Stability of nonautonomous differential equations:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2008
|
| Schriftenreihe: | Lecture notes in mathematics
1926 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 285 S. graph. Darst. |
| ISBN: | 9783540747741 3540747745 |
Internformat
MARC
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| 100 | 1 | |a Barreira, Luis |d 1968- |e Verfasser |0 (DE-588)128529717 |4 aut | |
| 245 | 1 | 0 | |a Stability of nonautonomous differential equations |c Luis Barreira ; Claudia Valls |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2008 | |
| 300 | |a XIV, 285 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in mathematics |v 1926 | |
| 650 | 4 | |a Differentialgleichungssystem - Nichtautonomes System - Stabilität | |
| 650 | 4 | |a Differential equations | |
| 650 | 4 | |a Lyapunov stability | |
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Datensatz im Suchindex
| _version_ | 1804137190551191552 |
|---|---|
| adam_text | Contents
Preface VII
1 Introduction 1
1.1 Exponential contractions 2
1.2 Exponential dichotomies and stable manifolds 4
1.3 Topological conjugacies 7
1.4 Center manifolds, symmetry and reversibility 10
1.5 Lyapunov regularity and stability theory 13
Part I Exponential dichotomies
2 Exponential dichotomies and basic properties 19
2.1 Nonuniform exponential dichotomies 19
2.2 Stable and unstable subspaces 22
2.3 Existence of dichotomies and ergodic theory 24
3 Robustness of nonuniform exponential dichotomies 27
3.1 Robustness in semi infinite intervals 27
3.1.1 Formulation of the results 27
3.1.2 Proofs 29
3.2 Stable and unstable subspaces 40
3.3 Robustness in the line 42
3.4 The case of strong dichotomies 49
Part II Stable manifolds and topological conjugacies
4 Lipschitz stable manifolds 55
4.1 Setup and standing assumptions 55
4.2 Existence of Lipschitz stable manifolds 56
XII Contents
4.3 Nonuniformly hyperbolic trajectories 59
4.4 Proof of the existence of stable manifolds 61
4.4.1 Preliminaries 61
4.4.2 Solution on the stable direction 62
4.4.3 Behavior under perturbations of the data 64
4.4.4 Reduction to an equivalent problem 67
4.4.5 Construction of the stable manifolds 69
4.5 Existence of Lipschitz unstable manifolds 71
5 Smooth stable manifolds in R™ 75
5.1 C1 stable manifolds 75
5.2 Nonuniformly hyperbolic trajectories 77
5.3 Example of a C1 flow with stable manifolds 78
5.4 Proof of the C1 regularity 79
5.4.1 A priori control of derivatives and auxiliary estimates .. 79
5.4.2 Lyapunov norms 83
5.4.3 Existence of an invariant family of cones 86
5.4.4 Construction and continuity of the stable spaces 92
5.4.5 Behavior of the tangent sets 95
5.4.6 C1 regularity of the stable manifolds 100
5.5 Ck stable manifolds 102
5.6 Proof of the Ck regularity 106
5.6.1 Method of proof 106
5.6.2 Linear extension of the vector field 107
5.6.3 Characterization of the stable spaces 110
5.6.4 Tangential component of the extension Ill
5.6.5 Ck regularity of the stable manifolds 116
6 Smooth stable manifolds in Banach spaces 119
6.1 Existence of smooth stable manifolds 120
6.2 Nonuniformly hyperbolic trajectories 121
6.3 Proof of the existence of smooth stable manifolds 123
6.3.1 Functional spaces 123
6.3.2 Derivatives of compositions 125
6.3.3 A priori control of the derivatives 127
6.3.4 Holder regularity of the top derivatives 129
6.3.5 Solution on the stable direction 133
6.3.6 Behavior under perturbations of the data 136
6.3.7 Construction of the stable manifolds 138
7 A nonautonomous Grobman Hartman theorem 145
7.1 Conjugacies for flows 145
7.2 Conjugacies for maps 146
7.2.1 Setup 147
7.2.2 Existence of topological conjugacies 149
Contents XIII
