Nonlinear Dynamics and Chaos / Edition 2

Nonlinear Dynamics and Chaos / Edition 2

by J. M. T. Thompson, H. B. Stewart
     
 

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ISBN-10: 0471876453

ISBN-13: 9780471876458

Pub. Date: 03/06/2002

Publisher: Wiley

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. It explores a breadth of topics -- ranging from the basic concepts to applications in the physical sciences -- is highly illustrated, and written in a clear and comprehensible style. Nonlinear

Overview

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. It explores a breadth of topics -- ranging from the basic concepts to applications in the physical sciences -- is highly illustrated, and written in a clear and comprehensible style. 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.

Product Details

ISBN-13:
9780471876458
Publisher:
Wiley
Publication date:
03/06/2002
Edition description:
REV
Pages:
460
Product dimensions:
6.26(w) x 9.02(h) x 1.22(d)

Table of Contents

Prefacexi
Preface to the First Editionxv
Acknowledgements from the First Editionxxi
1Introduction1
1.1Historical background1
1.2Chaotic dynamics in Duffing's oscillator3
1.3Attractors and bifurcations8
Part IBasic Concepts of Nonlinear Dynamics
2An overview of nonlinear phenomena15
2.1Undamped, unforced linear oscillator15
2.2Undamped, unforced nonlinear oscillator17
2.3Damped, unforced linear oscillator18
2.4Damped, unforced nonlinear oscillator20
2.5Forced linear oscillator21
2.6Forced nonlinear oscillator: periodic attractors22
2.7Forced nonlinear oscillator: chaotic attractor24
3Point attractors in autonomous systems26
3.1The linear oscillator26
3.2Nonlinear pendulum oscillations34
3.3Evolving ecological systems41
3.4Competing point attractors45
3.5Attractors of a spinning satellite47
4Limit cycles in autonomous systems50
4.1The single attractor50
4.2Limit cycle in a neural system51
4.3Bifurcations of a chemical oscillator55
4.4Multiple limit cycles in aeroelastic galloping58
4.5Topology of two-dimensional phase space61
5Periodic attractors in driven oscillators62
5.1The Poincare map62
5.2Linear resonance64
5.3Nonlinear resonance66
5.4The smoothed variational equation71
5.5Variational equation for subharmonics72
5.6Basins of attraction by mapping techniques73
5.7Resonance of a self-exciting system76
5.8The ABC of nonlinear dynamics79
6Chaotic attractors in forced oscillators80
6.1Relaxation oscillations and heartbeat80
6.2The Birkhoff-Shaw chaotic attractor82
6.3Systems with nonlinear restoring force93
7Stability and bifurcations of equilibria and cycles106
7.1Liapunov stability and structural stability106
7.2Centre manifold theorem109
7.3Local bifurcations of equilibrium paths111
7.4Local bifurcations of cycles123
7.5Basin changes at local bifurcations126
7.6Prediction of incipient instability128
Part IIIterated Maps as Dynamical Systems
8Stability and bifurcation of maps135
8.1Introduction135
8.2Stability of one-dimensional maps138
8.3Bifurcations of one-dimensional maps139
8.4Stability of two-dimensional maps149
8.5Bifurcations of two-dimensional maps156
8.6Basin changes at local bifurcations of limit cycles158
9Chaotic behaviour of one- and two-dimensional maps161
9.1General outline161
9.2Theory for one-dimensional maps164
9.3Bifurcations to chaos167
9.4Bifurcation diagram of one-dimensional maps170
9.5Henon map174
Part IIIFlows, Outstructures, and Chaos
10The geometry of recurrence183
10.1Finite-dimensional dynamical systems183
10.2Types of recurrent behaviour187
10.3Hyperbolic stability types for equilibria195
10.4Hyperbolic stability types for limit cycles200
10.5Implications of hyperbolic structure205
11The Lorenz system207
11.1A model of thermal convection207
11.2First convective instability209
11.3The chaotic attractor of Lorenz214
11.4Geometry of a transition to chaos222
12Rossler's band229
12.1The simply folded band in an autonomous system229
12.2Return map and bifurcations233
12.3Smale's horseshoe map238
12.4Transverse homoclinic trajectories243
12.5Spatial chaos and localized buckling246
13Geometry of bifurcations249
13.1Local bifurcations249
13.2Global bifurcations in the phase plane258
13.3Bifurcations of chaotic attractors266
Part IVApplications in the Physical Sciences
14Subharmonic resonances of an offshore structure285
14.1Basic equation and non-dimensional form286
14.2Analytical solution for each domain288
14.3Digital computer program289
14.4Resonance response curves290
14.5Effect of damping294
14.6Computed phase projections296
14.7Multiple solutions and domains of attraction298
15Chaotic motions of an impacting system302
15.1Resonance response curve302
15.2Application to moored vessels306
15.3Period-doubling and chaotic solutions306
16Escape from a potential well313
16.1Introduction313
16.2Analytical formulation314
16.3Overview of the steady-state response319
16.4The two-band chaotic attractor324
16.5Resonance of the steady states328
16.6Transients and basins of attraction333
16.7Homoclinic phenomena340
16.8Heteroclinic phenomena346
16.9Indeterminate bifurcations352
Appendix359
Illustrated Glossary369
Bibliography402
Online Resources428
Index429

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