Random Dynamical Systems

Front cover
......Page 1
Title
......Page 5
ISBN......Page 6
Preface
......Page 7
Contents......Page 13
Part I - Random Dynamical Systems and Their Generators
......Page 19
1.1 Definition of a Random Dynamical System......Page 21
1.2 Local Random Dynamical Systems......Page 29
1.3 Perfection of a Crude Cocycle......Page 33
1.4 Invariant Measures for Measurable Random Dynamical Systems
......Page 39
1.5.1 Polish State Space......Page 44
1.5.2 Compact Metric State Space......Page 48
1.6 Invariant Measures on Random Sets......Page 50
1.7 markov measures
......Page 55
1.8 Invariant Measures for Local RDS......Page 58
1.9.1 Bundle RDS......Page 61
1.9.2 Isomorphisms of RDS......Page 63
Summary......Page 67
2.1.1 One-Sided Discrete Time......Page 68
2.1.2 Two-Sided Discrete Time......Page 70
2.1.3 RDS with Independent Increments......Page 71
2.2.1 RDS frorn Randorn Differential Equations......Page 75
2.2.2 The Memoryless Case......Page 82
2.2.3 Random Differential Equations from RDS......Page 84
2.2.4 The Manifold Case......Page 85
2.3.1 Introduction. Two Cultures......Page 86
2.3.2 Semimartingales and Dynamical Systems:Stochastic Calculus for Two-Sided Time......Page 89
2.3.3 Semimartingale Helices with Spatial Parameter......Page 96
2.3.4 RDS from Stochastic Differential Equations......Page 100
2.3.5 Stochastic Differential Equations from RDS......Page 105
2.3.6 White Noise......Page 109
2.3.7 An Example......Page 116
2.3.8 The Manifold Case......Page 119
2.3.9 RDS with Independent Increments......Page 121
Part II - Multiplicative Ergodic Theory
......Page 127
3.1 Introduction......Page 129
3.2.1 Deterministic Theory of Lyapunov Exponents......Page 131
3.2.2 Singular Values......Page 135
3.2.3 Exterior Powers......Page 136
3.3 The Furstenberg-Kesten Theorem......Page 139
3.3.2 The Furstenberg-Kesten Theorem for One-Sided Time......Page 140
3.3.3 The Furstenberg-Kesten Theorem for Two-Sided Tilne......Page 148
3.4.1 The MET for One-Sided Time......Page 152
3.4.2 The MET for Two-Sided Time......Page 171
3.4.3 Examples......Page 177
4.1.1 Tempered Random Variables......Page 181
4.1.2 Lyapunov Cohomology......Page 184
4.2.1 Linearization of a c' RDS......Page 190
4.2.2 The MET for RDS on Manifolds......Page 192
4.2.3 Random Differential Equations......Page 197
4.2.4 Stochastic Differential Equations......Page 199
4.3 Random Lyapunov Metrics and Norms......Page 204
4.3.1 The Control of Non-Uniformity in the MET......Page 205
4.3.2 Random Scalar Products......Page 209
4.3.3 Random Riemannian Metrics on Manifolds......Page 215
5.1 Inverse and Adjoint......Page 219
5.2 The MET on Linear Subbundles and Quotient Spaces......Page 224
5.3.1 Exterior Powers......Page 229
5.3.2 Volume and Determinant......Page 231
5.3.3 Angles......Page 233
5.4 Tensor Product......Page 236
5.6.1 Representation......Page 239
5.6.2 Invariant Measure in the Hyperbolic Case......Page 241
5.6.3 Time Reversibility and Iterated Function Systems......Page 249
Summary......Page 253
6.1 Cocycles on Lie Groups and Their Homogeneous Spaces......Page 254
6.1.1 Group-Valued Co cycles and Their Generators......Page 255
6.1.2 Cocycles Induced by Actions......Page 256
6.2 RDS Induced on S^d-1 and P^d-1
......Page 259
6.2.1 Invariant Measures......Page 260
