ields as systems biology, evolutionary game theory, and sociophysics. To give

    you a taste of these recent developments, I’ve added about twenty substantial

    new exercises that I hope will entice you to learn more. The ields and applica-

    tions include (with the associated exercises listed in parentheses):

    Animal behavior: calling rhythms of Japanese tree frogs (8.6.9)

    Classical mechanics: driven pendulum with quadratic damping (8.5.5)

    Ecology: predator-prey model; periodic harvesting (7.2.18, 8.5.4)

    Evolutionary biology: survival of the ittest (2.3.5, 6.4.8)

    Evolutionary game theory: rock-paper-scissors (6.5.20, 7.3.12)

    Linguistics: language death (2.3.6)

    Prebiotic chemistry: hypercycles (6.4.10)

    Psychology and literature: love dynamics in Gone with the Wind (7.2.19)

    Macroeconomics: Keynesian cross model of a national economy (6.4.9)

    Mathematics: repeated exponentiation (10.4.11)

    Neuroscience: binocular rivalry in visual perception (8.1.14, 8.2.17)

    Sociophysics: opinion dynamics (6.4.11, 8.1.15)

    Systems biology: protein dynamics (3.7.7, 3.7.8)

    Thanks to my colleagues Danny Abrams, Bob Behringer, Dirk Brockmann,Michael Elowitz, Roy Goodman, Jeff Hasty, Chad Higdon-Topaz, Mogens

    Jensen, Nancy Kopell, Tanya Leise, Govind Menon, Richard Murray, Mary

    Strogatz-CROPPED2.pdf 9 5232014 8:40:05 AMX PREFACE TO THE SECOND EDITION

    Silber, Jim Sochacki, Jean-Luc Thiffeault, John Tyson, Chris Wiggins, and

    Mary Lou Zeeman for their suggestions about possible new exercises. I am

    especially grateful to Bard Ermentrout for devising the exercises about

    Japanese tree frogs (8.6.9) and binocular rivalry (8.1.14, 8.2.17), and to Jordi

    Garcia-Ojalvo for sharing his exercises about systems biology (3.7.7, 3.7.8).

    In all other respects, the aims, organization, and text of the irst edition

    have been left intact, except for a few corrections and updates here and there.

    Thanks to all the teachers and students who wrote in with suggestions.

    It has been a pleasure to work with Sue Caulield, Priscilla McGeehon, and

    Cathleen Tetro at Westview Press. Many thanks for your guidance and atten-

    tion to detail.

    Finally, all my love goes out to my wife Carole, daughters Leah and Jo, and

    dog Murray, for putting up with my distracted air and making me laugh.

    Steven H. Strogatz

    Ithaca, New York

    2014

    Strogatz-CROPPED2.pdf 10 5232014 8:40:05 AMXI PREFACE TO THE FIRST EDITION

    PREFACE TO THE FIRST EDITION

    This textbook is aimed at newcomers to nonlinear dynamics and chaos, espe-

    cially students taking a irst course in the subject. It is based on a one-semester

    course I’ve taught for the past several years at MIT. My goal is to explain the

    mathematics as clearly as possible, and to show how it can be used to under-

    stand some of the wonders of the nonlinear world.

    The mathematical treatment is friendly and informal, but still careful.

    Analyti cal methods, concrete examples, and geometric intuition are stressed.

    The theory is developed systematically, starting with irst-order differential

    equations and their bifurcations, followed by phase plane analysis, limit cycles

    and their bifurcations, and culminating with the Lorenz equations, chaos, iter-

    ated maps, period doubling, renormalization, fractals, and strange attractors.

    A unique feature of the book is its emphasis on applications. These include

    me chanical vibrations, lasers, biological rhythms, superconducting circuits,insect outbreaks, chemical oscillators, genetic control systems, chaotic water-

    wheels, and even a technique for using chaos to send secret messages. In each

    case, the sci entiic background is explained at an elementary level and closely

    integrated with the mathematical theory.

    Prerequisites

    The essential prerequisite is single-variable calculus, including curve-

    sketch ing, Taylor series, and separable differential equations. In a few places,multivari able calculus (partial derivatives, Jacobian matrix, divergence theo-

    rem) and linear algebra (eigenvalues and eigenvectors) are used. Fourier analy-

    sis is not assumed, and is developed where needed. Introductory physics is used

    throughout. Other scientiic prerequisites would depend on the applications

    considered, but in all cases, a irst course should be adequate preparation.

    Strogatz-CROPPED2.pdf 11 5232014 8:40:05 AMXII PREFACE TO THE FIRST EDITION

    Possible Courses

    The book could be used for several types of courses:

    A broad introduction to nonlinear dynamics, for students with no prior

    expo sure to the subject. (This is the kind of course I have taught.) Here

    one goes straight through the whole book, covering the core material at

    the beginning of each chapter, selecting a few applications to discuss in

    depth and giving light treatment to the more advanced theoretical topics

    or skipping them alto gether. A reasonable schedule is seven weeks on

    Chapters 1-8, and ive or six weeks on Chapters 9-12. Make sure there’s

    enough time left in the semester to get to chaos, maps, and fractals.

    A traditional course on nonlinear ordinary differential equations, but

    with more emphasis on appl ications and less on perturbation theory than

    usual. Such a course would focus on Chapters 1-8.

    A modern course on bifurcations, chaos, fractals, and their applications,for students who have already been exposed to phase plane analysis.

    Topics would be selected mainly from Chapters 3, 4, and 8-12.

    For any of these courses, the students should be assigned homework from

    the exercises at the end of each chapter. They could also do computer projects;

    build chaotic circuits and mechanical systems; or look up some of the refer-

    ences to get a taste of current research. This can be an exciting course to teach,as well as to take. I hope you enjoy it.

    Conventions

    Equations are numbered consecutively within each section. For instance,when we’re working in Section 5.4, the third equation is called (3) or Equation

    (3), but elsewhere it is called (5.4.3) or Equation (5.4.3). Figures, examples, and

    exercises are always called by their full names, e.g., Exercise 1.2.3. Examples

    and proofs end with a loud thump, denoted by the symbol ■.

