Small perturbations of chaotic dynamical systems

Abstract

CONTENTS Introduction 1. Chaotic dynamical systems 1.1. A stochastic attractor. Hyperbolic systems. The method of symbolic dynamics 1.2. Dynamical systems with singularities. Piecewise stretching maps. The operator approach 1.3. The Li-Yorke chaos 2. Random perturbations of stochastic attractors 2.1. Random perturbations of hyperbolic systems 2.2. Random perturbations of systems with singularities 2.3. Stabilization of unstable invariant measures 2.4. The “typical” (“generic”) property and properties of ε- and ε-a-trajectories 2.5. The most probable trajectories 3. Space discretization in chaotic systems 3.1. Definitions and basic examples 3.2. Properties of the periodic trajectories under discretization 3.3. Statistical probability under discretizations 3.4. Stability of stochastic attractors under space discretizations 3.5. Chaos under a partial space discretization 4. Time discretization in dynamical systems 4.1. Chaos under time discretizations References