6.2.2 Furstenberg-Khasminskii Formulas......Page 269
6.2.3 Spectrum and Splitting......Page 278
6.3.1 Invariant Measures......Page 281
6.3.2 Furstenberg-Khasminskii Formulas......Page 284
6.4.1 Sphere Bundle and Projective Bundle......Page 287
6.4.2 Grassmannian Bundles......Page 291
6.5.1 The Concept of Rotation Number of a Plane......Page 295
6.5.2 Rotation Numbers for RDE......Page 303
6.5.3 Rotation Numbers for SDE......Page 316
Part III - Smooth Random Dynamical Systems
......Page 321
Summary......Page 323
7.1 The Problem of Invariant Manifolds......Page 324
7.2 Reductions and Preparations......Page 326
7.2.1 Reductions......Page 327
7.2.2 Preparations......Page 329
7.3 Global Invariant Manifolds......Page 335
7.3.1 Construction of Unstable Manifolds......Page 336
7.3.2 Construction of Stable Manifolds......Page 355
7.3.3 Construction of Center Manifolds......Page 358
7.3.4 The Continuous Time Case......Page 361
7.3.5 Higher Regularity......Page 364
7.3.6 Final Global Invariant Manifold Theorem......Page 378
7.4.1 Invariant Foliations......Page 379
7.4.2 Topological Decoupling......Page 386
7.4.3 Hartman-Grobman Theorem......Page 391
7.5 Local Invariant Manifolds......Page 397
7.5.1 Local Manifolds for Discrete Time......Page 398
7.5.2 Dynamical Characterization and Globalization......Page 404
7.5.3 Local Manifolds for Continuous Time......Page 411
7.6 Examples......Page 414
8.1 Deterministic Prerequisites......Page 423
8.2.1 The Random Cohomological Equation......Page 430
8.2.2 Nonresonant Case......Page 435
8.2.3 Resonant Case......Page 437
8.3 Normal Forms for Random Differential Equations......Page 438
8.3.1 The Random Cohomological Equation......Page 439
8.3.2 Nonresonant Case......Page 443
8.3.3 Resonant Case......Page 444
8.3.4 Examples......Page 448
8.4.1 The Reduction Procedure......Page 451
8.4.2 Parametrized RDE......Page 457
8.4.3 Small Noise: A Case Study......Page 460
8.5 Normal Forms for Stochastic Differential Equations......Page 464
8.5.1 The Random Cohomological Equation......Page 465
8.5.2 Nonresonant Case......Page 468
8.5.3 Small Noise Case......Page 479
9.1 Introduction......Page 483
9.2.1 Definition of a Stochastic Bifurcation Point......Page 486
9.2.2 The Phenomenological Approach......Page 489
9.3.1 Transcritical Bifurcation......Page 495
9.3.2 Pitchfork Bifurcation......Page 498
9.3.4 A General Criterion for Pitchfork Bifurcation......Page 500
9.3.5 Real Noise Case......Page 507
9.3.6 Discrete Time......Page 508
9.4.1 Introduction. Completeness. Linearization......Page 509
9.4.2 Hopf Bifurcation
......Page 515
9.4.3 Pitchfork Bifurcation......Page 528
9.5.1 Baxendale's Sufficient Conditions for D-Bifurcation and Associated P-Bifurcation......Page 536
9.5.2 Further Studies......Page 547
Part IV - Appendices......Page 551
A.1 Ergodic Theory
......Page 553
A.2 Stochastic Processes and Dynamical Systems......Page 560
A.3 Stationary Processes......Page 563
A.4 Markov Processes......Page 566
B.1 Two-Parameter Flows on a Manifold......Page 569
B.2 Spaces of Functions in R^d
......Page 570
B.3 Differential Equations in R^d
......Page 573
B.4 Autonomous Case: Dynamical Systems......Page 575
B.5 Vector Fields and Flows on Manifolds......Page 578
References......Page 581
Index......Page 599
Back cover
......Page 606