    Acknowledgments

    Thanks to the National Science Foundation for inancial support. For help

    with the book, thanks to Diana Dabby, Partha Saha, and Shinya Watanabe

    (students); Jihad Touma and Rodney Worthing (teaching assistants); Andy

    Christian, Jim Crutchield, Kevin Cuomo, Frank DeSimone, Roger Eckhardt,Dana Hobson, and Thanos Siapas (for providing igures); Bob Devaney, Irv

    Epstein, Danny Kaplan, Willem Malkus, Charlie Marcus, Paul Matthews,Strogatz-CROPPED2.pdf 12 5232014 8:40:05 AMXIII PREFACE TO THE FIRST EDITION

    Arthur Mattuck, Rennie Mirollo, Peter Renz, Dan Rockmore, Gil Strang,Howard Stone, John Tyson, Kurt Wiesenfeld, Art Winfree, and Mary Lou

    Zeeman (friends and colleagues who gave ad vice); and to my editor Jack

    Repcheck, Lynne Reed, Production Supervisor, and all the other helpful peo-

    ple at Addison-Wesley. Finally, thanks to my family and Elisabeth for their

    love and encouragement.

    Steven H. Strogatz

    Cambridge, Massachusetts

    1994

    Strogatz-CROPPED2.pdf 13 5232014 8:40:05 AM1 1.0 CHAOS, FRACTALS, AND DYNAMICS

    1

    OVERVIEW

    1.0 Chaos, Fractals, and Dynamics

    There is a tremendous fascination today with chaos and fractals. James Gleick’s

    book Chaos (Gleick 1987) was a bestseller for months—an amazing accomplish-

    ment for a book about mathematics and science. Picture books like The Beauty

    of Fractals by Peitgen and Richter (1986) can be found on coffee tables in living

    rooms everywhere. It seems that even nonmathematical people are captivated by

    the ininite patterns found in fractals (Figure 1.0.1). Perhaps most important of

    all, chaos and fractals represent hands-on mathematics that is alive and changing.

    You can turn on a home computer and create stunning mathematical images that

    no one has ever seen before.

    The aesthetic appeal of chaos and fractals may explain why so many people

    have become intrigued by these ideas. But maybe you feel the urge to go deeper—

    to learn the mathematics behind the pictures, and to see how the ideas can be

    applied to problems in science and engi-

    neering. If so, this is a textbook for you.

    The style of the book is informal (as

    you can see), with an emphasis on con-

    crete examples and geometric thinking,rather than proofs and abstract argu-

    ments. It is also an extremely “applied”

    book—virtually every idea is illustrated

    by some application to science or engi-

    neering. In many cases, the applications

    are drawn from the recent research liter-

    ature. Of course, one problem with such

    an applied approach is that not every-

    one is an expert in physics and biology

    Figure 1.0.1

    Strogatz-CROPPED2.pdf 15 5232014 8:40:05 AM2 OVERVIEW

    and luid mechanics . . so the science as well as the mathematics will need to be

    explained from scratch. But that should be fun, and it can be instructive to see the

    connections among different ields.

    Before we start, we should agree about something: chaos and fractals are part

    of an even grander subject known as dynamics. This is the subject that deals with

    change, with systems that evolve in time. Whether the system in question settles

    down to equilibrium, keeps repeating in cycles, or does something more com-

    plicated, it is dynamics that we use to analyze the behavior. You have probably

    been exposed to dynamical ideas in various places—in courses in differential

    equations, classical mechanics, chemical kinetics, population biology, and so on.

    Viewed from the perspective of dynamics, all of these subjects can be placed in a

    common framework, as we discuss at the end of this chapter.

    Our study of dynamics begins in earnest in Chapter 2. But before digging in,we present two overviews of the subject, one historical and one logical. Our treat-

    ment is intuitive; careful deinitions will come later. This chapter concludes with

    a “dynamical view of the world,” a framework that will guide our studies for the

    rest of the book.

    1.1 Capsule History of Dynamics

    Although dynamics is an interdisciplinary subject today, it was originally a branch

    of physics. The subject began in the mid-1600s, when Newton invented differen-

    tial equations, discovered his laws of motion and universal gravitation, and com-

    bined them to explain Kepler’s laws of planetary motion. Speciically, Newton

    solved the two-body problem—the problem of calculating the motion of the earth

    around the sun, given the inverse-square law of gravitational attraction between

    them. Subsequent generations of mathematicians and physicists tried to extend

    Newton’s analytical methods to the three-body problem (e.g., sun, earth, and

    moon) but curiously this problem turned out to be much more dificult to solve.

    After decades of effort, it was eventually realized that the three-body problem was

    essentially impossible to solve, in the sense of obtaining explicit formulas for the

    motions of the three bodies. At this point the situation seemed hopeless.

    The breakthrough came with the work of Poincaré in the late 1800s. He intro-

    duced a new point of view that emphasized qualitative rather than quantitative

    questions. For example, instead of asking for the exact positions of the planets at

    all times, he asked “Is the solar system stable forever, or will some planets even-

    tually ly off to ininity?” Poincaré developed a powerful geometric approach to

    analyzing such questions. That approach has lowered into the modern subject

    of dynamics, with applications reaching far beyond celestial mechanics. Poincaré

    was also the irst person to glimpse the possibility of chaos, in which a determinis-

    tic system exhibits aperiodic behavior that depends sensitively on the initial condi-

    tions, thereby rendering long-term prediction impossible.

    Strogatz-CROPPED2.pdf 16 5232014 8:40:05 AM3 1.1 CAPSULE HISTORY OF DYNAMICS

    But chaos remained in the background in the irst half of the twentieth century;

    instead dynamics was largely concerned with nonlinear oscillators and their appli-

    cations in physics and engineering. Nonlinear oscillators played a vital role in the

    development of such technologies as radio, radar, phase-locked loops, and lasers.

    On the theoretical side, nonlinear oscillators also stimulated the invention of new

    mathematical techniques—pioneers in this area include van der Pol, Andronov,Littlewood, Cartwright, Levinson, and Smale. Meanwhile, in a separate develop-

    ment, Poincaré’s geometric methods were being extended to yield a much deeper

    understanding of classical mechanics, thanks to the work of Birkhoff and later

    Kolmogorov, Arnol’d, and Moser.

    The invention of the high-speed computer in the 1950s was a watershed in the

    history of dynamics. The computer allowed one to experiment with equations in

    a way that was impossible before, and thereby to develop some intuition about

    nonlinear systems. Such experiments led to Lorenz’s discovery in 1963 of chaotic

    motion on a strange attractor. He studied a simpliied model of convection rolls in

    the atmosphere to gain insight into the notorious unpredictability of the weather.

    Lorenz found that the solutions to his equations never settled down to equilibrium

    or to a periodic state—instead they continued to oscillate in an irregular, aperi-

    odic fashion. Moreover, if he started his simulations from two slightly different

    initial conditions, the resulting behaviors would soon become totally different.

