Random dynamical systems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2003
|
| Edition: | Corrected 2nd printing |
| Series: | Springer monographs in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 586 S. Ill., graph. Darst. |
| ISBN: | 9783540637585 3540637583 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Titel: Random dynamical systems
Autor: Arnold, Ludwig
Jahr: 2003
Contents
Part I. Random Dynamical Systems and Their Generators
Chapter 1. Basic Definitions. Invariant Measures 3
1.1 Definition of a Random Dynamical System 3
1.2 Local RDS 11
1.3 Perfection of a Crude Cocycle 15
1.4 Invariant Measures for Measurable RDS 21
1.5 Invariant Measures for Continuous RDS 26
1.5.1 Polish State Space 26
1.5.2 Compact Metric State Space 30
1.6 Invariant Measures on Random Sets 32
1.7 Markov Measures 37
1.8 Invariant Measures for Local RDS 40
1.9 RDS on Bundles. Isomorphisms 43
1.9.1 Bundle RDS 43
1.9.2 Isomorphisms of RDS 45
Chapter 2. Generation 49
2.1 Discrete Time: Products of Random Mappings 50
2.1.1 One-Sided Discrete Time 50
2.1.2 Two-Sided Discrete Time 52
2.1.3 RDS with Independent Increments 53
2.2 Continuous Time 1: Random Differential Eqs 57
2.2.1 RDS from Random Differential Equations 57
2.2.2 The Memoryless Case 64
2.2.3 Random Differential Equations from RDS 66
2.2.4 The Manifold Case 67
2.3 Continuous Time 2: Stochastic Differential Eqs 68
2.3.1 Introduction. Two Cultures 68
2.3.2 Semimartingales and Dynamical Systems:
Stochastic Calculus for Two-Sided Time 71
2.3.3 Semimartingale Helices with Spatial Parameter 78
XII Contents
2.3.4 RDS from Stochastic Differential Equations 82
2.3.5 Stochastic Differential Equations from RDS 87
2.3.6 White Noise 91
2.3.7 An Example 98
2.3.8 The Manifold Case 101
2.3.9 RDS with Independent Increments 103
Part II. Multiplicative Ergodic Theory
Chapter 3- The Multiplicative Ergodic Theorem
in Euclidean Space Ill
3.1 Introduction Ill
3.2 Lyapunov Exponents 113
3.2.1 Deterministic Theory of Lyapunov Exponents 113
3.2.2 Singular Values 117
3.2.3 Exterior Powers 118
3.3 Furstenberg-Kesten Theorem 121
3.3.1 The Subadditive Ergodic Theorem 122
3.3.2 The Furstenberg-Kesten Theorem for One-Sided Time 122
3.3.3 The Furstenberg-Kesten Theorem for Two-Sided Time 130
3.4 Multiplicative Ergodic Theorem 134
3.4.1 The MET for One-Sided Time 134
3.4.2 The MET for Two-Sided Time 153
3.4.3 Examples 159
Chapter 4. The Multiplicative Ergodic Theorem
on Bundles and Manifolds 163
4.1 Temperedness. Lyapunov Cohomology 163
4.1.1 Tempered Random Variables 163
4.1.2 Lyapunov Cohomology 166
4.2 The MET on Manifolds 172
4.2.1 Linearization of a C1 RDS 172
4.2.2 The MET for RDS on Manifolds 174
4.2.3 Random Differential Equations 179
4.2.4 Stochastic Differential Equations 181
4.3 Random Lyapunov Metrics and Norms 186
4.3.1 The Control of Non-Uniformity in the MET 187
4.3.2 Random Scalar Products 191
4.3.3 Random Riemannian Metrics on Manifolds 197
Contents XIII
Chapter 5. The MET for Related Linear and Afflne RDS 201
5.1 Inverse and Adjoint 201
5.2 The MET on Linear Subbundles 206
5.3 Exterior Powers, Volume, Angle 211
5.3.1 Exterior Powers 211
5.3.2 Volume and Determinant 213
5.3.3 Angles 215
5.4 Tensor Product 218
5.5 Manifold Versions 221
5.6 Affine RDS 221
5.6.1 Representation 221
5.6.2 Invariant Measure in the Hyperbolic Case 223
5.6.3 Time Reversibility and Iterated Function Systems .... 231
Chapter 6. RDS on Homogeneous Spaces
of the General Linear Group 235
6.1 Cocycles on Lie Groups 236
6.1.1 Group-Valued Cocycles and Their Generators 237
6.1.2 Cocycles Induced by Actions 238
6.2 RDS Induced on S4 1 and P* 1 241
6.2.1 Invariant Measures 242
6.2.2 Furstenberg-Khasminskii Formulas 251
6.2.3 Spectrum and Splitting 260
6.3 RDS on Grassmannians 263
6.3.1 Invariant Measures 263
6.3.2 Furstenberg-Khasminskii Formulas 266
6.4 Manifold Versions 269
6.4.1 Sphere Bundle and Projective Bundle 269
6.4.2 Grassmannian Bundles 273
6.5 Rotation Numbers 277
6.5.1 The Concept of Rotation Number of a Plane 277
6.5.2 Rotation Numbers for RDE 285
6.5.3 Rotation Numbers for SDE 298
Part III. Smooth Random Dynamical Systems
Chapter 7. Invariant Manifolds 305
7.1 The Problem of Invariant Manifolds 306
7.2 Reductions and Preparations 308
7.2.1 Reductions 309
7.2.2 Preparations 311
7.3 Global Invariant Manifolds 317
XIV Contents
7.3.1 Construction of Unstable Manifolds 318
7.3.2 Construction of Stable Manifolds 337
7.3.3 Construction of Center Manifolds 340
7.3.4 The Continuous Time Case 343
7.3.5 Higher Regularity 346
7.3.6 Final Global Invariant Manifold Theorem 360
7.4 Hartman-Grobman Theorem 361
7.4.1 Invariant Foliations 361
7.4.2 Topological Decoupling 368
7.4.3 Hartman-Grobman Theorem 373
7.5 Local Invariant Manifolds 379
7.5.1 Local Manifolds for Discrete Time 380
