Smooth Dynamical Systems (eBook)

Front Cover 1
Smooth Dynamical Systems 4
Copyright Page 5
Contents 6
Preface 10
Introduction 12
I. The simple pendulum 13
II. A dissipative system 16
III. The spherical pendulum 18
IV. Vector fields and dynamical systems 21
Chapter 1. Some Simple Examples 23
I. Flows and homeomorphisms 23
II. Orbits 24
III. Examples of dynamical systems 25
IV. Constructing systems 28
V. Properties of orbits 34
Appendix 1 37
Chapter 2. Equivalent Systems 39
I. Topological conjugacy 39
II. Homeomorphisms of the circle 40
III. Flow equivalence and topological equivalence 42
IV. Local equivalence 49
V. Limit sets of flows 52
VI. Limit sets of homeomorphisms 57
VII. Non-wandering sets 57
Appendix 2 61
Chapter 3. Integration of Vector Fields 68
I. Vector fields 69
II. Velocity vector fields and integral flows 71
III. Ordinary differential equations 74
IV. Local integrals 78
V. Global integrals 83
Appendix 3 87
Chapter 4. Linear Systems 91
I. Linear flows on Rn 93
II. Linear automorphisms of Rn 99
III. The spectrum of a linear endomorphism . 101
IV. Hyperbolic linear automorphisms 102
V. Hyperbolic linear vector fields 108
Appendix 4 114
Chapter 5. Linearization 120
I. Regular points 122
II. Hartman's theorem 124
III. Hartman's theorem for flows 128
IV. Hyperbolic closed orbits 130
Appendix 5 137
Chapter 6. Stable Manifolds 154
I. The stable manifold at a hyperbolic fixed point of a diffeomorphism 155
II. Stable manifold theory for flows 162
III. The generalized stable manifold theorem 165
Appendix 6 171
Chapter 7. Stable Systems 172
I. Low dimensional systems 173
II. Anosov systems 179
III. Characterization of structural stability 182
IV. Density 191
V. Omega stability 196
VI. Bifurcation 202
Appendix A. Theory of Manifolds 207
Appendix B. Map Spaces 236
Appendix C. The Contraction Mapping Theorem 250
Bibliography 257
Subject Index 264