## Nonlinear Dynamics and Chaos: Geometrical Methods for Engineers and ScientistsNonlinear Dynamics and Chaos: Geometrical Methods for Engineers and Scientists J. M. T. Thompson, FRS, University College, London H. B. Stewart, Brookhaven National Laboratory This book is the first comprehensive, systematic account of nonlinear dynamics and chaos, one of the fastest-growing disciplines of applicable mathematics. It is highly illustrated and written in a clear, comprehensible style, progressing gently from the most elementary to the most advanced ideas while requiring little previous knowledge of mathematics. Examples of applications to a wide variety of scientific fields introduce concepts of instabilities, bifurcations and catastrophes, and particular attention is given to the vital new ideas of chaotic behaviour and unpredictability in deterministic systems. This is a book for systems analysts, for mathematicians, and for all those in any field of science or technology who use computers to model systems which change over time. Contents Preface 1 Introduction Part I Basic Concepts of Nonlinear Dynamics 2 An overview of nonlinear phenomena; 3 Point attractors in autonomous systems; 4 Limit cycles in autonomous systems; 5 Periodic attractors in driven oscillators; 6 Chaotic attractors in forced oscillators; 7 Stability and bifurcations of equilibria and cycles Part II Iterated Maps as Dynamical Systems 8 Stability and bifurcation of maps; 9 Chaotic behaviour of one- and two-dimensional maps Part III Flows, Outstructures, and Chaos 10 The geometry of recurrence; 11 The Lorenz system; 12 Rössler?s band; 13 Geometry of bifurcation Part IV Applications in the Physical Sciences 14 Subharmonic resonances of an offshore structure; 15 Chaotic motions of an impacting system; 16 The particle accelerator and Hamiltonian dynamics; 17 Experimental observations of order and chaos References and Bibliography Index |

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### Contents

Introduction | 1 |

An overview of nonlinear phenomena | 15 |

Point attractors in autonomous systems | 27 |

Copyright | |

16 other sections not shown

### Common terms and phrases

amplitude asymptotic basins Belousov-Zhabotinsky reaction centre manifold chaos chaotic attractor chaotic motion Chapter complex control parameter control-phase convection coordinates corresponding curve damping diagram differential equations dimension dimensional discontinuous divergence dynamical system eigenvalues eigenvectors equilibrium point example final motions fixed point flip bifurcation flow fold forced oscillator forcing cycle frequency function geometric Henon Henon map Hopf bifurcation horseshoe horseshoe map illustrated in Figure inset intersection invariant manifolds iterated limit cycle linear oscillator logistic map Lorenz system nonlinear oscillator observed one-dimensional maps origin outset path pendulum period-doubling periodic motion periodic orbits perturbation phase portrait phase projection phase space plane Poincare map Poincare section qualitative quasi-periodic ratio recurrent behaviour region response saddle cycle saddle point saddle-node sequence shown in Figure shows solution spiral starting conditions steady steady-state structurally stable studied subharmonic supercritical three-dimensional phase space tion topological transient transition transverse two-dimensional typical undamped unforced unstable vector field waveform zero