Geometric Methods for Discrete Dynamical Systems [Hardcover]

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. The book will be a useful reference for mathematicians, scientists, and engineers studying this field, and an ideal text for graduate courses in dynamical systems.

1. Examples
2. Dynamical Systems
3. Hyperbolic Fixed Points
4. Isolated Invariant Sets and Isolating Blocks
5. The Conley Index
6. Symplectic Maps
7. Invariant Measures
Appendix A Metric Spaces
Appendix B Numerical Methods for Ordinary Differential Equations
Appendix C Tangent Bundles, Manifolds, and Differential Forms
Appendix D Symplectic Manifolds
Appendix E Algebraic Topology
References
Index

This book addresses the iterative processes used to approximate solutions to ordinary and partial differential equations. . .The first of seven chapters presents examples of dynamical systems and mapping the iterative processes and the second gives basic definitions and behavior of dynamical system orbits. The following chapters treat the stable manifold, invariant sets, the Conley index, and symplectic maps. The last chapter introduces invariant means, including the Poincar? theorem. --Bulletin of the American Meteorological Society

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. Contents: Examples / Dynamical Systems / Hyperbolic Fixed Points / l³D