7.3 Holder regularity of the conjugacies 155
7.3.1 Main statement 155
7.3.2 Lyapunov norms 156
7.3.3 Proof of the Holder regularity 157
7.4 Proofs of the results for flows 162
7.4.1 Reduction to discrete time 162
7.4.2 Proofs 164
Part III Center manifolds, symmetry and reversibility
8 Center manifolds in Banach spaces 171
8.1 Standing assumptions 171
8.2 Existence of center manifolds 173
8.3 Proof of the existence of center manifolds 176
8.3.1 Functional spaces 176
8.3.2 Lipschitz property of the derivatives 179
8.3.3 Solution on the central direction 184
8.3.4 Reduction to an equivalent problem 187
8.3.5 Construction of the center manifolds 191
9 Reversibility and equivariance in center manifolds 197
9.1 Reversibility for nonautonomous equations 197
9.1.1 The notion of reversibility 197
9.1.2 Relation with the autonomous case 199
9.1.3 Nonautonomous reversible equations 201
9.2 Reversibility in center manifolds 202
9.2.1 Formulation of the main result 202
9.2.2 Auxiliary results 206
9.2.3 Proof of the reversibility 212
9.3 Equivariance for nonautonomous equations 213
9.4 Equivariance in center manifolds 214
Part IV Lyapunov regularity and stability theory
10 Lyapunov regularity and exponential dichotomies 219
10.1 Lyapunov exponents and regularity 219
10.2 Existence of nonuniform exponential dichotomies 222
10.3 Bounds for the regularity coefficient 226
10.3.1 Lower bound 226
10.3.2 Upper bound in the triangular case 227
10.3.3 Reduction to the triangular case 232
10.4 Characterizations of regularity 235
10.5 Equations with negative Lyapunov exponents 241
XIV Contents
10.5.1 Lipschitz stable manifolds 242
10.5.2 Smooth stable manifolds 243
10.6 Measure preserving flows 244
11 Lyapunov regularity in Hilbert spaces 249
11.1 The notion of regularity 249
11.2 Upper triangular reduction 252
11.3 Regularity coefficient and Perron coefficient 254
11.4 Characterizations of regularity 256
11.5 Lower and upper bounds for the coefficients 259
12 Stability of nonautonomous equations in Hilbert spaces.... 265
12.1 Setup 265
12.2 Stability results 267
12.3 Smallness of the perturbation 268
12.4 Norm estimates for the evolution operators 270
12.5 Proofs of the stability results 273
References 277
Index 283
|
| adam_txt |
Contents
Preface VII
1 Introduction 1
1.1 Exponential contractions 2
1.2 Exponential dichotomies and stable manifolds 4
1.3 Topological conjugacies 7
1.4 Center manifolds, symmetry and reversibility 10
1.5 Lyapunov regularity and stability theory 13
Part I Exponential dichotomies
2 Exponential dichotomies and basic properties 19
2.1 Nonuniform exponential dichotomies 19
2.2 Stable and unstable subspaces 22
2.3 Existence of dichotomies and ergodic theory 24
3 Robustness of nonuniform exponential dichotomies 27
3.1 Robustness in semi infinite intervals 27
3.1.1 Formulation of the results 27
3.1.2 Proofs 29
3.2 Stable and unstable subspaces 40
3.3 Robustness in the line 42
3.4 The case of strong dichotomies 49
Part II Stable manifolds and topological conjugacies
4 Lipschitz stable manifolds 55
4.1 Setup and standing assumptions 55
4.2 Existence of Lipschitz stable manifolds 56
XII Contents
4.3 Nonuniformly hyperbolic trajectories 59
4.4 Proof of the existence of stable manifolds 61
4.4.1 Preliminaries 61
4.4.2 Solution on the stable direction 62
4.4.3 Behavior under perturbations of the data 64
4.4.4 Reduction to an equivalent problem 67
4.4.5 Construction of the stable manifolds 69
4.5 Existence of Lipschitz unstable manifolds 71
5 Smooth stable manifolds in R™ 75
5.1 C1 stable manifolds 75
5.2 Nonuniformly hyperbolic trajectories 77
5.3 Example of a C1 flow with stable manifolds 78
5.4 Proof of the C1 regularity 79
