This IMA Volume in Mathematics and its Applications PATTERN FORMATION IN CONTINUOUS AND COUPLED SYSTEMS is based on the proceedings of a workshop with the same title, but goes be yond the proceedings by presenting a series of mini-review articles that sur vey, and provide an introduction to, interesting problems in the field. The workshop was an integral part of the 1997-98 IMA program on "EMERG ING APPLICATIONS OF DYNAMICAL SYSTEMS." I would like to thank Martin Golubitsky, University of Houston (Math ematics) Dan Luss, University of Houston (Chemical Engineering), and Steven H. Strogatz, Cornell University (Theoretical and Applied Mechan ics) for their excellent work as organizers of the meeting and for editing the proceedings. I also take this opportunity to thank the National Science Foundation (NSF), and the Army Research Office (ARO), whose financial support made the workshop possible. Willard Miller, Jr., Professor and Director v PREFACE Pattern formation has been studied intensively for most of this cen tury by both experimentalists and theoreticians, and there have been many workshops and conferences devoted to the subject. In the IMA workshop on Pattern Formation in Continuous and Coupled Systems held May 11-15, 1998 we attempted to focus on new directions in the patterns literature."

The goal of this third edition of Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering is the same as previous editions: to provide a good foundation - and a joyful experience - for anyone who’d like to learn about nonlinear dynamics and chaos from an applied perspective.

The presentation stresses analytical methods, concrete examples, and geometric intuition. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors.

The prerequisites are comfort with multivariable calculus and linear algebra, as well as a first course in physics. Ideas from probability, complex analysis, and Fourier analysis are invoked, but they're either worked out from scratch or can be safely skipped (or accepted on faith).

Changes to this edition include substantial exercises about conceptual models of climate change, an updated treatment of the SIR model of epidemics, and amendments (based on recent research) about the Selkov model of oscillatory glycolysis. Equations, diagrams, and every word has been reconsidered and often revised. There are also about 50 new references, many of them from the recent literature.

The most notable change is a new chapter. Chapter 13 is about the Kuramoto model.

The Kuramoto model is an icon of nonlinear dynamics. Introduced in 1975 by the Japanese physicist Yoshiki Kuramoto, his elegant model is one of the rare examples of a high-dimensional nonlinear system that can be solved by elementary means.

Students and teachers have embraced the book in the past, its general approach and framework continue to be sound.