7.5.2 Dynamical Characterization and Globalization 386
7.5.3 Local Manifolds for Continuous Time 393
7.6 Examples 396
Chapter 8. Normal Forms 405
8.1 Deterministic Prerequisites 405
8.2 Normal Forms for Random Diffeomorphisms 412
8.2.1 The Random Cohomological Equation 412
8.2.2 Nonresonant Case 417
8.2.3 Resonant Case 419
8.3 Normal Forms for RDE 420
8.3.1 The Random Cohomological Equation 421
8.3.2 Nonresonant Case 425
8.3.3 Resonant Case 426
8.3.4 Examples 430
8.4 Normal Form and Center Manifold 433
8.4.1 The Reduction Procedure 433
8.4.2 Parametrized RDE 439
8.4.3 Small Noise: A Case Study 442
8.5 Normal Forms for SDE 446
8.5.1 The Random Cohomological Equation 447
8.5.2 Nonresonant Case 450
8.5.3 Small Noise Case 461
Chapter 9. Bifurcation Theory 465
9.1 Introduction 465
9.2 What is Stochastic Bifurcation? 468
9.2.1 Definition of a Stochastic Bifurcation Point 468
9.2.2 The Phenomenological Approach 471
9.3 Dimension One 477
9.3.1 Transcritical Bifurcation 477
9.3.2 Pitchfork Bifurcation 480
Contents XV
9.3.3 Saddle-Node Case 482
9.3.4 A General Criterion for Pitchfork Bifurcation 482
9.3.5 Real Noise Case 489
9.3.6 Discrete Time 490
9.4 The Noisy Duffing-van der Pol Oscillator 491
9.4.1 Introduction. Completeness. Linearization 491
9.4.2 Hopf Bifurcation 497
9.4.3 Pitchfork Bifurcation 510
9.5 General Dimension. Further Studies 518
9.5.1 Baxendale s Sufficient Conditions for D-Bifurcation
and Associated P-Bifurcation 518
9.5.2 Further Studies 529
Part IV. Appendices
Appendix A. Measurable Dynamical Systems 535
A.I Ergodic Theory 535
A.2 Stochastic Processes and Dynamical Systems 542
A.3 Stationary Processes 545
A.4 Markov Processes 548
Appendix B. Smooth Dynamical Systems 551
B.I Two-Parameter Flows on a Manifold 551
B.2 Spaces of Functions in Rd 552
B.3 Differential Equations in Kd 555
B.4 Autonomous Case: Dynamical Systems 557
B.5 Vector Fields and Flows on Manifolds 560
References 563
Index 581
|
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| discipline | Mathematik |
| edition | Corrected 2nd printing |
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| indexdate | 2024-07-09T21:35:48Z |
| institution | BVB |
| isbn | 9783540637585 3540637583 |
| language | English |
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| spelling | Arnold, Ludwig 1937- Verfasser (DE-588)14200684X aut Random dynamical systems Ludwig Arnold Corrected 2nd printing Berlin [u.a.] Springer 2003 XV, 586 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Ergodic theory Random dynamical systems Stochastic differential equations Dynamisches System (DE-588)4013396-5 gnd rswk-swf Differenzierbares dynamisches System (DE-588)4137931-7 gnd rswk-swf Ergodische Kette (DE-588)4402921-4 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf Zufälliges dynamisches System (DE-588)4335207-8 gnd rswk-swf Zufälliges dynamisches System (DE-588)4335207-8 s DE-604 Ergodische Kette (DE-588)4402921-4 s Differenzierbares dynamisches System (DE-588)4137931-7 s Dynamisches System (DE-588)4013396-5 s Stochastische Differentialgleichung (DE-588)4057621-8 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017382110&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Arnold, Ludwig 1937- Random dynamical systems Ergodic theory Random dynamical systems Stochastic differential equations Dynamisches System (DE-588)4013396-5 gnd Differenzierbares dynamisches System (DE-588)4137931-7 gnd Ergodische Kette (DE-588)4402921-4 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd Zufälliges dynamisches System (DE-588)4335207-8 gnd |
| subject_GND | (DE-588)4013396-5 (DE-588)4137931-7 (DE-588)4402921-4 (DE-588)4057621-8 (DE-588)4335207-8 |
| title | Random dynamical systems |
| title_auth | Random dynamical systems |
| title_exact_search | Random dynamical systems |
| title_full | Random dynamical systems Ludwig Arnold |
| title_fullStr | Random dynamical systems Ludwig Arnold |
| title_full_unstemmed | Random dynamical systems Ludwig Arnold |
| title_short | Random dynamical systems |
| title_sort | random dynamical systems |
| topic | Ergodic theory Random dynamical systems Stochastic differential equations Dynamisches System (DE-588)4013396-5 gnd Differenzierbares dynamisches System (DE-588)4137931-7 gnd Ergodische Kette (DE-588)4402921-4 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd Zufälliges dynamisches System (DE-588)4335207-8 gnd |
| topic_facet | Ergodic theory Random dynamical systems Stochastic differential equations Dynamisches System Differenzierbares dynamisches System Ergodische Kette Stochastische Differentialgleichung Zufälliges dynamisches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017382110&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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