5.4.1 A priori control of derivatives and auxiliary estimates . 79
5.4.2 Lyapunov norms 83
5.4.3 Existence of an invariant family of cones 86
5.4.4 Construction and continuity of the stable spaces 92
5.4.5 Behavior of the tangent sets 95
5.4.6 C1 regularity of the stable manifolds 100
5.5 Ck stable manifolds 102
5.6 Proof of the Ck regularity 106
5.6.1 Method of proof 106
5.6.2 Linear extension of the vector field 107
5.6.3 Characterization of the stable spaces 110
5.6.4 Tangential component of the extension Ill
5.6.5 Ck regularity of the stable manifolds 116
6 Smooth stable manifolds in Banach spaces 119
6.1 Existence of smooth stable manifolds 120
6.2 Nonuniformly hyperbolic trajectories 121
6.3 Proof of the existence of smooth stable manifolds 123
6.3.1 Functional spaces 123
6.3.2 Derivatives of compositions 125
6.3.3 A priori control of the derivatives 127
6.3.4 Holder regularity of the top derivatives 129
6.3.5 Solution on the stable direction 133
6.3.6 Behavior under perturbations of the data 136
6.3.7 Construction of the stable manifolds 138
7 A nonautonomous Grobman Hartman theorem 145
7.1 Conjugacies for flows 145
7.2 Conjugacies for maps 146
7.2.1 Setup 147
7.2.2 Existence of topological conjugacies 149
Contents XIII
7.3 Holder regularity of the conjugacies 155
7.3.1 Main statement 155
7.3.2 Lyapunov norms 156
7.3.3 Proof of the Holder regularity 157
7.4 Proofs of the results for flows 162
7.4.1 Reduction to discrete time 162
7.4.2 Proofs 164
Part III Center manifolds, symmetry and reversibility
8 Center manifolds in Banach spaces 171
8.1 Standing assumptions 171
8.2 Existence of center manifolds 173
8.3 Proof of the existence of center manifolds 176
8.3.1 Functional spaces 176
8.3.2 Lipschitz property of the derivatives 179
8.3.3 Solution on the central direction 184
8.3.4 Reduction to an equivalent problem 187
8.3.5 Construction of the center manifolds 191
9 Reversibility and equivariance in center manifolds 197
9.1 Reversibility for nonautonomous equations 197
9.1.1 The notion of reversibility 197
9.1.2 Relation with the autonomous case 199
9.1.3 Nonautonomous reversible equations 201
9.2 Reversibility in center manifolds 202
9.2.1 Formulation of the main result 202
9.2.2 Auxiliary results 206
9.2.3 Proof of the reversibility 212
9.3 Equivariance for nonautonomous equations 213
9.4 Equivariance in center manifolds 214
Part IV Lyapunov regularity and stability theory
10 Lyapunov regularity and exponential dichotomies 219
10.1 Lyapunov exponents and regularity 219
10.2 Existence of nonuniform exponential dichotomies 222
10.3 Bounds for the regularity coefficient 226
10.3.1 Lower bound 226
10.3.2 Upper bound in the triangular case 227
10.3.3 Reduction to the triangular case 232
10.4 Characterizations of regularity 235
10.5 Equations with negative Lyapunov exponents 241
XIV Contents
10.5.1 Lipschitz stable manifolds 242
10.5.2 Smooth stable manifolds 243
10.6 Measure preserving flows 244
11 Lyapunov regularity in Hilbert spaces 249
11.1 The notion of regularity 249
11.2 Upper triangular reduction 252
11.3 Regularity coefficient and Perron coefficient 254
11.4 Characterizations of regularity 256
11.5 Lower and upper bounds for the coefficients 259
12 Stability of nonautonomous equations in Hilbert spaces. 265
12.1 Setup 265
12.2 Stability results 267
12.3 Smallness of the perturbation 268
12.4 Norm estimates for the evolution operators 270
12.5 Proofs of the stability results 273
References 277
Index 283 |
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| author | Barreira, Luis 1968- Valls, Claudia 1973- |
| author_GND | (DE-588)128529717 (DE-588)1026700426 |
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| callnumber-subject | QA - Mathematics |
