Preface vi

Preface to the First Edition xv

Acknowledgements from the First Edition xxi

1 Introduction 1

1.1 Historical background 1

1.2 Chaotic dynamics in Duffing's oscillator 3

1.3 Attractors and bifurcations 8

Part I Basic Concepts of Nonlinear Dynamics

2 An overview of nonlinear phenomena 15

2.1 Undamped, unforced linear oscillator 15

2.2 Undamped, unforced nonlinear oscillator 17

2.3 Damped, unforced linear oscillator 18

2.4 Damped, unforced nonlinear oscillator 20

2.5 Forced linear oscillator 21

2.6 Forced nonlinear oscillator: periodic attractors 22

2.7 Forced nonlinear oscillator: chaotic attractor 24

3 Point attractors in autonomous systems 26

3.1 The linear oscillator 26

3.2 Nonlinear pendulum oscillations 34

3.3 Evolving ecological systems 41

3.4 Competing point attractors 45

3.5 Attractors of a spinning satellite 47

4 Limit cycles in autonomous systems 50

4.1 The single attractor 50

4.2 Limit cycle in a neural system 51

4.3 Bifurcations of a chemical oscillator 55

4.4 Multiple limit cycles in aeroelastic galloping 58

4.5 Topology of two-dimensional phase space 61

5 Periodic attractors in driven oscillators 62

5.1 The Poincare map 62

5.2 Linear resonance 64

5.3 Nonlinear resonance 66

5.4 The smoothed variational equation 71

5.5 Variational equation for subharmonics 72

5.6 Basins ofattraction by mapping techniques 73

5.7 Resonance ofa self-exciting system 76

5.8 The ABC ofnonlinear dynamics 79

6 Chaotic attractors in forced oscillators 80

6.1 Relaxation oscillations and heartbeat 80

6.2 The Birkhoff±Shaw chaotic attractor 82

6.3 Systems with nonlinear restoring force 93

7 Stability and bifurcations of equilibria and cycles 106

7.1 Liapunov stability and structural stability 106

7.2 Centre manifold theorem 109

7.3 Local bifurcations of equilibrium paths 111

7.4 Local bifurcations of cycles 123

7.5 Basin changes at local bifurcations 126

7.6 Prediction ofincipient instability 128

Part II Iterated Maps as Dynamical Systems

8 Stability and bifurcation of maps 135

8.1 Introduction 135

8.2 Stability of one-dimensional maps 138

8.3 Bifurcations of one-dimensional maps 139

8.4 Stability of two-dimensional maps 149

8.5 Bifurcations of two-dimensional maps 156

8.6 Basin changes at local bifurcations of limit cycles 158

9 Chaotic behaviour of one- and two-dimensional maps 161

9.1 General outline 161

9.2 Theory for one-dimensional maps 164

9.3 Bifurcations to chaos 167

9.4 Bifurcation diagram of one-dimensional maps 170

9.5 HeÂnon map 174

Part III Flows, Outstructures, and Chaos

10 The geometry of recurrence 183

10.1 Finite-dimensional dynamical systems 183

10.2 Types ofrecurrent behaviour 187

10.3 Hyperbolic stability types for equilibria 195

10.4 Hyperbolic stability types for limit cycles 200

10.5 Implications ofhyperbolic structure 205

11 The Lorenz system 207

11.1 A model ofthermal convection 207

11.2 First convective instability 209

11.3 The chaotic attractor ofLorenz 214

11.4 Geometry ofa transition to chaos 222

1 2 RoÈssler's band 229

12.1 The simply folded band in an autonomous system 229

12.2 Return map and bifurcations 233

12.3 Smale's horseshoe map 238

12.4 Transverse homoclinic trajectories 243

12.5 Spatial chaos and localized buckling 246

13 Geometry of bifurcations 249

13.1 Local bifurcations 249

13.2 Global bifurcations in the phase plane 258

13.3 Bifurcations of chaotic attractors 266

Part IV Applications in the Physical Sciences

14 Subharmonic resonances of an offshore structure 285

14.1 Basic equation and non-dimensional form 286

14.2 Analytical solution for each domain 288

14.3 Digital computer program 289

14.4 Resonance response curves 290

14.5 Effect of damping 294

14.6 Computed phase projections 296

14.7 Multiple solutions and domains ofattraction 298

15 Chaotic motions of an impacting system 302

15.1 Resonance response curve 302

15.2 Application to moored vessels 306

15.3 Period-doubling and chaotic solutions 306

16 Escape from a potential well 313

16.1 Introduction 313

16.2 Analytical formulation 314

16.3 Overview ofthe steady-state response 319

16.4 The two-band chaotic attractor 324

16.5 Resonance ofthe steady states 328

16.6 Transients and basins ofattraction 333

16.7 Homoclinic phenomena 340

16.8 Heteroclinic phenomena 346

16.9 Indeterminate bifurcations 352

Appendix 359

Illustrated Glossary 369

Bibliography 402

Online Resources 428

Index 429

Covering one of the fastest growing areas of applied mathematics, Nonlinear Dynamics and Chaos: Second Edition, is a fully updated edition of this highly respected text. Covering a breadth of topics, ranging from the basic concepts to applications in the physical sciences, the book is highly illustrated and written in a clear and comprehensible style.

• Provides a self-contained introduction to the theory and applications of nonlinear dynamics and chaos.
• Introduces the concepts of instabilities, bifurcations, and catastrophes.
• Each idea is carefully explained and supported by examples.
• Features many applications to a wide variety of scientific fields.
• Includes an illustrated glossary of geometrical dynamics.
• Features a supplementary bibliography of further reading.
• Assumes minimal background knowledge.

Nonlinear Dynamics and Chaos: Second Edition provides an excellent introduction to the subject for students of mathematics, engineering, physics and applied science. It will also appeal to the many researchers who work with computer models of systems that change over time.