    The implication was that the system was inherently unpredictable—tiny errors

    in measuring the current state of the atmosphere (or any other chaotic system)

    would be ampliied rapidly, eventually leading to embarrassing forecasts. But

    Lorenz also showed that there was structure in the chaos—when plotted in three

    dimensions, the solutions to his equations fell onto a butterly-shaped set of points

    (Figure?1.1.1). He argued that this set had to be “an ininite complex of surfaces”—

    today we would regard it as an example of a fractal.

    x

    z

    Figure 1.1.1

    Strogatz-CROPPED2.pdf 17 5232014 8:40:05 AM4 OVERVIEW

    Lorenz’s work had little impact until the 1970s, the boom years for chaos.

    Here are some of the main developments of that glorious decade. In 1971, Ruelle

    and Takens proposed a new theory for the onset of turbulence in luids, based

    on abstract considerations about strange attractors. A few years later, May found

    examples of chaos in iterated mappings arising in population biology, and wrote

    an inluential review article that stressed the pedagogical importance of studying

    simple nonlinear systems, to counterbalance the often misleading linear intuition

    fostered by traditional education. Next came the most surprising discovery of all,due to the physicist Feigenbaum. He discovered that there are certain universal

    laws governing the transition from regular to chaotic behavior; roughly speaking,completely different systems can go chaotic in the same way. His work established

    a link between chaos and phase transitions, and enticed a generation of physicists

    to the study of dynamics. Finally, experimentalists such as Gollub, Libchaber,Swinney, Linsay, Moon, and Westervelt tested the new ideas about chaos in exper-

    iments on luids, chemical reactions, electronic circuits, mechanical oscillators,and semiconductors.

    Although chaos stole the spotlight, there were two other major developments in

    dynamics in the 1970s. Mandelbrot codiied and popularized fractals, produced

    magniicent computer graphics of them, and showed how they could be applied in

    a variety of subjects. And in the emerging area of mathematical biology, Winfree

    applied the geometric methods of dynamics to biological oscillations, especially

    circadian (roughly 24-hour) rhythms and heart rhythms.

    By the 1980s many people were working on dynamics, with contributions too

    numerous to list. Table 1.1.1 summarizes this history.

    1.2 The Importance of Being Nonlinear

    Now we turn from history to the logical structure of dynamics. First we need to

    introduce some terminology and make some distinctions.

    There are two main types of dynamical systems: differential equations and iter-

    ated maps (also known as difference equations). Differential equations describe

    the evolution of systems in continuous time, whereas iterated maps arise in prob-

    lems where time is discrete. Differential equations are used much more widely in

    science and engineering, and we shall therefore concentrate on them. Later in the

    book we will see that iterated maps can also be very useful, both for providing sim-

    ple examples of chaos, and also as tools for analyzing periodic or chaotic solutions

    of differential equations.

    Now conining our attention to differential equations, the main distinction is

    between ordinary and partial differential equations. For instance, the equation for

    a damped harmonic oscillator

    Strogatz-CROPPED2.pdf 18 5232014 8:40:05 AM5 1.2 THE IMPORTANCE OF BEING NONLINEAR

    m dx

    dt

    b

    dx

    dt

    kx

    2

    2

    0 ++= (1)

    is an ordinary differential equation, because it involves only ordinary derivatives

    dx dt and d

    2

    x dt

    2

    . That is, there is only one independent variable, the time t. In

    contrast, the heat equation

    = ?

    u

    t

    u

    x

    2

    2

    Dynamics — A Capsule History

    1666 Newton Invention of calculus, explanation of planetary

    motion

    1700s Flowering of calculus and classical mechanics

    1800s Analytical studies of planetary motion

    1890s Poincaré Geometric approach, nightmares of chaos

    1920–1950 Nonlinear oscillators in physics and engineering,invention of radio, radar, laser

    1920–1960 Birkhoff Complex behavior in Hamiltonian mechanics

    Kolmogorov

    Arnol’d

    Moser

    1963 Lorenz Strange attractor in simple model of convection

    1970s Ruelle Takens Turbulence and chaos

    May Chaos in logistic map

    Feigenbaum Universality and renormalization, connection

    between chaos and phase transitions

    Experimental studies of chaos

    Winfree Nonlinear oscillators in biology

    Mandelbrot Fractals

    1980s Widespread interest in chaos, fractals, oscillators,and their applications

    Table 1.1.1

    Strogatz-CROPPED2.pdf 19 5232014 8:40:05 AM6 OVERVIEW

    is a partial differential equation—it has both time t and space x as independent

    variables. Our concern in this book is with purely temporal behavior, and so we

    deal with ordinary differential equations almost exclusively.

    A very general framework for ordinary differential equations is provided by the

    system

     …

    

     …

    xfx x

    xfx x

    n

    nn n

    111

    1

    (, , )

    (, , ).

    (2)

    Here the overdots denote differentiation with respect to t. Thus  xdxdt ii

    w . The

    variables x1, … , xn might represent concentrations of chemicals in a reactor, pop-

    ulations of different species in an ecosystem, or the positions and velocities of the

    planets in the solar system. The functions f1, … , fn are determined by the problem

    at hand.

    For example, the damped oscillator (1) can be rewritten in the form of (2),thanks to the following trick: we introduce new variables x1 x and xx 2  . Then

     xx 1 2 , from the deinitions, and

      xx

    b

    m

    x

    k

    m

    x

    b

    m

    x

    k

    m

    x

    2

    21

    ==? ?

    =? ?

    from the deinitions and the governing equation (1). Hence the equivalent system

    (2) is

    

    

    xx

    x

    b

    m

    x

    k

    m

    x

    1

    221

    =

    =? ?

    2

    .

    This system is said to be linear, because all the xi

    on the right-hand side appear

    to the irst power only. Otherwise the system would be nonlinear. Typical nonlinear

    terms are products, powers, and functions of the xi

    , such as x1 x2, ( x1)

    3

    , or cos x2.

    For example, the swinging of a pendulum is governed by the equation

     x

    g

    L

    x += sin , 0

    where x is the angle of the pendulum from vertical, g is the acceleration due to

    gravity, and L is the length of the pendulum. The equivalent system is nonlinear:

    Strogatz-CROPPED2.pdf 20 5232014 8:40:05 AM7 1.2 THE IMPORTANCE OF BEING NONLINEAR

    

    

    xx

    x

    g

    L

    x

    12

    21

    =

    =? sin.

