Nonlinear Dynamics and Chaos

John Michael Tutill Thompson, born on 7 June 1937 in Cottingham, England, is an Honorary Fellow in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge. He is married with two children.H. B. Stewart is the author of Nonlinear Dynamics and Chaos, 2nd Edition, published by Wiley.

"... much more extensive than before." (The Mathematical Review, March 2004)"The fully updated second edition provides a self-contained introduction to the theory and applications of nonlinear dynamics and chaos." (International Journal of Environmental Analytical Chemistry, Vol.84, No.14 – 15, 10 – 20 December 2004)

• Preface viPreface to the First Edition xvAcknowledgements from the First Edition xxi1 Introduction 11.1 Historical background 11.2 Chaotic dynamics in Duffing's oscillator 31.3 Attractors and bifurcations 8Part I Basic Concepts of Nonlinear Dynamics2 An overview of nonlinear phenomena 152.1 Undamped, unforced linear oscillator 152.2 Undamped, unforced nonlinear oscillator 172.3 Damped, unforced linear oscillator 182.4 Damped, unforced nonlinear oscillator 202.5 Forced linear oscillator 212.6 Forced nonlinear oscillator: periodic attractors 222.7 Forced nonlinear oscillator: chaotic attractor 243 Point attractors in autonomous systems 263.1 The linear oscillator 263.2 Nonlinear pendulum oscillations 343.3 Evolving ecological systems 413.4 Competing point attractors 453.5 Attractors of a spinning satellite 474 Limit cycles in autonomous systems 504.1 The single attractor 504.2 Limit cycle in a neural system 514.3 Bifurcations of a chemical oscillator 554.4 Multiple limit cycles in aeroelastic galloping 584.5 Topology of two-dimensional phase space 615 Periodic attractors in driven oscillators 625.1 The Poincare map 625.2 Linear resonance 645.3 Nonlinear resonance 665.4 The smoothed variational equation 715.5 Variational equation for subharmonics 725.6 Basins ofattraction by mapping techniques 735.7 Resonance ofa self-exciting system 765.8 The ABC ofnonlinear dynamics 796 Chaotic attractors in forced oscillators 806.1 Relaxation oscillations and heartbeat 806.2 The Birkhoff±Shaw chaotic attractor 826.3 Systems with nonlinear restoring force 937 Stability and bifurcations of equilibria and cycles 1067.1 Liapunov stability and structural stability 1067.2 Centre manifold theorem 1097.3 Local bifurcations of equilibrium paths 1117.4 Local bifurcations of cycles 1237.5 Basin changes at local bifurcations 1267.6 Prediction ofincipient instability 128Part II Iterated Maps as Dynamical Systems8 Stability and bifurcation of maps 1358.1 Introduction 1358.2 Stability of one-dimensional maps 1388.3 Bifurcations of one-dimensional maps 1398.4 Stability of two-dimensional maps 1498.5 Bifurcations of two-dimensional maps 1568.6 Basin changes at local bifurcations of limit cycles 1589 Chaotic behaviour of one- and two-dimensional maps 1619.1 General outline 1619.2 Theory for one-dimensional maps 1649.3 Bifurcations to chaos 1679.4 Bifurcation diagram of one-dimensional maps 1709.5 HeÂnon map 174Part III Flows, Outstructures, and Chaos10 The geometry of recurrence 18310.1 Finite-dimensional dynamical systems 18310.2 Types ofrecurrent behaviour 18710.3 Hyperbolic stability types for equilibria 19510.4 Hyperbolic stability types for limit cycles 20010.5 Implications ofhyperbolic structure 20511 The Lorenz system 20711.1 A model ofthermal convection 20711.2 First convective instability 20911.3 The chaotic attractor ofLorenz 21411.4 Geometry ofa transition to chaos 2221 2 RoÈssler's band 22912.1 The simply folded band in an autonomous system 22912.2 Return map and bifurcations 23312.3 Smale's horseshoe map 23812.4 Transverse homoclinic trajectories 24312.5 Spatial chaos and localized buckling 24613 Geometry of bifurcations 24913.1 Local bifurcations 24913.2 Global bifurcations in the phase plane 25813.3 Bifurcations of chaotic attractors 266Part IV Applications in the Physical Sciences14 Subharmonic resonances of an offshore structure 28514.1 Basic equation and non-dimensional form 28614.2 Analytical solution for each domain 28814.3 Digital computer program 28914.4 Resonance response curves 29014.5 Effect of damping 29414.6 Computed phase projections 29614.7 Multiple solutions and domains ofattraction 29815 Chaotic motions of an impacting system 30215.1 Resonance response curve 30215.2 Application to moored vessels 30615.3 Period-doubling and chaotic solutions 30616 Escape from a potential well 31316.1 Introduction 31316.2 Analytical formulation 31416.3 Overview ofthe steady-state response 31916.4 The two-band chaotic attractor 32416.5 Resonance ofthe steady states 32816.6 Transients and basins ofattraction 33316.7 Homoclinic phenomena 34016.8 Heteroclinic phenomena 34616.9 Indeterminate bifurcations 352Appendix 359Illustrated Glossary 369Bibliography 402Online Resources 428Index 429