3rd Edition

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

By Steven H Strogatz Copyright 2024
    616 Pages
    by Chapman & Hall

    616 Pages
    by Chapman & Hall

    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.

    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


    Author Index

    Subject Index


    Steven Strogatz is the Schurman Professor of Applied Mathematics at Cornell University. His honors include MIT's highest teaching prize, a lifetime achievement award for the communication of mathematics to the general public, and membership in the American Academy of Arts and Sciences. His research on a wide variety of nonlinear systems from synchronized fireflies to small-world networks has been featured in the pages of Scientific American, Nature, Discover, Business Week, and The New York Times.