    Nonlinearity makes the pendulum equation very dificult to solve analytically.

    The usual way around this is to fudge, by invoking the small angle approximation

    sin x x x for x  1. This converts the problem to a linear one, which can then be

    solved easily. But by restricting to small x, we’re throwing out some of the physics,like motions where the pendulum whirls over the top. Is it really necessary to make

    such drastic approximations?

    It turns out that the pendulum equation can be solved analytically, in terms of

    elliptic functions. But there ought to be an easier way. After all, the motion of the

    pendulum is simple: at low energy, it swings back and forth, and at high energy

    it whirls over the top. There should be some way of extracting this information

    from the system directly. This is the sort of problem we’ll learn how to solve, using

    geometric methods.

    Here’s the rough idea. Suppose we happen to know a solution to the pendulum

    system, for a particular initial condition. This solution would be a pair of func-

    tions x1(t) and x2(t), representing the position and velocity of the pendulum. If we

    construct an abstract space with coordinates (x1, x2), then the solution ( x1(t), x2(t))

    corresponds to a point moving along a curve in this space (Figure 1.2.1).

    x2

    (x1(t),x2(t))

    (x1(0),x2(0))

    x1

    Figure 1.2.1

    This curve is called a trajectory, and the space is called the phase space for the

    system. The phase space is completely illed with trajectories, since each point can

    serve as an initial condition.

    Our goal is to run this construction in reverse: given the system, we want to

    draw the trajectories, and thereby extract information about the solutions. In

    Strogatz-CROPPED2.pdf 21 5232014 8:40:05 AM8 OVERVIEW

    many cases, geometric reasoning will allow us to draw the trajectories without

    actually solving the system!

    Some terminology: the phase space for the general system (2) is the space with

    coordinates x1, … , xn. Because this space is n-dimensional, we will refer to (2) as

    an n-dimensional system or an nth-order system. Thus n represents the dimension

    of the phase space.

    Nonautonomous Systems

    You might worry that (2) is not general enough because it doesn’t include any

    explicit time dependence. How do we deal with time-dependent or nonautonomous

    equations like the forced harmonic oscillator mx bx kx F t   ++= cos ? In this

    case too there’s an easy trick that allows us to rewrite the system in the form (2). We

    let x1 x and xx 2  as before but now we introduce x3 t. Then  x3 1 and so

    the equivalent system is

    

    

    

    xx

    x

    m

    kx bx Fx

    x

    12

    21 23

    3

    1

    1

    =

    =?+

    =

    (cos) (3)

    which is an example of a three-dimensional system. Similarly, an nth-order

    time-dependent equation is a special case of an (n  1)-dimensional system. By

    this trick, we can always remove any time dependence by adding an extra dimen-

    sion to the system.

    The virtue of this change of variables is that it allows us to visualize a phase

    space with trajectories frozen in it. Otherwise, if we allowed explicit time depen-

    dence, the vectors and the trajectories would always be wiggling—this would ruin

    the geometric picture we’re trying to build. A more physical motivation is that the

    state of the forced harmonic osci l lator is truly three-dimensional: we need to know

    three numbers, x,  x , and t, to predict the future, given the present. So a three-di-

    mensional phase space is natural.

    The cost, however, is that some of our terminology is nontraditional. For exam-

    ple, the forced harmonic oscillator would traditionally be regarded as a second-or-

    der linear equation, whereas we will regard it as a third-order nonlinear system,since (3) is nonlinear, thanks to the cosine term. As we’ll see later in the book,forced oscillators have many of the properties associated with nonlinear systems,and so there are genuine conceptual advantages to our choice of language.

    Why Are Nonlinear Problems So Hard?

    As we’ve mentioned earlier, most nonlinear systems are impossible to solve

    analytically. Why are nonlinear systems so much harder to analyze than linear

    ones? The essential difference is that linear systems can be broken down into parts.

    Then each part can be solved separately and inally recombined to get the answer.

    Strogatz-CROPPED2.pdf 22 5232014 8:40:05 AM9 1.3 A DYNAMICAL VIEW OF THE WORLD

    This idea allows a fantastic simpliication of complex problems, and underlies

    such methods as normal modes, Laplace transforms, superposition arguments,and Fourier analysis. In this sense, a linear system is precisely equal to the sum of

    its parts.

    But many things in nature don’t act this way. Whenever parts of a system inter-

    fere, or cooperate, or compete, there are nonlinear interactions going on. Most of

    everyday life is nonlinear, and the principle of superposition fails spectacularly.

    If you listen to your two favorite songs at the same time, you won’t get double

    the pleasure! Within the realm of physics, nonlinearity is vital to the operation

    of a laser, the formation of turbulence in a luid, and the superconductivity of

    Josephson junctions.

    1.3 A Dynamical View of the World

    Now that we have established the ideas of nonlinearity and phase space, we can

    present a framework for dynamics and its applications. Our goal is to show the

    logical structure of the entire subject. The framework presented in Figure 1.3.1 will

    guide our studies thoughout this book.

    The framework has two axes. One axis tells us the number of variables needed

    to characterize the state of the system. Equivalently, this number is the dimension

    of the phase space. The other axis tells us whether the system is linear or nonlinear.

    For example, consider the exponential growth of a population of organisms.

    This system is described by the irst-order differential equation  xrx where x

    is the population at time t and r is the growth rate. We place this system in the

    column labeled “n 1” because one piece of information—the current value of the

    population x—is suficient to predict the population at any later time. The system

    is also classiied as linear because the differential equation

     xrx is linear in x.

    As a second example, consider the swinging of a pendulum, governed by

     x

    g

    L

    x += sin . 0

    In contrast to the previous example, the state of this system is given by two vari-

    ables: its current angle x and angular velocity  x . (Think of it this way: we need

    the initial values of both x and  x to determine the solution uniquely. For example,if we knew only x, we wouldn’t know which way the pendulum was swinging.)

    Because two variables are needed to specify the state, the pendulum belongs in

    the n 2 column of Figure 1.3.1. Moreover, the system is nonlinear, as discussed

    in the previous section. Hence the pendulum is in the lower, nonlinear half of the

    n 2 column.

