Geometric Methods for Discrete Dynamical Systems
Robert W. Easton
- Newyork Oxford University Press 1998
- 157p
1 Examples; A. Logistic Maps; B. Graphical Analysis; C. Hénon Maps; D. The Standard Map Family; E. Arnold's Circle Maps; F. Quadratic Maps; G. Duffing's Equation; H. Interesting Maps; I. Problems; J. Further Reading; 2 Dynamical Systems; A. Discrete and Continuous Dynamical Systems; B. Omega Limit Sets; C. Epsilon Chains; D. The Conley Decomposition Theorem; E. Directed Graphs; F. Local Analysis of Orbits; G. Summary; H. Problems; I. Further Reading; 3 Hyperbolic Fixed Points; A. Linearization; B. Stable and Unstable Manifolds; C. Shadowing and Structural Stability. D. The Hartman- Grobman TheoremE. Smale's Horseshoe Map and Symbolic Dynamics; F. Hyperbolic Invariant Sets; G. Trellis Structure and Resonance Zones; H. Topological Entropy; I. Problems; J. Further Reading; 4 Isolated Invariant Sets and Isolating Blocks; A. Attracting Sets; B. Isolated Invariant Sets and Isolating Blocks; C. Constructing Isolating Blocks; D. Basic Sets; E. Symbolic Dynamics; F. Filtrations of Isolated Invariant Sets; G. Stacks of Isolating Blocks; H. Calculating Directed Graphs; I. Further Reading; 5 The Conley Index; A. The Conley Index of an Isolating Block. B. Continuation of Isolated Invariant SetsC. The Homology Conley Index; D. References; 6 Symplectic Maps; A. Linear Symplectic Maps; B. Classical Mechanics; C. Variational Principles; D. Generating Functions; E. Symplectic Integrators; F. Separatrix Movement; G. Normal Forms; H. Problems; I. Further Reading; 7 Invariant Measures; A. Measure Spaces; B. Invariant Measures; C. Further Reading; Appendix A: Metric Spaces; A. Definitions; B. The Hausdorff Metric; C. Fractals; Appendix B: Numerical Methods for Ordinary Differential Equations. Appendix C: Tangent Bundles, Manifolds, and Differential FormsAppendix D: Symplectic Manifolds; Appendix E: Algebraic Topology; References
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. The theory examines errors which arise from round-off in numerical simulations, from the inexactness of mathematical models used to describe physical processes, and from the effects of external controls. The author provides an introduction accessible to beginning graduate students and emphasizing geometric aspects of the theory. Conley's ideas about rough orbits and chain-recurrence play a central role in the treatment.