Geometric Methods for Discrete Dynamical Systems

Geometric Methods for Discrete Dynamical Systems

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This book is for those interested in dynamical systems. It assumes a solid undergraduate training in mathematics. Geometrical methods are developed to study the process of iteration, which involves taking the output of a function and feeding it back as input. Iteration processes are used to produce fractals and wavelets, and to numerically approximate solutions to ordinary and partical differential equations. Each iteration procedure generates a discrete dynamical system. These systems are at the heart of many numerical algorithms. Essentially all mathematical models of evolving physical systems can be viewed as discrete dynamical systems. This book attempts to present the fundamental ideas of discrete dynamical systems as clearly and geometrically as possible. Illustrative examples of dynamical systems are presented in the first chapter. The second chapter gives a review of the typology of metric spaces. The third presents basic results and establishes a philosophy of dynamics which is strongly influenced by the work of Charles Conley. The stable manifold and local structural stability theorems are presented in the fourth chapter. Invariant sets and isolating blocks are defined in the fifth. The sixth develops what is called the Conley Index in the context of discrete dynamics, and the final chpater covers measure-preserving and symplectic maps. The book would be suitable for use as a main text for a graduate course in dynamical systems, and as a reference for engineers and more

Product details

  • Hardback | 172 pages
  • 162.8 x 241.6 x 20.1mm | 453.6g
  • Oxford University Press Inc
  • New York, United States
  • English
  • New.
  • 38 line figures
  • 0195085450
  • 9780195085457

Review quote

"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 Poincare 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"show more

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 Topologyshow more