    One can continue to classify systems in this way, and the result will be some-

    thing like the framework shown here. Admittedly, some aspects of the picture are

    Strogatz-CROPPED2.pdf 23 5232014 8:40:05 AM10 OVERVIEW

    Continuum

    Exponential growth

    RC circuit

    Radioactive decay

    Oscillations

    Linear oscillator

    Civil engineering,Collective phenomena

    Coupled harmonic oscillators

    Waves and patterns

    Elasticity

    Mass and spring

    structures

    Solid-state physics Wave equations

    RLC circuit

    2-body problem

    (Kepler, Newton)

    Electrical engineering Molecular dynamics

    Equilibrium statistical

    mechanics

    Electromagnetism (Maxwell)

    Quantum mechanics

    (Schr?dinger, Heisenberg, Dirac)

    Heat and diffusion

    Acoustics

    Viscous luids

    The frontier

    Chaos

    Spatio-temporal complexity

    Fixed points

    Pendulum

    Strange attractors

    Coupled nonlinear oscillators

    Nonlinear waves (shocks, solitons)

    Bifurcations Anharmonic oscillators

    (Lorenz) Lasers, nonlinear optics Plasmas

    Overdamped systems,relaxational dynamics

    Limit cycles

    Biological oscillators

    3-body problem (Poincaré)

    Chemical kinetics

    Nonequilibrium statistical

    mechanics

    Earthquakes

    General relativity (Einstein)

    Logistic equation

    for single species

    (neurons, heart cells)

    Predator-prey cycles

    Nonlinear electronics

    (van der Pol, Josephson)

    Iterated maps (Feigenbaum)

    Fractals

    (Mandelbrot)

    Forced nonlinear oscillators

    (Levinson, Smale)

    Nonlinear solid-state physics

    (semiconductors)

    Josephson arrays

    Heart cell synchronization

    Neural networks

    Quantum ield theory

    Reaction-diffusion,biological and chemical waves

    Fibrillation

    Practical uses of chaos

    Quantum chaos ?

    Immune system

    Ecosystems

    Economics

    Turbulent luids (Navier-Stokes)

    Life

    Number of variables

    Nonlinearity

    Linear

    Nonlinear

    Growth, decay, or

    equilibrium

    n = 1 n = 2 n ≥ 3 n >> 1

    Epilepsy

    Figure 1.3.1

    Strogatz-CROPPED2.pdf 24 5232014 8:40:05 AM11 1.3 A DYNAMICAL VIEW OF THE WORLD

    debatable. You might think that some topics should be added, or placed differ-

    ently, or even that more axes are needed—the point is to think about classifying

    systems on the basis of their dynamics.

    There are some striking patterns in Figure 1.3.1. All the simplest systems occur

    in the upper left-hand corner. These are the small linear systems that we learn

    about in the irst few years of college. Roughly speaking, these linear systems

    exhibit growth, decay, or equilibrium when n 1, or osci l lations when n??2. The

    italicized phrases in Figure 1.3.1 indicate that these broad classes of phenomena

    irst arise in this part of the diagram. For example, an RC circuit has n 1 and

    cannot oscillate, whereas an RLC circuit has n 2 and can oscillate.

    The next most familiar part of the picture is the upper right-hand corner. This

    is the domain of classical applied mathematics and mathematical physics where

    the linear partial differential equations live. Here we ind Maxwell’s equations

    of electricity and magnetism, the heat equation, Schr?dinger’s wave equation in

    quantum mechanics, and so on. These partial differential equations involve an

    ininite “continuum” of variables because each point in space contributes addi-

    tional degrees of freedom. Even though these systems are large, they are tractable,thanks to such linear techniques as Fourier analysis and transform methods.

    In contrast, the lower half of Figure 1.3.1—the nonlinear half—is often ignored

    or deferred to later courses. But no more! In this book we start in the lower left cor-

    ner and systematically head to the right. As we increase the phase space dimension

    from n 1 to n 3, we encounter new phenomena at every step, from ixed points

    and bifurcations when n 1, to nonlinear oscillations when n 2, and inally

    chaos and fractals when n 3. In all cases, a geometric approach proves to be

    very powerful, and gives us most of the information we want, even though we usu-

    ally can’t solve the equations in the traditional sense of inding a formula for the

    answer. Our journey will also take us to some of the most exciting parts of modern

    science, such as mathematical biology and condensed-matter physics.

    You’ll notice that the framework also contains a region forbiddingly marked

    “The frontier.” It’s like in those old maps of the world, where the mapmakers

    wrote, “Here be dragons” on the unexplored parts of the globe. These topics are

    not completely unexplored, of course, but it is fair to say that they lie at the limits

    of current understanding. The problems are very hard, because they are both large

    and nonlinear. The resulting behavior is typically complicated in both space and

    time, as in the motion of a turbulent luid or the patterns of electrical activity in a

    ibrillating heart. Toward the end of the book we will touch on some of these prob-

    lems—they will certainly pose challenges for years to come.

    Strogatz-CROPPED2.pdf 25 5232014 8:40:05 AM Part I

    ONE-DIMENSIONAL FLOWS

    Strogatz-CROPPED2.pdf 27 5232014 8:40:06 AM15 2.0 INTRODUCTION

    2

    FLOWS ON THE LINE

    2.0 Introduction

    In Chapter 1, we introduced the general system

    

    

    

    xf x x

    xf x x

    n

    nn n

    111

    1

    ( , ... , )

    ( , ... , )

    and mentioned that its solutions could be visualized as trajectories lowing through

    an n-dimensional phase space with coordinates ( x1, … , xn). At the moment, this

    idea probably strikes you as a mind-bending abstraction. So let’s start slowly,beginning here on earth with the simple case n??1. Then we get a single equation

    of the form

     xf x .

    Here x ( t ) is a real-valued function of time t, and f ( x ) is a smooth real-valued func-

    tion of x. We’ll call such equations one-dimensional or irst-order systems.

    Before there’s any chance of confusion, let’s dispense with two fussy points of

    terminology:

    1. The word system is being used here in the sense of a dynamical system,not in the classical sense of a collection of two or more equations. Thus

    a single equation can be a “system.”

    2. We do not allow f to depend explicitly on time. Time-dependent or

    “nonautonomous” equations of the form  xf x t (, ) are more com-

    plicated, because one needs two pieces of information, x and t, to pre-

    dict the future state of the system. Thus

     xf x t (, )

    should really be

    regarded as a two-dimensional or second-order system, and will there-

    fore be discussed later in the book.

    Strogatz-CROPPED2.pdf 29 5232014 8:40:06 AM16 FLOWS ON THE LINE

    2.1 A Geometric Way of Thinking

    Pictures are often more helpful than formulas for analyzing nonlinear systems.

    Here we illustrate this point by a simple example. Along the way we will introduce

    one of the most basic techniques of dynamics: interpreting a differential equation

    as a vector ield.

