Shadowing in Dynamical Systems : Lecture Notes in Mathematics - Sergei Yu Pilyugin

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Shadowing in Dynamical Systems

By: Sergei Yu Pilyugin, S. y. Pilyugin

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Paperback | 1 October 1999

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296 Pages

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24.13 x 15.88 x 1.91

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This book is an introduction to the theory of shadowing of approximate trajectories in dynamical systems by exact ones. This is the first book completely devoted to the theory of shadowing. It shows the importance of shadowing theory for both the qualitative theory of dynamical systems and the theory of numerical methods. Shadowing Methods allow us to estimate differences between exact and approximate solutions on infinite time intervals and to understand the influence of error terms. The book is intended for specialists in dynamical systems, for researchers and graduate students in the theory of numerical methods.

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Shadowing Near an Invariant Set	p. 1
Basic Definitions	p. 1
Shadowing Near a Hyperbolic Set for a Diffeomorphism	p. 9
Hyperbolic Sets	p. 10
The Classical Shadowing Lemma	p. 14
Shadowing for a Family of Approximate Trajectories	p. 22
The Method of Bowen	p. 27
Shadowing for Mappings of Banach Spaces	p. 34
Shadowing for a Sequence of Mappings	p. 35
Conditions of Uniqueness	p. 40
Application to the Classical Shadowing Lemma	p. 43
Theorems of Chow-Lin-Palmer and Steinlein-Walther	p. 45
Finite-Dimensional Case	p. 53
Limit Shadowing	p. 63
Limit Shadowing Property	p. 64
L[subscript p]-Shadowing	p. 68
The Sacker-Sell Spectrum and Weighted Shadowing	p. 71
Asymptotic Pseudotrajectories	p. 84
Shadowing for Flows	p. 88
Topologically Stable, Structurally Stable, and Generic Systems	p. 103
Shadowing and Topological Stability	p. 103
Shadowing in Structurally Stable Systems	p. 109
The Case of a Flow	p. 109
The Case of a Diffeomorphism	p. 145
Shadowing in Two-Dimensional Diffeomorphisms	p. 158
C[superscript 0]-Genericity of Shadowing for Homeomorphisms	p. 172
Systems with Special Structure	p. 173
One-Dimensional Systems	p. 173
Linear and Linearly Induced Systems	p. 182
Lattice Systems	p. 189
Global Attractors for Evolution Systems	p. 202
Numerical Applications of Shadowing	p. 219
Finite Shadowing	p. 219
Periodic Shadowing for Flows	p. 223
Approximation of Spectral Characteristics	p. 233
Evaluation of Upper Lyapunov Exponents	p. 233
Approximation of the Morse Spectrum	p. 238
Discretizations of PDEs	p. 244
Shadowing in Discretizations	p. 245
Discretization Errors on Unbounded Time Intervals	p. 255
References	p. 259
Index	p. 269
Table of Contents provided by Blackwell. All Rights Reserved.

Shadowing in Dynamical Systems

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Paperback

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