Nonlinear Dynamics and Chaos

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