    Consider the following nonlinear differential equation:

     xx sin . (1 )

    To emphasize our point about formulas versus pictures, we have chosen one of

    the few nonlinear equations that can be solved in closed form. We separate the

    variables and then integrate:

    dt

    dx

    x

    sin

    ,which implies

    txdx

    xxC

    =

    =? + +

    ∫ csc

    csc cot .

    ln

    To evaluate the constant C, suppose that x??x0 at t??0. Then C?? ln | csc x0 

    cot?x0 |. Hence the solution is

    t

    xx

    xx

    = +

    +

    ln

    csc cot

    csc cot

    .

    00

    (2)

    This result is exact, but a headache to interpret. For example, can you answer the

    following questions?

    1. Suppose x0?

    ?Q 4; describe the qualitative features of the solution x ( t )

    for all t > 0. In particular, what happens as t l d
?

    2. For an arbitrary initial condition x0, what is the behavior of x ( t ) as

    t?l
d
?

    Think about these questions for a while, to see that formula (2) is not transparent.

    In contrast, a graphical analysis of (1) is clear and simple, as shown in

    Figure?2.1.1. We think of t as time, x as the position of an imaginary particle mov-

    ing along the real line, and  x as the velocity of that particle. Then the differential

    equation  xx sin represents a vector ield on the line: it dictates the velocity vec-

    tor  x at each x. To sketch the vector ield, it is convenient to plot  x versus x, and

    then draw arrows on the x-axis to indicate the corresponding velocity vector at

    each x. The arrows point to the right when  x0 and to the left when  x 0 .

    Strogatz-CROPPED2.pdf 30 5232014 8:40:06 AM17 2.1 A GEOMETRIC WAY OF THINKING

    x˙

    x

    π 2π

    Figure 2.1.1

    Here’s a more physical way to think about the vector ield: imagine that luid

    is lowing steadily along the x-axis with a velocity that varies from place to place,according to the rule  xx sin . As shown in Figure?2.1.1, the low is to the right

    when  x0

    and to the left when  x 0. At points where  x 0, there is no low;

    such points are therefore called ixed points. You can see that there are two kinds

    of ixed points in Figure?2.1.1: solid black dots represent stable ixed points (often

    called attractors or sinks, because the low is toward them) and open circles repre-

    sent unstable ixed points (also known as repellers or sources).

    Armed with this picture, we can now easily understand the solutions to the dif-

    ferential equation  xx sin . We just start our imaginary particle at x0 and watch

    how it is carried along by the low.

    This approach allows us to answer the questions above as follows:

    1. Figure? 2.1.1 shows that a particle starting at x0??Q 4 moves to the

    right faster and faster until it crosses x??Q 2 (where sin?x reaches

    its maximum). Then the particle starts slowing down and eventually

    approaches the stable ixed point x??Q from the left. Thus, the quali-

    tative form of the solution is as shown in Figure?2.1.2.

    Note that the curve is concave up at irst, and then concave down;

    this corresponds to the initial acceleration for x  Q 2, followed by the

    deceleration toward x??Q.

    2. The same reasoning applies to any initial condition x0. Figure? 2.1.1

    shows that if  x0 initially, the particle heads to the right and asymp-

    totically approaches the near-

    est stable ixed point. Similarly,if  x 0 initially, the particle

    approaches the nearest stable

    ixed point to its left. If  x 0,then x remains constant. The

    qualitative form of the solu-

    tion for any initial condition is

    sketched in Figure?2.1.3.

    x

    4

    t

    – π

    π

    Figure 2.1.2

    Strogatz-CROPPED2.pdf 31 5232014 8:40:06 AM18 FLOWS ON THE LINE

    x

    t 0

    π

    π

    2π

    2π

    Figure 2.1.3

    In all honesty, we should admit that a picture can’t tell us certain quantitative

    things: for instance, we don’t know the time at which the speed  x

    is greatest. But

    in many cases qualitative information is what we care about, and then pictures are

    ine.

    2.2 Fixed Points and Stability

    The ideas developed in the last section can be extended to any one-dimensional

    system  xf x . We just need to draw the graph of f ( x ) and then use it to sketch

    the vector ield on the real line (the x-axis in Figure?2.2.1).

    x˙

    x

    f (x)

    Figure 2.2.1

    Strogatz-CROPPED2.pdf 32 5232014 8:40:06 AM19 2.2 FIXED POINTS AND STABILITY

    As before, we imagine that a luid is lowing along the real line with a local velocity

    f ( x ). This imaginary luid is called the phase luid, and the real line is the phase

    space. The low is to the right where f ( x ) > 0 and to the left where f ( x )  0. To

    ind the solution to  xfx starting from an arbitrary initial condition x0, we

    place an imaginary particle (known as a phase point) at x0 and watch how it is

    carried along by the low. As time goes on, the phase point moves along the x-axis

    according to some function x ( t ). This function is called the trajectory based at x0,and it represents the solution of the differential equation starting from the initial

    condition x0. A picture like Figure?2.2.1, which shows all the qualitatively different

    trajectories of the system, is called a phase portrait.

    The appearance of the phase portrait is controlled by the ixed points x, deined

    by f ( x)??0; they correspond to stagnation points of the low. In Figure?2.2.1, the

    solid black dot is a stable ixed point (the local low is toward it) and the open dot

    is an unstable ixed point (the low is away from it).

    In terms of the original differential equation, ixed points represent equilibrium

    solutions (sometimes called steady, constant, or rest solutions, since if x??x ini-

    tially, then x ( t )??x for all time). An equilibrium is deined to be stable if all suf-

    iciently small disturbances away from it damp out in time. Thus stable equilibria

    are represented geometrically by stable ixed points. Conversely, unstable equilib-

    ria, in which disturbances grow in time, are represented by unstable ixed points.

    EXAMPLE 2.2.1:

    Find all ixed points for  xx =? 2

    1, and classify their stability.

    Solution: Here f ( x )??x2

    – 1. To ind the ixed points, we set f ( x)??0 and solve

    for x. Thus x??o1. To determine stability, we plot x2

    –
1 and then sketch the

    vector ield (Figure?2.2.2). The low is to the right where x2

    – 1 > 0 and to the left

    where x2

    – 1  0. Thus x??–1 is stable, and x??1 is unstable. ?

    x˙

    x

    f (x) = x2

    1

    Figure 2.2.2

    Strogatz-CROPPED2.pdf 33 5232014 8:40:06 AM20 FLOWS ON THE LINE

    Note that the deinition of stable equilibrium is based on small disturbances;

    certain large disturbances may fail to decay. In Example 2.2.1, all small distur-

    bances to x?? –1 will decay, but a large disturbance that sends x to the right

    of x??1 will not decay—in fact, the phase point will be repelled out to d. To

    emphasize this aspect of stability, we sometimes say that x??–1 is locally stable,but not globally stable.

