Geometric methods for discrete dynamical systems

Robert W. Easton0195085450, 9780195085457, 9781423735236

This book looks at dynamics as an iteration process where the output of a function is fed back as an input to determine the evolution of an initial state over time.

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Table of contents :
Cover ……Page 1
OXFORD ENGINEERING SCIENCE SERIES ……Page 2
RECENT TITLES IN THE OXFORD ENGINEERING SCIENCE SERIES ……Page 3
Title page ……Page 4
Date-line ……Page 5
Dedication ……Page 6
Preface ……Page 8
Contents ……Page 12
1 Examples ……Page 18
A. Logistic Maps ……Page 19
B. Graphical Analysis ……Page 20
C. Henon Maps ……Page 24
D. The Standard Map Family ……Page 27
E. Arnold’s Circle Maps ……Page 28
F. Quadratic Maps ……Page 29
G. Duffing’s Equation ……Page 30
H. Interesting Maps ……Page 31
J. Further Reading ……Page 32
A. Discrete and Continuous Dynamical Systems ……Page 33
B. Omega Limit Sets ……Page 36
C. Epsilon Chains ……Page 40
D. The Conley Decomposition Theorem ……Page 43
E. Directed Graphs ……Page 45
F. Local Analysis of Orbits ……Page 46
H. Problems ……Page 48
I. Further Reading ……Page 49
A. Linearization ……Page 50
B. Stable and Unstable Manifolds ……Page 54
C. Shadowing and Structural Stability ……Page 64
D. The Hartman- Grobman Theorem ……Page 66
E. Smale’s Horseshoe Map and Symbolic Dynamics ……Page 69
F. Hyperbolic Invariant Sets ……Page 73
G. Trellis Structure and Resonance Zones ……Page 74
H. Topological Entropy ……Page 85
I. Problems ……Page 86
J. Further Reading ……Page 87
A. Attracting Sets ……Page 88
B. Isolated Invariant Sets and Isolating Blocks ……Page 92
C. Constructing Isolating Blocks ……Page 97
E. Symbolic Dynamics ……Page 98
F. Filtrations of Isolated Invariant Sets ……Page 100
G. Stacks of Isolating Blocks ……Page 101
H. Calculating Directed Graphs ……Page 103
I. Further Reading ……Page 107
A. The Conley Index of an Isolating Block ……Page 108
B. Continuation of Isolated Invariant Sets ……Page 111
C. The Homology Conley Index ……Page 113
D. References ……Page 117
A. Linear Symplectic Maps ……Page 118
B. Classical Mechanics ……Page 122
C. Variational Principles ……Page 125
D. Generating Functions ……Page 128
E. Symplectic Integrators ……Page 132
F. Separatrix Movement ……Page 133
G. Normal Forms ……Page 136
I. Further Reading ……Page 140
A. Measure Spaces ……Page 141
B. Invariant Measures ……Page 142
C. Further Reading ……Page 145
A. Definitions ……Page 146
B. The Hausdorff Metric ……Page 149
C. Fractals ……Page 150
Appendix B Numerical Methods for Ordinary Differential Equations ……Page 152
Appendix C Tangent Bundles, Manifolds, and Differential Forms ……Page 157
Appendix D Symplectic Manifolds ……Page 162
Appendix E Algebraic Topology ……Page 164
References ……Page 167
Index ……Page 170