Table of Contents

Preface to the Third Edition

Preface to the Second Edition

Preface to the First Edition

Chapter 1 Overview

1.0 Chaos, Fractals, and Dynamics

1.1 Capsule History of Dynamics

1.2 The Importance of Being Nonlinear

1.3 A Dynamical View of the World

Part I One-Dimensional Flows

Chapter 2 Flows on the Line

2.0 Introduction

2.1 A Geometric Way of Thinking

2.2 Fixed Points and Stability

2.3 Population Growth

2.4 Linear Stability Analysis

2.5 Existence and Uniqueness

2.6 Impossibility of Oscillations

2.7 Potentials

2.8 Solving Equations on the Computer

Exercises for Chapter 2

Chapter 3 Bifurcations

3.0 Introduction

3.1 Saddle-Node Bifurcation

3.2 Transcritical Bifurcation

3.3 Laser Threshold

3.4 Pitchfork Bifurcation

3.5 Overdamped Bead on a Rotating Hoop

3.6 Imperfect Bifurcations and Catastrophes

3.7 Insect Outbreak

Exercises for Chapter 3

Chapter 4 Flows on the Circle

4.0 Introduction

4.1 Examples and Definitions

4.2 Uniform Oscillator

4.3 Nonuniform Oscillator

4.4 Overdamped Pendulum

4.5 Fireflies

4.6 Superconducting Josephson Junctions

Exercises for Chapter 4

Part II Two-Dimensional Flows

Chapter 5 Linear Systems

5.0 Introduction

5.1 Definitions and Examples

5.2 Classification of Linear Systems

5.3 Love Affairs

Exercises for Chapter 5

Chapter 6 Phase Plane

6.0 Introduction

6.1 Phase Portraits

6.2 Existence, Uniqueness, and Topological Consequences

6.3 Fixed Points and Linearization

6.4 Rabbits versus Sheep

6.5 Conservative Systems

6.6 Reversible Systems

6.7 Pendulum

6.8 Index Theory

Exercises for Chapter 6

Chapter 7 Limit Cycles

7.0 Introduction

7.1 Examples

7.2 Ruling Out Closed Orbits

7.3 Poincaré−Bendixson Theorem

7.4 Liénard Systems

7.5 Relaxation Oscillations

7.6 Weakly Nonlinear Oscillators

Exercises for Chapter 7

Chapter 8 Bifurcations Revisited

8.0 Introduction

8.1 Saddle-Node, Transcritical, and Pitchfork Bifurcations

8.2 Hopf Bifurcations

8.3 Oscillating Chemical Reactions

8.4 Global Bifurcations of Cycles

8.5 Hysteresis in the Driven Pendulum and Josephson Junction

8.6 Coupled Oscillators and Quasiperiodicity

8.7 Poincaré Maps

Exercises for Chapter 8

Part III Chaos

Chapter 9 Lorenz Equations

9.0 Introduction

9.1 A Chaotic Waterwheel

9.2 Simple Properties of the Lorenz Equations

9.3 Chaos on a Strange Attractor

9.4 Lorenz Map

9.5 Exploring Parameter Space

9.6 Using Chaos to Send Secret Messages

Exercises for Chapter 9

Chapter 10 One-Dimensional Maps

10.0 Introduction

10.1 Fixed Points and Cobwebs

10.2 Logistic Map: Numerics

10.3 Logistic Map: Analysis

10.4 Periodic Windows

10.5 Liapunov Exponent

10.6 Universality and Experiments

10.7 Renormalization

Exercises for Chapter 10

Chapter 11 Fractals

11.0 Introduction

11.1 Countable and Uncountable Sets

11.2 Cantor Set

11.3 Dimension of Self-Similar Fractals

11.4 Box Dimension

11.5 Pointwise and Correlation Dimensions

Exercises for Chapter 11

Chapter 12 Strange Attractors

12.0 Introduction

12.1 The Simplest Examples

12.2 Hénon Map

12.3 Rössler System

12.4 Chemical Chaos and Attractor Reconstruction

12.5 Forced Double-Well Oscillator

Exercises for Chapter 12

Part IV Collective Behavior

Chapter 13 Kuramoto Model

13.0 Introduction

13.1 Governing Equations

13.2 Visualization and the Order Parameter

13.3 Mean-Field Coupling and Rotating Frame

13.4 Steady State

13.5 Self-Consistency

13.6 Remaining Questions

Exercises for Chapter 13

Answers to Selected Exercises

References

Author Index

Subject Index

This IMA Volume in Mathematics and its Applications PATTERN FORMATION IN CONTINUOUS AND COUPLED SYSTEMS is based on the proceedings of a workshop with the same title, but goes be yond the proceedings by presenting a series of mini-review articles that sur vey, and provide an introduction to, interesting problems in the field. The workshop was an integral part of the 1997-98 IMA program on "EMERG ING APPLICATIONS OF DYNAMICAL SYSTEMS." I would like to thank Martin Golubitsky, University of Houston (Math ematics) Dan Luss, University of Houston (Chemical Engineering), and Steven H. Strogatz, Cornell University (Theoretical and Applied Mechan ics) for their excellent work as organizers of the meeting and for editing the proceedings. I also take this opportunity to thank the National Science Foundation (NSF), and the Army Research Office (ARO), whose financial support made the workshop possible. Willard Miller, Jr., Professor and Director v PREFACE Pattern formation has been studied intensively for most of this cen tury by both experimentalists and theoreticians, and there have been many workshops and conferences devoted to the subject. In the IMA workshop on Pattern Formation in Continuous and Coupled Systems held May 11-15, 1998 we attempted to focus on new directions in the patterns literature."

Over the past three years I have grown accustomed to the puzzled look which appears on people's faces when they hear that I am a mathematician who studies sleep. They wonder, but are usually too polite to ask, what does mathematics have to do with sleep? Instead they ask the questions that fascinate us all: Why do we have to sleep? How much sleep do we really need? Why do we dream? These questions usually spark a lively discussion leading to the exchange of anecdotes, last night's dreams, and other personal information. But they are questions about the func tion of sleep and, interesting as they are, I shall have little more to say about them here. The questions that have concerned me deal instead with the timing of sleep. For those of us on a regular schedule, questions of timing may seem vacuous. We go to bed at night and get up in the morning, going through a cycle of sleeping and waking every 24 hours. Yet to a large extent, the cycle is imposed by the world around us."