ISBN: 3-540-66299-5
TITLE: Shadowing in Dynamical Systems
AUTHOR: Pilyugin, Sergei Yu.
TOC:

Chapter 1. Shadowing Near an Invariant Set 1
1.1 Basic Definitions 1
1.2 Shadowing Near a Hyperbolic Set for a Diffeomorphism 9
1.2.1 Hyperbolic Sets 10
1.2.2 The Classical Shadowing Lemma 14
1.2.3 Shadowing for a Family of Approximate Trajectories 22
1.2.4 The Method of Bowen 27
1.3 Shadowing for Mappings of Banach Spaces 34
1.3.1 Shadowing for a Sequence of Mappings 35
1.3.2 Conditions of Uniqueness 40
1.3.3 Application to the Classical Shadowing Lemma 43
1.3.4 Theorems of Chow-Lin-Palmer and Steinlein-Walther 45
1.3.5 Finite-Dimensional Case 53
1.4 Limit Shadowing 63
1.4.1 Limit Shadowing Property 64
1.4.2 L_p-Shadowing 68
1.4.3 The Sacker-Sell Spectrum and Weighted Shadowing 71
1.4.4 Asymptotic Pseudotrajectories 84
1.5 Shadowing for Flows 88
Chapter 2. Topologically Stable, Structurally Stable, and Generic Systems 103
2.1 Shadowing and Topological Stability 103
2.2 Shadowing in Structurally Stable Systems 109
2.2.1 The Case of a Flow 109
2.2.2 The Case of a Diffeomorphism 145
2.3 Shadowing in Two-Dimensional Diffeomorphisms 158
2.4 C_O-Genericity of Shadowing for Homeomorphisms 172
Chapter 3. Systems with Special Structure 173
3.1 One-Dimensional Systems 173
3.2 Linear and Linearly Induced Systems 182
3.3 Lattice Systems 189
3.4 Global Attractors for Evolution Systems 202
Chapter 4. Numerical Applications of Shadowing 219
4.1 Finite Shadowing 219
4.2 Periodic Shadowing for Flows 223
4.3 Approximation of Spectral Characteristics 233
4.3.1 Evaluation of Upper Lyapunov Exponents 233
4.3.2 Approximation of the Morse Spectrum 238
4.4 Discretizations of PDEs 244
4.4.1 Shadowing in Discretizations 245
4.4.2 Discretization Errors on Unbounded Time Intervals 255
References 259
Index 269
END