    EXAMPLE 2.2.2:

    Consider the electrical circuit shown in Figure?2.2.3. A resistor R and a capacitor

    C are in series with a battery of constant dc voltage V0. Suppose that the switch

    is closed at t??0, and that there is no charge on the capacitor initially. Let Q ( t )

    denote the charge on the capacitor at time t p 0. Sketch the graph of?Q ( t ).

    Solution: This type of circuit problem

    is probably fami l iar to you. It is governed

    by linear equations and can be solved

    analytically, but we prefer to illustrate

    the geometric approach.

    First we write the circuit equations.

    As we go around the circuit, the total

    voltage drop must equal zero; hence

    –V0? RI  Q C??0, where I is the cur-

    rent lowing through the resistor. This

    current causes charge to accumulate on

    the capacitor at a rate

     QI . Hence

    + + =

    ==?

    VRQQC

    QfQ V

    R

    Q

    RC

    0

    0

    0 

    

    or

    .

    The graph of f ( Q ) is a straight line with a negative slope (Figure?2.2.4). The corre-

    sponding vector ield has a ixed point where f ( Q )??0, which occurs at Q??CV0.

    The low is to the right where f ( Q ) > 0 and

    to the left where f ( Q )??0. Thus the low is

    always toward Q—it is a stable ixed point.

    In fact, it is globally stable, in the sense that it

    is approached from all initial conditions.

    To sket ch Q ( t ), we start a phase point at

    the origin of Figure? 2.2.4 and imagine how

    it would move. The low carries the phase

    point monotonically toward Q. Its speed

     Q

    I

    R

    C

    V0

    +

    Figure 2.2.3

    Q ˙

    f (Q)

    Q

    Q

    Figure 2.2.4

    Strogatz-CROPPED2.pdf 34 5232014 8:40:06 AM21 2.3 POPULATION GROWTH

    decreases linearly as it approaches the ixed point; therefore Q ( t ) is increasing and

    concave down, as shown in Figure?2.2.5. ?

    EXAMPLE 2.2.3:

    Sketch the phase portrait corresponding

    to  xx x =?cos , and determine the sta-

    bility of all the ixed points.

    Solution: One approach would be to

    plot the function f ( x )??x – cos?x and

    then sketch the associated vector ield.

    This method is valid, but it requires you

    to igure out what the graph of x – cos x

    looks like.

    There’s an easier solution, which exploits the fact that we know how to graph

    y??x and y??cos x separately. We plot both graphs on the same axes and then

    observe that they intersect in exactly one point (Figure?2.2.6).

    y = cos x

    y = x

    x

    x

    Figure 2.2.6

    This intersection corresponds to a ixed point, since x??cos x and therefore

    f ( x)??0. Moreover, when the line lies above the cosine curve, we have x > cos

    x and so  x0: the low is to the right. Similarly, the low is to the left where the

    line is below the cosine curve. Hence x is the only ixed point, and it is unstable.

    Note that we can classify the stability of x, even though we don’t have a formula

    for x itself! ?

    2.3 Population Growth

    The simplest model for the growth of a population of organisms is

     NrN ,where N ( t ) is the population at time t, and r 0 is the growth rate. This model

    Q

    t

    CV0

    Figure 2.2.5

    Strogatz-CROPPED2.pdf 35 5232014 8:40:06 AM22 FLOWS ON THE LINE

    predicts exponential growth:

    N ( t )??N0ert

    , where N0 is the

    population at t??0.

    Of course such exponen-

    tial growth cannot go on for-

    ever. To model the effects of

    overcrowding and limited

    resources, population biolo-

    gists and demographers often

    assume that the per capita

    growth rate

     NN decreases

    when N becomes suficiently

    large, as shown in Figure?2.3.1.

    For small N, the growth

    rate equals r, just as before.

    However, for populations

    larger than a certain carrying

    capacity K, the growth rate

    actually becomes negative; the

    death rate is higher than the

    birth rate.

    A mathematically conve-

    nient way to incorporate these ideas is to assume that the per capita growth rate

     NN decreases linearly with N (Figure?2.3.2).

    This leads to the logistic equation

     NrN N

    K

    =? ?

    ?

    ? ?

    1

    irst suggested to describe the growth of human populations by Verhulst in 1838.

    This equation can be solved analytically (Exercise 2.3.1) but once again we prefer

    a graphical approach. We plot

     N versus N to see what the vector ield looks like.

    Note that we plot only N p 0, since it makes no sense to think about a negative

    population (Figure?2.3.3). Fixed points occur at N??0 and N??K, as found by

    setting

     N 0 and solving for N. By looking at the low in Figure?2.3.3, we see that

    N??0 is an unstable ixed point and N??K is a stable ixed point. In biological

    terms, N??0 is an unstable equilibrium: a small population will grow exponen-

    tially fast and run away from N??0. On the other hand, if N is disturbed slightly

    from K, the disturbance will decay monotonically and N ( t ) l K as t l d.

    In fact, Figure?2.3.3 shows that if we start a phase point at any N0 > 0, it will

    always low toward N??K. Hence the population always approaches the carrying

    capacity.

    The only exception is if N0??0; then there’s nobody around to start reproducing,and so N??0 for all time. (The model does not allow for spontaneous generation!)

    Growth rate

    r

    K N

    Figure 2.3.1

    Growth rate

    r

    K N

    Figure 2.3.2

    Strogatz-CROPPED2.pdf 36 5232014 8:40:06 AM23 2.3 POPULATION GROWTH

    N ˙

    K2 K

    N

    Figure 2.3.3

    Figure? 2.3.3 also allows us to deduce the qualitative shape of the solutions.

    For example, if N0  K 2, the phase point moves faster and faster until it crosses

    N??K 2, where the parabola in Figure?2.3.3 reaches its maximum. Then the phase

    point slows down and eventually creeps toward N??K. In biological terms, this

    means that the population initially grows in an accelerating fashion, and the graph

    of N ( t ) is concave up. But after N??K 2, the derivative

     N

    begins to decrease, and

    so N ( t ) is concave down as it asymptotes to the hor izontal l ine N??K (Figure?2.3.4).