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| classification_tum | MAT 344f MAT 345f |
| ctrlnum | (OCoLC)254924142 (DE-599)DNB985199768 |
| dewey-full | 515.392 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.392 |
| dewey-search | 515.392 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV022948528 |
| illustrated | Illustrated |
| index_date | 2024-07-02T19:01:06Z |
| indexdate | 2024-07-09T21:08:21Z |
| institution | BVB |
| isbn | 9783540747741 3540747745 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016153044 |
| oclc_num | 254924142 |
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| owner_facet | DE-824 DE-91G DE-BY-TUM DE-29 DE-355 DE-BY-UBR DE-83 DE-11 DE-188 |
| physical | XIV, 285 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Barreira, Luis 1968- Verfasser (DE-588)128529717 aut Stability of nonautonomous differential equations Luis Barreira ; Claudia Valls Berlin [u.a.] Springer 2008 XIV, 285 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1926 Differentialgleichungssystem - Nichtautonomes System - Stabilität Differential equations Lyapunov stability Stabilität (DE-588)4056693-6 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Nichtautonomes System (DE-588)4480820-3 gnd rswk-swf Differentialgleichungssystem (DE-588)4121137-6 gnd rswk-swf Differentialgleichungssystem (DE-588)4121137-6 s Nichtautonomes System (DE-588)4480820-3 s Stabilität (DE-588)4056693-6 s DE-604 Differentialgleichung (DE-588)4012249-9 s b DE-604 Valls, Claudia 1973- Verfasser (DE-588)1026700426 aut Lecture notes in mathematics 1926 (DE-604)BV000676446 1926 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016153044&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Barreira, Luis 1968- Valls, Claudia 1973- Stability of nonautonomous differential equations Lecture notes in mathematics Differentialgleichungssystem - Nichtautonomes System - Stabilität Differential equations Lyapunov stability Stabilität (DE-588)4056693-6 gnd Differentialgleichung (DE-588)4012249-9 gnd Nichtautonomes System (DE-588)4480820-3 gnd Differentialgleichungssystem (DE-588)4121137-6 gnd |
| subject_GND | (DE-588)4056693-6 (DE-588)4012249-9 (DE-588)4480820-3 (DE-588)4121137-6 |
| title | Stability of nonautonomous differential equations |
| title_auth | Stability of nonautonomous differential equations |
| title_exact_search | Stability of nonautonomous differential equations |
| title_exact_search_txtP | Stability of nonautonomous differential equations |
| title_full | Stability of nonautonomous differential equations Luis Barreira ; Claudia Valls |
| title_fullStr | Stability of nonautonomous differential equations Luis Barreira ; Claudia Valls |
| title_full_unstemmed | Stability of nonautonomous differential equations Luis Barreira ; Claudia Valls |
| title_short | Stability of nonautonomous differential equations |
| title_sort | stability of nonautonomous differential equations |
| topic | Differentialgleichungssystem - Nichtautonomes System - Stabilität Differential equations Lyapunov stability Stabilität (DE-588)4056693-6 gnd Differentialgleichung (DE-588)4012249-9 gnd Nichtautonomes System (DE-588)4480820-3 gnd Differentialgleichungssystem (DE-588)4121137-6 gnd |
| topic_facet | Differentialgleichungssystem - Nichtautonomes System - Stabilität Differential equations Lyapunov stability Stabilität Differentialgleichung Nichtautonomes System Differentialgleichungssystem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016153044&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT barreiraluis stabilityofnonautonomousdifferentialequations AT vallsclaudia stabilityofnonautonomousdifferentialequations |