Chaos near resonance:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
New York ; Berlin ; Heidelberg ; Barcelona ; Hong Kong ; London
Springer
1999
|
| Series: | Applied mathematical sciences
138 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 401 - 420 |
| Physical Description: | XVI, 427 S. graph. Darst. |
| ISBN: | 0387986979 |
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| 100 | 1 | |a Haller, György |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Chaos near resonance |c G. Haller |
| 264 | 1 | |a New York ; Berlin ; Heidelberg ; Barcelona ; Hong Kong ; London |b Springer |c 1999 | |
| 300 | |a XVI, 427 S. |b graph. Darst. | ||
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| 500 | |a Literaturverz. S. 401 - 420 | ||
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| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Chaotic behavior in systems | |
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Record in the Search Index
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Contents
Preface vii
1 Concepts From Dynamical Systems 1
1.1 Flows, Maps, and Dynamical Systems I
1.2 Ordinary Differential Equations as Dynamical Systems . 3
1.3 Liouville's Theorem 5
1.4 Structural Stability and Bifurcation 6
1.5 Hamiltonian Systems 7
1.6 Poincare Cartan Integral Invariant 10
1.7 Generating Functions 11
1.8 Infinite Dimensional Hamiltonian Systems 12
1.9 Symplectic Reduction 13
1.10 Integrable Systems 15
1.11 KAM Theory and Whiskered Tori 16
1.12 Invariant Manifolds 19
1.13 Stable and Unstable Manifolds 21
1.14 Stable and Unstable Foliations 24
1.15 Strong Stable and Unstable Manifolds 26
1.16 Weak Hyperbolicity 27
1.17 Homoclinic Orbits and Homoclinic Manifolds 28
1.18 Singular Perturbations and Slow Manifolds 29
1.19 Exchange Lemma 31
1.20 Exchange Lemma and Observability 33
1.21 Normal Forms 34
1.22 Averaging Methods 36
xii Contents
1.23 Lambda Lemma and the Homoclinic Tangle 40
1.24 Smale Horseshoes and Symbolic Dynamics 41
1.25 Chaos 45
1.26 Hyperbolic Sets, Transient Chaos, and Strange Attractors . 47
1.27 Melnikov Methods 48
1.28 Silnikov Orbits 51
2 Chaotic Jumping Near Resonances: Finite Dimensional Systems 56
2.1 Resonances and Slow Manifolds 56
2.1.1 The Main Examples 57
2.2 Assumptions and Definitions 58
2.2.1 An Important Class of ODEs 58
2.2.2 /V Chains of Homoclinic Orbits 62
2.2.3 Partially Slow Manifolds 65
2.2.4 /V Pulse Homoclinic Orbits 66
2.3 Passage Lemmas 68
2.3.1 Fenichel Normal Form 68
2.3.2 Entry Conditions and Passage Time 72
2.3.3 Local Estimates 75
2.4 Tracking Lemmas 86
2.4.1 The Local Map 87
2.4.2 The Global Map 88
2.4.3 A Note on the Purely Hamiltonian Case 90
2.5 Energy Lemmas 91
2.5.1 Energy as a Coordinate 91
2.5.2 Energy of Entry Points 92
2.5.3 Improved Local Estimates 95
2.5.4 Energy of Projected Entry Points 96
2.6 Existence of Multipulse Orbits 98
2.6.1 Main Ideas 99
2.6.2 Existence Theorem 103
2.6.3 Remarks on Applications of the Main Theorem . 108
2.6.4 The Most Frequent Case: Chain Independent
Energy Functions 110
2.6.5 Formulation With Other Invariants Ill
2.7 Disintegration of Invariant Manifolds Through Jumping . 114
2.8 Dissipative Chaos: Generalized Silnikov Orbits 116
2.9 Hamiltonian Chaos: Homoclinic Tangles 123
2.9.1 Orbits Homoclinic to Invariant Spheres 125
2.9.2 The Case of n = 0: Orbits Heteroclinic
toSlowm Tori 126
2.9 3 The Case of n = 0, m = 1: Orbits Homoclinic to Slow
Periodic Solutions !27
2.9.4 Resonant Energy Functions 131
2.9.5 Phase Shifts of Opposite Sign 135
Contents xiii
2.10 Universal Homoclinic Bifurcations
in Hamiltonian Applications 136
2.11 Heteroclinic Jumping Between Slow Manifolds 141
2.11.1 Partially Broken Heteroclinic Structures 142
2.11.2 Cat's Eyes Heteroclinic Structures 145
2.12 Partially Slow Manifolds of Higher Codimension 147
2.12.1 Setup 147
2.12.2 Passage Lemmas 151
2.12.3 Tracking Lemmas 151
2.12.4 Energy Lemmas 152
2.12.5 Existence Theorem for Multipulse Orbits 152
2.12.6 Multipulse Silnikov Manifolds 155
2.13 Bibliographical Notes 157
3 Chaos Due to Resonances in Physical Systems 159
