Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering

750 Pages·2008·14.14 MB·English

by  Steven H. Strogatz

NONLINEAR
DYNAMICS AND
CHAOS

NONLINEAR
DYNAMICS AND
CHAOS

With Applications to
Physics, Biology, Chemistry,
and Engineering

STEVEN H. STROGATZ
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Copyright © 1994 by Perseus Books Publishing, LLC

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or
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without the prior written permission of the publisher.

Cataloging-in-Publication Data is available from the Library of Congress

ISBN-13 978-0-7382-0453-6
ISBN-10 0-7382-0453-6

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Text design by Joyce C. Weston
Set in 10-point Times by Compset, Inc.

Cover art is a computer-generated picture of a twisted scroll ring, from Winfree and Strogatz (1984) with
permission. Scroll rings are self-sustaining sources of waves in diverse excitable media, including heart
muscle, neural tissue, and excitable chemical reactions (Winfree and Strogatz 1984, Winfree 1987b).

First paperback printing, December 2000

EBC 30 29 28 27
CONTENTS

Preface

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
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
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
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
Part II. Two-Dimensional Flows
5. Linear Systems
5.0 Introduction
5.1 Definitions and Examples
5.2 Classification of Linear Systems
5.3 Love Affairs
Exercises
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
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 Oscillators
7.6 Weakly Nonlinear Oscillators
Exercises
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
Part III. Chaos
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
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
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
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
Answers to Selected Exercises
References
Author Index
Subject Index
PREFACE

This textbook is aimed at newcomers to nonlinear dynamics and chaos,
especially students taking a first course in the subject. It is based on a one-
semester course I’ve taught for the past several years at MIT and Cornell. My
goal is to explain the mathematics as clearly as possible, and to show how it can
be used to understand some of the wonders of the nonlinear world.
The mathematical treatment is friendly and informal, but still careful.
Analytical methods, concrete examples, and geometric intuition are stressed. 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.
A unique feature of the book is its emphasis on applications. These include
mechanical vibrations, lasers, biological rhythms, superconducting circuits,
insect outbreaks, chemical oscillators, genetic control systems, chaotic
waterwheels, and even a technique for using chaos to send secret messages. In
each case, the scientific background is explained at an elementary level and
closely integrated with the mathematical theory.
Prerequisites

The essential prerequisite is single-variable calculus, including curve-
sketching, Taylor series, and separable differential equations. In a few places,
multivariable calculus (partial derivatives, Jacobian matrix, divergence theorem)
and linear algebra (eigenvalues and eigenvectors) are used. Fourier analysis is
not assumed, and is developed where needed. Introductory physics is used
throughout. Other scientific prerequisites would depend on the applications
considered, but in all cases, a first course should be adequate preparation.