Geometric Methods for Discrete Dynamical Systems: (50 Oxford Engineering Science Series)

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.

Table of Contents:
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

Review :
"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 Poincaré 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 / Isolated Invariant Sets and Isolating Blocks / The Conley Index / Symplectic Maps / Invariant Measures."--Bulletin of Math Books "This book provides a nice introduction to the theory of dynamical systems. The first chapter starts with the discussion of examples which play a fundamental role. Some of these examples can be traced back to physical situations. The author explains some of the fundamental ideas of the modern theory of dynamical systems. He explains carefully why the behaviour of individual solutions is less important than the knowledge of the behaviour of most solutions. . . . Altogether the book is carefully written, the main ideas are well motivated and presented. . . . The book is suited for an introductory course in dynamical systems . . ."--Signa "This introduction to discrete dynamical systems starts from a discussion of a series of fundamental examples . . . These are used to introduce the principal notions and tools in dynamical systems . . . Proofs are given in a two-dimensional setting, but the methods easily generalize to higher dimensions. The general aim of this book is to present an introduction to the theory of isolated invariant sets and the discrete Conley index. The approach to the discrete Conley index presented here is different from the original one given by M. Mrozek. The main objects are the isolating blocks for the isolated invariant set. The Conley index of an isolating block is introduced and the Conley index for and isolated invariant set is defined by taking the direct limit of the indices of a sequence of isolating blocks which converge to the isolated invariant set. . . . The last two chapters present some basic facts from symplectic dynamics and invariant measure theory."--Mathematical Reviews "This book is a concise and rigorous introduction to the theory of dynamical systems, plunging right into the basic abstract concepts. The book is accessible for high-level mathematics students with a prerequisite understanding of linear algebra and functions of several variables, and an advanced background in analysis and some related subjects. There are examples and a lucid treatment of qualitative ideas. This is a very fine book, clearly written with a lot of basic subjects thoroughly discussed. The book is useful as background material for all of us and it is very suitable for a seminar on dynamical systems theory." - F. Verhuist, Boekbesprekingen "Robert W. Easton's Geometric Methods for Discrete Dynamical Systems can be used as a reference for mathematicians and as a supplement or text for standard mathematics graduate courses in dynamial systems. . . .this book is a useful reference for geometric and topographical aspects of dynamical systems theory, and it should help these points of view to gain a wider audeince among theoretical and applied non-linear dynamicists." SIAM Review "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 Poincaré 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 / Isolated Invariant Sets and Isolating Blocks / The Conley Index / Symplectic Maps / Invariant Measures."--Bulletin of Math Books "This book provides a nice introduction to the theory of dynamical systems. The first chapter starts with the discussion of examples which play a fundamental role. Some of these examples can be traced back to physical situations. The author explains some of the fundamental ideas of the modern theory of dynamical systems. He explains carefully why the behaviour of individual solutions is less important than the knowledge of the behaviour of most solutions. . . . Altogether the book is carefully written, the main ideas are well motivated and presented. . . . The book is suited for an introductory course in dynamical systems . . ."--Signa "This introduction to discrete dynamical systems starts from a discussion of a series of fundamental examples . . . These are used to introduce the principal notions and tools in dynamical systems . . . Proofs are given in a two-dimensional setting, but the methods easily generalize to higher dimensions. The general aim of this book is to present an introduction to the theory of isolated invariant sets and the discrete Conley index. The approach to the discrete Conley index presented here is different from the original one given by M. Mrozek. The main objects are the isolating blocks for the isolated invariant set. The Conley index of an isolating block is introduced and the Conley index for and isolated invariant set is defined by taking the direct limit of the indices of a sequence of isolating blocks which converge to the isolated invariant set. . . . The last two chapters present some basic facts from symplectic dynamics and invariant measure theory."--Mathematical Reviews "This book is a concise and rigorous introduction to the theory of dynamical systems, plunging right into the basic abstract concepts. The book is accessible for high-level mathematics students with a prerequisite understanding of linear algebra and functions of several variables, and an advanced background in analysis and some related subjects. There are examples and a lucid treatment of qualitative ideas. This is a very fine book, clearly written with a lot of basic subjects thoroughly discussed. The book is useful as background material for all of us and it is very suitable for a seminar on dynamical systems theory." - F. Verhuist, Boekbesprekingen "Robert W. Easton's Geometric Methods for Discrete Dynamical Systems can be used as a reference for mathematicians and as a supplement or text for standard mathematics graduate courses in dynamial systems. . . .this book is a useful reference for geometric and topographical aspects of dynamical systems theory, and it should help these points of view to gain a wider audeince among theoretical and applied non-linear dynamicists." SIAM Review