Random dynamical systems

Table of Contents

Part I. Random Dynamical Systems and Their Generators
Chapter 1. Basic Definitions. Invariant
Measures
1.1 Definition of a Random Dynamical System
1.2 Local RDS
1.3 Perfection of a Crude Cocycle
1.4 Invariant Measures for Measurable RDS
1.5 Invariant Measures for Continuous RDS
1.5.1 Polish State Space
1.5.2 Compact Metric State Space
1.6 Invariant Measures on Random Sets
1.7 Markov Measures
1.8 Invariant Measures for Local RDS
1.9 RDS on Bundles. Isomorphisms
1.9.1 Bundle RDS
1.9.2 Isomorphisms of RDS
Chapter 2. Generation
2.1 Discrete Time: Products of Random
Mappings
2.1.1 One-Sided Discrete Time
2.1.2 Two-Sided Discrete Time
2.1.3 RDS with Independent Increments
2.2 Continuous Time 1: Random
Differential Eqs.
2.2.1 RDS from Random Differential Equations
2.2.2 The Memoryless Case
2.2.3 Random Differential Equations from RDS
2.2.4 The Manifold Case
2.3 Continuous Time 2: Stochastic
Differential Eqs.
2.3.1 Introduction. Two Cultures
2.3.2 Semimartingales and Dynamical
Systems: Stochastic Calculus for Two-Sided Time
2.3.3 Semimartingale Helices with Spatial Parameter
2.3.4 RDS from Stochastic Differential Equations
2.3.5 Stochastic Differential Equations from RDS
2.3.6 White Noise
2.3.7 An Example
2.3.8 The Manifold Case
2.3.9 RDS with Independent Increments
Part II. Multiplicative Ergodic Theory
Chapter 3. The Multiplicative Ergodic
Theorem in Euclidean Space
3.1 Introduction
3.2 Lyapunov Exponents
3.2.1 Deterministic Theory of Lyapunov Exponents
3.2.2 Singular Values
3.2.3 Exterior Powers
3.3 Furstenberg-Kesten Theorem
3.3.1 The Subadditive Ergodic Theorem
3.3.2 The Furstenberg-Kesten Theorem for One-Sided Time
3.3.3 The Furstenberg-Kesten Theorem for Two-Sided Time
3.4 Multiplicative Ergodic Theorem
3.4.1 The MET for One-Sided Time
3.4.2 The MET for Two-Sided Time
3.4.3 Examples
Chapter 4. The Multiplicative Ergodic
Theorem on Bundles and Manifolds
4.1 Temperedness. Lyapunov Cohomology
4.1.1 Tempered Random Variables
4.1.2 Lyapunov Cohomology
4.2 The MET on Manifolds
4.2.1 Linearization of a C(1) RDS
4.2.2 The MET for RDS on Manifolds
4.2.3 Random Differential Equations
4.2.4 Stochastic Differential Equations
4.3 Random Lyapunov Metrics and Norms
4.3.1 The Control of Non-Uniformity in
the MET
4.3.2 Random Scalar Products
4.3.3 Random Riemannian Metrics on
Manifolds
Chapter 5. The MET for Related Linear and Affine RDS
5.1 Inverse and Adjoint
5.2 The MET on Linear Subbundles
5.3 Exterior Powers, Volume, Angle
5.3.1 Exterior Powers
5.3.2 Volume and Determinant
5.3.3 Angles
5.4 Tensor Product
5.5 Manifold Versions
5.6 Affine RDS
5.6.1 Representation
5.6.2 Invariant Measure in the
Hyperbolic Case
5.6.3 Time Reversibility and Iterated Function Systems
Chapter 6. RDS on Homogeneous Spaces of the
General Linear Group
6.1 Cocycles on Lie Groups
6.1.1 Group-Valued Cocycles and Their Generators
6.1.2 Cocycles Induced by Actions
6.2 RDS Induced on S(d-1) and P(d-1)
6.2.1 Invariant Measures
6.2.2 Furstenberg-Khasminskii Formulas
6.2.3 Spectrum and Splitting
6.3 RDS on Grassmannians
6.3.1 Invariant Measures
6.3.2 Furstenberg-Khasminskii Formulas
6.4 Manifold Versions
6.4.1 Sphere Bundle and Projective Bundle
6.4.2 Grassmannian Bundles
6.5 Rotation Numbers
6.5.1 The Concept of Rotation Number of a Plane
6.5.2 Rotation Numbers for RDE
6.5.3 Rotation Numbers for SDE
Part III. Smooth Random Dynamical Systems
Chapter 7. Invariant Manifolds
7.1 The Problem of Invariant Manifolds
7.2 Reductions and Preparations
7.2.1 Reductions
7.2.2 Preparations
7.3 Global Invariant Manifolds
7.3.1 Construction of Unstable Manifolds
7.3.2 Construction of Stable Manifolds
7.3.3 Construction of Center Manifolds
7.3.4 The Continuous Time Case
7.3.5 Higher Regularity
7.3.6 Final Global Invariant Manifold Theorem
7.4 Hartman-Grobman Theorem
7.4.1 Invariant Foliations
7.4.2 Topological Decoupling
7.4.3 Hartman-Grobman Theorem
7.5 Local Invariant Manifolds
7.5.1 Local Manifolds for Discrete Time
7.5.2 Dynamical Characterization and Globalization
7.5.3 Local Manifolds for Continuous Time
7.6 Examples
Chapter 8. Normal Forms
8.1 Deterministic Prerequisites
8.2 Normal Forms for Random Diffeomorphisms
8.2.1 The Random Cohomological Equation
8.2.2 Nonresonant Case
8.2.3 Resonant Case
8.3 Normal Forms for RDE
8.3.1 The Random Cohomological Equation
8.3.2 Nonresonant Case
8.3.3 Resonant Case
8.3.4 Examples
8.4 Normal Form and Center Manifold
8.4.1 The Reduction Procedure
8.4.2 Parametrized RDE
8.4.3 Small Noise: A Case Study
8.5 Normal Forms for SDE
8.5.1 The Random Cohomological Equation
8.5.2 Nonresonant Case
8.5.3 Small Noise Case
Chapter 9. Bifurcation Theory
9.1 Introduction
9.2 What is Stochastic Bifurcation?
9.2.1 Definition of a Stochastic Bifurcation Point
9.2.2 The Phenomenological Approach
9.3 Dimension One
9.3.1 Transcritical Bifurcation
9.3.2 Pitchfork Bifurcation
9.3.3 Saddle-Node Case
9.3.4 A General Criterion for Pitchfork Bifurcation
9.3.5 Real Noise Case
9.3.6 Discrete Time
9.4 The Noisy Duffing-van der Pol Oscillator
9.4.1 Introduction, Completeness,
Linearization
9.4.2 Hopf Bifurcation
9.4.3 Pitchfork Bifurcation
9.5 General Dimension. Further Studies
9.5.1 Baxendale's Sufficient Conditions
for D-Bifurcation and Associated P-Bifurcation
9.5.2 Further Studies
Part IV. Appendices
Appendix A. Measurable Dynamical Systems
A.1 Ergodic Theory
A.2 Stochastic Processes and Dynamical Systems
A.3 Stationary Processes
A.4 Markov Processes
Appendix B. Smooth Dynamical Systems
B.1 Two-Parameter Flows on a Manifold
B.2 Spaces of Functions in R(d)
B.3 Differential Equations in R(d)
B.4 Autonomous Case: Dynamical Systems
B.5 Vector Fields and Flows on Manifolds
References
Index