    Thus the graph of N ( t ) is S-shaped or sigmoid for N0  K 2.

    N

    K2

    K

    t

    Figure 2.3.4

    Something qualitatively different occurs if the initial condition N0 lies between

    K 2 and K; now the solutions are decelerating from the start. Hence these solutions

    are concave down for all t. If the population initially exceeds the carrying capacity

    ( N0 > K ), then N ( t ) decreases toward N??K and is concave up. Finally, if N0??0

    or N0??K, then the population stays constant.

    Critique of the Logistic Model

    Before leaving this example, we should make a few comments about the biolog-

    ical validity of the logistic equation. The algebraic form of the model is not to be

    taken literally. The model should really be regarded as a metaphor for populations

    that have a tendency to grow from zero population up to some carrying capacity K.

    Strogatz-CROPPED2.pdf 37 5232014 8:40:06 AM24 FLOWS ON THE LINE

    Originally a much stricter interpretation was proposed, and the model was

    argued to be a universal law of growth (Pearl 1927). The logistic equation was

    tested in laboratory experiments in which colonies of bacteria, yeast, or other

    simple organisms were grown in conditions of constant climate, food supply, and

    absence of predators. For a good review of this literature, see Krebs (1972, pp.

    190–200). These experiments often yielded sigmoid growth curves, in some cases

    with an impressive match to the logistic predictions.

    On the other hand, the agreement was much worse for fruit lies, lour beetles,and other organisms that have complex life cycles involving eggs, larvae, pupae,and adults. In these organisms, the predicted asymptotic approach to a steady

    carrying capacity was never observed—instead the populations exhibited large,persistent luctuations after an initial period of logistic growth. See Krebs (1972)

    for a discussion of the possible causes of these luctuations, including age structure

    and time-delayed effects of overcrowding in the population.

    For further reading on population biology, see Pielou (1969) or May (1981).

    Edelstein–Keshet (1988) and Murray (2002, 2003) are excellent textbooks on math-

    ematical biology in general.

    2.4 Linear Stability Analysis

    So far we have relied on graphical methods to determine the stability of ixed

    points. Frequently one would like to have a more quantitative measure of stability,such as the rate of decay to a stable ixed point. This sort of information may be

    obtained by linearizing about a ixed point, as we now explain.

    Let x be a ixed point, and let I ( t )??x ( t ) – x be a small perturbation away

    from x. To see whether the perturbation grows or decays, we derive a differential

    equation for I. Differentiation yields

      I =?= d

    dt

    xx x (),since x is constant. Thus   II == = + xfx fx ( ). Now using Taylor’s expan-

    sion we obtain

    f ( x  I )??f ( x)  I f ′ ( x)  O ( I2) ,where O ( I2) denotes quadratically small terms in I. Finally, note that f ( x)??0

    since x is a ixed point. Hence

     II I =+ fx O ′() ( ).

    2

    Now if f ′( x) v 0, the O ( I2) terms are negligible and we may write the

    approximation

    Strogatz-CROPPED2.pdf 38 5232014 8:40:06 AM25 2.4 LINEAR STABILITY ANALYSIS

     II ≈ ′ fx () .

    This is a linear equation in I, and is called the linearization about x. It shows that

    the perturbation I ( t ) grows exponentially if f ′( x) 0 and decays if f ′ ( x)  0. If

    f ′( x)??0, the O ( I2) terms are not negligible and a nonlinear analysis is needed

    to determine stability, as discussed in Example 2.4.3 below.

    The upshot is that the slope f ′( x) at the ixed point determines its stability. If

    you look back at the earlier examples, you’ll see that the slope was always negative

    at a stable ixed point. The importance of the sign of f ′( x) was clear from our

    graphical approach; the new feature is that now we have a measure of how stable

    a ixed point is—that’s determined by the magnitude of f ′( x). This magnitude

    plays the role of an exponential growth or decay rate. Its reciprocal 1 | f ′( x)| is

    a characteristic time scale; it determines the time required for x ( t ) to vary signii-

    cantly in the neighborhood of x.

    EXAMPLE 2.4.1:

    Using linear stability analysis, determine the stability of the ixed points for

     xx sin .

    Solution: The ixed points occur where f ( x )??sin?x??0. Thus x??kQ, where

    k is an integer. Then

    ′ == ?

    ?

    ? ?

    fx k

    k

    k

    () cos

    ,Q

    1 even

    1, odd.

    Hence x is unstable if k is even and stable if k is odd. This agrees with the results

    shown in Figure?2.1.1. ?

    EXAMPLE 2.4.2:

    Classify the ixed points of the logistic equation, using linear stability analysis, and

    ind the characteristic time scale in each case.

    Solution: Here fN rN N

    K =? 1 , with ixed points N??0 and N??K. Then

    ′ =? fN r rN

    K

    2

    and so f ′(0)??r and f ′( K )??–r. Hence N??0 is unstable and

    N??K is stable, as found earlier by graphical arguments. In either case, the char-

    acteristic time scale is 11 fN r ′() . ?

    EXAMPLE 2.4.3:

    What can be said about the stability of a ixed point when f ′( x)??0 ?

    Solution: Nothing can be said in general. The stability is best determined on

    a case-by-case basis, using graphical methods. Consider the following examples:

    (a)  xx =? 3

    (b)  xx 3

    (c)  xx 2

    (d)  x 0

    Strogatz-CROPPED2.pdf 39 5232014 8:40:06 AM26 FLOWS ON THE LINE

    Each of these systems has a ixed point x??0 with f ′( x)??0. However the stabil-

    ity is different in each case. Figure?2.4.1 shows that (a) is stable and (b) is unstable.

    Case (c) is a hybrid case we’ll call half-stable, since the ixed point is attracting from

    the left and repelling from the right. We therefore indicate this type of ixed point

    by a half-illed circle. Case (d) is a whole line of ixed points; perturbations neither

    grow nor decay.

    These examples may seem artiicial, but we will see that they arise naturally in the

    context of bifurcations—more about that later. ?

    2.5 Existence and Uniqueness

    Our treatment of vector ields has been very informal. In particular, we have taken

    a cavalier attitude toward questions of existence and uniqueness of solutions to

    the system  xf x . That’s in keeping with the “applied” spirit of this book.

    Nevertheless, we should be aware of what can go wrong in pathological cases.

    x ˙ x ˙

    x ˙ x ˙

    x

    x x

    (d)

    (b) (a)

    (c) ......