Smooth Dynamical Systems

By: M. C. Irwin, Inger Irwin

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Hardcover | 1 December 2000

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Hardcover
272 Pages

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22.23 x 15.24 x 1.91

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A reprint of this work, first published in 1980. The material covered continues to provide a basis for research in the mathematics of dynamical systems. It addresses: equivalent systems; integration of vector fields; linear systems, linearization and stable manifolds; stable systems; and more.

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Foreword	p. ix
Preface	p. xi
Introduction	p. 1
The simple pendulum	p. 2
A dissipative system	p. 5
The spherical pendulum	p. 7
Vector fields and dynamical systems	p. 10
Some Simple Examples	p. 12
Flows and homeomorphisms	p. 12
Orbits	p. 13
Examples of dynamical systems	p. 14
Constructing systems	p. 17
Properties of orbits	p. 23
Appendix 1	p. 26
Group actions	p. 26
Equivalent Systems	p. 28
Topological conjugacy	p. 28
Homeomorphisms of the circle	p. 29
Flow equivalence and topological equivalence	p. 31
Local equivalence	p. 38
Limit sets of flows	p. 41
Limit sets of homeomorphisms	p. 46
Non-wandering sets	p. 46
Appendix 2	p. 50
Two topological lemmas	p. 50
Oriented orbits in Hausdorff spaces	p. 51
Compactification	p. 54
Integration of Vector Fields	p. 57
Vector fields	p. 58
Velocity vector fields and integral flows	p. 60
Ordinary differential equations	p. 63
Local integrals	p. 67
Global integrals	p. 72
Appendix 3	p. 76
Integrals of perturbed vector fields	p. 76
First integrals	p. 78
Linear Systems	p. 80
Linear flows on R[superscript n]	p. 82
Linear automorphisms of R[superscript n]	p. 88
The spectrum of a linear endomorphism	p. 90
Hyperbolic linear automorphisms	p. 91
Hyperbolic linear vector fields	p. 97
Appendix 4	p. 103
Spectral Theory	p. 103
Linearization	p. 109
Regular points	p. 111
Hartman's theorem	p. 113
Hartman's theorem for flows	p. 117
Hyperbolic closed orbits	p. 119
Appendix 5	p. 126
Smooth linearization	p. 126
Liapunov stability	p. 128
The index of a fixed point	p. 133
Stable Manifolds	p. 143
The stable manifold at a hyperbolic fixed point of a diffeomorphism	p. 144
Stable manifold theory for flows	p. 151
The generalized stable manifold theorem	p. 154
Appendix 6	p. 160
Perturbed stable manifolds	p. 160
Stable Systems	p. 161
Low dimensional systems	p. 162
Anosov systems	p. 168
Characterization of structural stability	p. 171
Density	p. 180
Omega stability	p. 185
Bifurcation	p. 191
Theory of Manifolds	p. 196
Topological manifolds	p. 196
Smooth manifolds and maps	p. 200
Smooth vector bundles	p. 204
The tangent bundle	p. 208
Immersions, embeddings and submersions	p. 211
Sections of vector bundles	p. 213
Tensor bundles	p. 218
Riemannian manifolds	p. 219
Map Spaces	p. 225
Spaces of smooth maps	p. 226
Composition theorems	p. 227
Spaces of sections	p. 234
Spaces of dynamical systems	p. 237
The Contraction Mapping Theorem	p. 239
Bibliography	p. 246
Subject Index	p. 253
Table of Contents provided by Syndetics. All Rights Reserved.

Smooth Dynamical Systems

At a Glance

Hardcover

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