Normally hyperbolic invariant manifolds in dynamical systems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York u.a.
Springer
1994
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| Series: | Applied mathematical sciences
105 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 193 S. graph. Darst. |
| ISBN: | 354094205X 038794205X |
Staff View
MARC
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| 100 | 1 | |a Wiggins, Stephen |d ca. 20./21. Jh. |e Verfasser |0 (DE-588)1247764664 |4 aut | |
| 245 | 1 | 0 | |a Normally hyperbolic invariant manifolds in dynamical systems |c Stephen Wiggins |
| 264 | 1 | |a New York u.a. |b Springer |c 1994 | |
| 300 | |a IX, 193 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Applied mathematical sciences |v 105 | |
| 650 | 4 | |a Differentiable dynamical systems | |
| 650 | 4 | |a Hyperbolic spaces | |
| 650 | 4 | |a Invariant manifolds | |
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Record in the Search Index
| _version_ | 1804123998244569088 |
|---|---|
| adam_text | Contents
Preface v
1 Introduction, Motivation, and Background 1
1.1 Examples of the Use of Invariant Manifold Ideas and
Methods in Applications 1
1.2 Issues and Methods 4
1.2.1 Existence of Invariant Manifolds 5
1.2.2 The Persistence and Differentiability of Invariant
Manifolds Under Perturbaton 5
1.2.3 Behavior Near an Invariant Manifold—
Stable, Unstable, and Center Manifolds 6
1.2.4 More Refined Behavior Near Invariant Manifolds—
Foliations of Stable, Unstable, and Center Manifolds 7
1.2.5 The Liapunov Perron Method 7
1.2.6 Hadamard s Method—The Graph Transform .... 8
1.2.7 The Lie Transform, or Deformation Method 8
1.2.8 Irwin s Method 9
1.3 Motivational Examples 9
2 Background from the Theory of Differentiable Manifolds 21
2.1 The Definition of a Manifold and Examples 21
2.2 Derivatives, Tangent Spaces, Normal Spaces, and Other
Structures on Manifolds 28
2.2.1 The Normal Space at a Point 33
2.2.2 Tangent Bundles and Normal Bundles 34
2.2.3 The Neighborhood of a Manifold 36
2.3 Manifolds with Boundary 39
2.4 An Example 41
3 Persistence of Overflowing Invariant Manifolds—
Fenichel s Theorem 51
3.1 Overflowing Invariant Manifolds and Generalized Lyapunov
Type Numbers 56
3.1.1 Stability of M and Generalized Lyapunov Type
Numbers 57
viii Contents
3.1.2 Some Properties of Generalized Lyapunov Type
Numbers 62
3.2 Coordinates and Dynamics Near M 70
3.2.1 Local Coordinates Near M 70
3.2.2 Jiggling the Normal Bundle to Gain a Derivative,
and the Construction of a Neighborhood of M ... 72
3.2.3 Local Expressions for the Time T Map Generated by
the Flow 74
3.2.4 The Perturbed Vector Field and Flow 79
3.2.5 Smallness and Closeness Parameters 81
3.2.6 The Space of Sections of the Transversal Bundle . . 81
3.2.7 The Graph Transform 81
3.3 Statement and Proof of the Main Theorem 85
3.3.1 Differentiability of the Persisting Overflowing
Invariant Manifold 92
4 The Unstable Manifold of an Overflowing Invariant
Manifold 111
4.1 Generalized Lyapunov Type Numbers Ill
4.2 Local Coordinates Near M 114
4.3 The GR I Estimates 116
4.4 The Space of Sections of N e | u»_ V3 over TV | us_ V3 .... 119
4.5 The Unstable Manifold Theorem . ~! .* 120
4.5.1 Differentiability of WU(M) 128
4.5.2 Wu (M) Satisfies the Hypotheses of the Persistence
Theorem for Overflowing Invariant Manifolds .... 129
5 Foliations of Unstable Manifolds 131
5.1 Generalized Lypaunov Type Numbers 131
5.2 Local Coordinates Near M 134
5.3 The GR I Estimates 134
5.4 The Space of Families of w Dimensional Lipschitz Surfaces . 135
5.5 Sequences of Contraction Maps 136
5.6 The Theorem on Foliations of Unstable Manifolds 138
5.6.1 The Proof of the Theorem 139
5.7 Persistence of the Fibers Under Perturbations 157
6 Miscellaneous Properties and Results 159
6.1 Inflowing Invariant Manifolds 159