3.1 Oscillations of a Parametrically Forced Beam 160
3.1.1 The Mechanical Model 160
3.1.2 The Modal Approximation 162
3.1.3 The Integrable Limit 163
3.1.4 Homoclinic Bifurcations in the Purely Forced
Modal Equations 166
3.1.5 Structurally Stable Heteroclinic Connections for the
Forced Damped Beam 167
3.1.6 Chaos: Generalized Silnikov Orbits and Cycles for the
Forced Damped Beam 172
3.1.7 Numerical Study 176
3.2 Resonant Surface Wave Interactions 178
3.2.1 Derivation of the Amplitude Equations 179
3.2.2 Thee=0Limit 186
3.2.3 Chaotic Dynamics for e 0:
Generalized Silnikov Cycles 188
3.2.4 Passage to the Limit 6 = ^/JI 191
3.2.5 The Inclusion of the O(nv)
Time Dependent Terms 194
3.2.6 Comparison With the Simonelli Gollub Experiment . . 194
3.3 Chaotic Pitching of Nonlinear Vibration Absorbers 199
3.3.1 The Mechanical Model 199
3.3.2 A More General Class of Problems 200
3.4 Mechanical Systems With Widely Spaced Frequencies 202
3.4.1 A Two Mode Model 203
3.4.2 The Geometry of Energy Transfer 204
3.4.3 An Example 207
3.5 Irregular Particle Motion in the Atmosphere 211
3.5.1 The Model 211
3.5.2 Phase Space Geometry and Its Physical Meaning . 212
xiv Contents
3.6 Subharmonic Generation in an Optical Cavity 215
3.6.1 A Two Mode Model 215
3.6.2 The Ideal Cavity (e =0) 217
3.6.3 Chaotic Dynamics for e 0 218
3.7 Intermittent Bursting in Turbulent Boundary Layers 220
3.7.1 Modal Equations With Weak 0(2) ¦ D4
Symmetry Breaking 223
3.7.2 The Slow Manifold 224
3.7.3 Fast Heteroclinic Cycles 225
3.8 Further Problems 228
4 Resonances in Hamiltonian Systems 231
4.1 Resonant Equilibria 231
4.1.1 Birkhoff Normal Form 232
4.1.2 A Class of 1 : 2 : k Resonances 233
4.1.3 Geometry of the Normal Form 235
4.1.4 Homoclinic Orbits in the Two Degree of Freedom
Subsystem 241
4.1.5 Homoclinic Jumping in the Normal Form 244
4.1.6 Homoclinic Jumping and Chaos in the Full Problem . . 245
4.2 The Classical Water Molecule 248
4.2.1 The Normal Form 250
4.2.2 Homoclinic Chaos and Energy Transfer 251
4.3 Dynamics Near Intersecting Resonances 255
4.3.1 Arnold Diffusion in Near Integrable Systems 255
4.3.2 Cross Resonance Diffusion 257
4.3.3 Normal Form for Weak Strong Double Resonances . . 258
4.3.4 The Pendulum Type Hamiltonian 262
4.3.5 Dynamics in the Full Normal Form 269
4.4 An Example From Rigid Body Dynamics 271
4.5 Resonances in A Priori Unstable Systems 277
4.5.1 A Physical Example 278
4.5.2 Whiskered Tori 280
4.5.3 Resonances on Invariant Manifolds 282
4.5.4 Cross Resonance Diffusion, Homoclinic Bifurcations,
and Horseshoes 284
5 Chaotic Jumping Near Resonances: Infinite Dimensional Systems 286
5.1 The Main Examples 287
5.2 Assumptions and Definitions 288
5.2.1 The Phase Space and the Evolution Equation 288
5.2.2 Regularity and Geometric Assumptions 289
5.2.3 /V Chains of Homoclinic Orbits 295
5.3 Invariant Manifolds and Foliations 298
5.3.1 Partially Slow Manifold 298
Contents xv
5.3.2 Preliminary Normal Form 304
5.3.3 Smooth Foliations for W^iM^ ) and
W .(Me.k) ' 306
5.3.4 /V Pulse Homoclinic Orbits 309
5.4 Passage Lemmas 310
5.4.1 Fenichel Normal Form 311
5.4.2 Entry Conditions and Passage Time 314
5.4.3 Local Estimates 316
5.5 Tracking Lemmas 318
5.5.1 The Local Map 319
5.5.2 The Global Map 321
5.6 Energy Lemmas 321
5.6.1 Energy as a Coordinate 322
5.6.2 Energy of Entry Points 323
5.6.3 Improved Local Estimates 326
5.6.4 Energy of Projected Entry Points 328
5.7 Multipulse Homoclinic Orbits in Sobolev Spaces 329
5.7.1 Definitions and Notation 329
5.7.2 Existence Theorem 330
5.7.3 Remarks on Applications of the Main Theorem . 336
5.7.4 Chain Independent Energy Functions 336
5.7.5 Formulation With Other Invariants 336
5.8 Disintegration of Invariant Manifolds Through Jumping . 337
5.9 Generalized Silnikov Orbits 338
5.10 The Purely Hamiltonian Case 341
5.10.1 Universal Homoclinic Bifurcations 342
5.11 Homoclinic Jumping in the Perturbed NLS Equation 342
5.11.1 Homoclinic Tree in the Forced NLS
Equation (D = 0) 342
5.11.2 /V Pulse Orbits in the Damped Forced
NLS Equation 346
5.11.3 Silnikov Type Orbits in the Damped Forced
NLS Equation 348
5.12 Partially Slow Manifolds of Higher Codimension 354