6.2 Compact, Boundary less Invariant Manifolds 159
6.3 Boundary Modifications 160
6.4 Parameter Dependent Vector Fields 162
6.5 Continuation of Overflowing Invariant Manifolds—
The Size of the Perturbation 163
6.6 Discrete Time Dynamics, or Maps 163
Contents ix
7 Examples 165
7.1 Invariant Manifolds Near a Hyperbolic Fixed Point 165
7.2 Invariant Manifolds Near a Nonhyperbolic Fixed Point ... 166
7.3 Weak Hyperbolicity 168
7.4 Asymptotic Expansions for Invariant Manifolds 170
7.5 The Invariant Manifold Structure Associated with the Study
of Orbits Homoclinic to Resonances 171
7.5.1 The Analytic and Geometric Structure of the
Unperturbed Equations 172
7.5.2 The Analytic and Geometric Structure of the
Perturbed Equations 174
References 185
Index 191
|
| any_adam_object | 1 |
| author | Wiggins, Stephen ca. 20./21. Jh |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
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| discipline | Mathematik |
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| id | DE-604.BV009654904 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:38:40Z |
| institution | BVB |
| isbn | 354094205X 038794205X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006383996 |
| oclc_num | 844307690 |
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| physical | IX, 193 S. graph. Darst. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
| publishDateSort | 1994 |
| publisher | Springer |
| record_format | marc |
| series | Applied mathematical sciences |
| series2 | Applied mathematical sciences |
| spelling | Wiggins, Stephen ca. 20./21. Jh. Verfasser (DE-588)1247764664 aut Normally hyperbolic invariant manifolds in dynamical systems Stephen Wiggins New York u.a. Springer 1994 IX, 193 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences 105 Differentiable dynamical systems Hyperbolic spaces Invariant manifolds Invariante Mannigfaltigkeit (DE-588)4348147-4 gnd rswk-swf Dynamisches System (DE-588)4013396-5 gnd rswk-swf Invariante Mannigfaltigkeit (DE-588)4348147-4 s DE-604 Dynamisches System (DE-588)4013396-5 s Applied mathematical sciences 105 (DE-604)BV000005274 105 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006383996&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wiggins, Stephen ca. 20./21. Jh Normally hyperbolic invariant manifolds in dynamical systems Applied mathematical sciences Differentiable dynamical systems Hyperbolic spaces Invariant manifolds Invariante Mannigfaltigkeit (DE-588)4348147-4 gnd Dynamisches System (DE-588)4013396-5 gnd |
| subject_GND | (DE-588)4348147-4 (DE-588)4013396-5 |
| title | Normally hyperbolic invariant manifolds in dynamical systems |
| title_auth | Normally hyperbolic invariant manifolds in dynamical systems |
| title_exact_search | Normally hyperbolic invariant manifolds in dynamical systems |
| title_full | Normally hyperbolic invariant manifolds in dynamical systems Stephen Wiggins |
| title_fullStr | Normally hyperbolic invariant manifolds in dynamical systems Stephen Wiggins |
| title_full_unstemmed | Normally hyperbolic invariant manifolds in dynamical systems Stephen Wiggins |
| title_short | Normally hyperbolic invariant manifolds in dynamical systems |
| title_sort | normally hyperbolic invariant manifolds in dynamical systems |
| topic | Differentiable dynamical systems Hyperbolic spaces Invariant manifolds Invariante Mannigfaltigkeit (DE-588)4348147-4 gnd Dynamisches System (DE-588)4013396-5 gnd |
| topic_facet | Differentiable dynamical systems Hyperbolic spaces Invariant manifolds Invariante Mannigfaltigkeit Dynamisches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006383996&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005274 |
| work_keys_str_mv | AT wigginsstephen normallyhyperbolicinvariantmanifoldsindynamicalsystems |