5.12.1 Setup 354
5.12.2 Existence Theorem for Multipulse Orbits 358
5.12.3 Multipulse Silnikov Manifolds 362
5.13 Homoclinic Jumping in the CNLS System 363
5.13.1 Homoclinic Jumping in the Forced CNLS Equations
(«, = ft = 0) _ 364
5.13.2 /V Pulse Jumping Orbits in the Damped Forced
CNLS System (A = 0) 365
5.13.3 /V Pulse Silnikov Manifolds in the
Full CNLS System 366
5.14 Bibliographical Notes 369
xvi Contents
A Elements of Differential Geometry 371
A.I Manifolds 371
A.2 Tangent, Cotangent, and Normal Bundles 372
A.3 Transversality 374
A.4 Maps on Manifolds 375
A.5 Regular and Critical Points 377
A.6 Lie Derivative 377
A.7 Lie Algebras, Lie Groups, and Their Actions 378
A.8 Orbit Spaces 379
A.9 Infinite Dimensional Manifolds 380
A.10 Differential Forms 381
A.ll Maps and Differential Forms 383
A. 12 Exterior Derivative 384
A. 13 Closed and Exact Forms 385
A. 14 Lie Derivative of Forms 385
A. 15 Volume Forms and Orientation 385
A. 16 Symplectic Forms 386
A. 17 Poisson Brackets 387
A. 18 Integration on Manifolds and Stokes's Theorem 388
B Some Facts From Analysis 390
B.I Fourier Series 390
B.2 Gronwall Inequality 392
B.3 Banach and Hilbert Spaces 393
B.4 Differentiation and the Mean Value Theorem 394
B.5 Distributions and Generalized Derivatives 396
B.6 Sobolev Spaces 397
B.7 C°° Bump Functions 398
B.8 Factorization of Functions With a Zero 399
References 401
Symbol Index 421
Index 423 |
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| id | DE-604.BV012918112 |
| illustrated | Illustrated |
| indexdate | 2024-10-30T11:01:58Z |
| institution | BVB |
| isbn | 0387986979 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008793749 |
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| owner_facet | DE-91G DE-BY-TUM DE-634 DE-83 DE-188 DE-20 |
| physical | XVI, 427 S. graph. Darst. |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | Springer |
| record_format | marc |
| series | Applied mathematical sciences |
| series2 | Applied mathematical sciences |
| spelling | Haller, György Verfasser aut Chaos near resonance G. Haller New York ; Berlin ; Heidelberg ; Barcelona ; Hong Kong ; London Springer 1999 XVI, 427 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences 138 Literaturverz. S. 401 - 420 Resonanz - Chaotisches System - Dynamisches System Mathematik Chaotic behavior in systems Resonance Mathematics Resonanz (DE-588)4132123-6 gnd rswk-swf Dynamisches System (DE-588)4013396-5 gnd rswk-swf Chaotisches System (DE-588)4316104-2 gnd rswk-swf Resonanz (DE-588)4132123-6 s Chaotisches System (DE-588)4316104-2 s Dynamisches System (DE-588)4013396-5 s DE-604 Applied mathematical sciences 138 (DE-604)BV000005274 138 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008793749&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Haller, György Chaos near resonance Applied mathematical sciences Resonanz - Chaotisches System - Dynamisches System Mathematik Chaotic behavior in systems Resonance Mathematics Resonanz (DE-588)4132123-6 gnd Dynamisches System (DE-588)4013396-5 gnd Chaotisches System (DE-588)4316104-2 gnd |
| subject_GND | (DE-588)4132123-6 (DE-588)4013396-5 (DE-588)4316104-2 |
| title | Chaos near resonance |
| title_auth | Chaos near resonance |
| title_exact_search | Chaos near resonance |
| title_full | Chaos near resonance G. Haller |
| title_fullStr | Chaos near resonance G. Haller |
| title_full_unstemmed | Chaos near resonance G. Haller |
| title_short | Chaos near resonance |
| title_sort | chaos near resonance |
| topic | Resonanz - Chaotisches System - Dynamisches System Mathematik Chaotic behavior in systems Resonance Mathematics Resonanz (DE-588)4132123-6 gnd Dynamisches System (DE-588)4013396-5 gnd Chaotisches System (DE-588)4316104-2 gnd |
| topic_facet | Resonanz - Chaotisches System - Dynamisches System Mathematik Chaotic behavior in systems Resonance Mathematics Resonanz Dynamisches System Chaotisches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008793749&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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