Bifurcations and catastrophes: geometry of solutions to nonlinear problems
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | German English |
| Published: |
Berlin [u.a.]
Springer
2000
|
| Series: | Universitext
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VIII, 301 S. graph. Darst |
| ISBN: | 3540521186 |
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| 245 | 1 | 0 | |a Bifurcations and catastrophes |b geometry of solutions to nonlinear problems |c Michel Demazure |
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Record in the Search Index
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| adam_text | Table of Contents
Introduction 1
1. Local Inversion 13
1.1 Introduction 13
1.2 A Preliminary Statement 14
1.3 Partial Derivatives. Strictly Differentiable Functions 17
1.4 The Local Inversion Theorem: General Statement 19
1.5 Functions of Class Cr 20
1.6 The Local Inversion Theorem for C maps 24
1.7 Curvilinear Coordinates 26
1.8 Generalizations of the Local Inversion Theorem 29
2. Submanifolds 31
2.1 Introduction 31
2.2 Definitions of Submanifolds 32
2.3 First Examples 36
2.4 Tangent Spaces of a Submanifold 40
2.5 Transversality: Intersections 43
2.6 Transversality: Inverse Images 45
2.7 The Implicit Function Theorem 47
2.8 Diffeomorphisms of Submanifolds 50
2.9 Parametrizations, Immersions and Embeddings 52
2.10 Proper Maps; Proper Embeddings 55
2.11 From Submanifolds to Manifolds 58
2.12 Some History 60
3. Transversality Theorems 63
3.1 Introduction 63
3.2 Countability Properties in Topology 65
3.3 Negligible Subsets 69
3.4 The Complement of the Image of a Submanifold 70
3.5 Sard s Theorem 74
3.6 Critical Points, Submersions and the Geometrical Form
of Sard s Theorem 75
VI Table of Contents
3.7 The Transversality Theorem: Weak Form 78
3.8 Jet Spaces 81
3.9 The Thom Transversality Theorem 83
3.10 Some History 86
4. Classification of Differentiable Functions 87
4.1 Introduction 87
4.2 Taylor Formulae Without Remainder 88
4.3 The Problem of Classification of Maps 90
4.4 Critical Points: the Hessian Form 93
4.5 The Morse Lemma 96
4.6 Bifurcations of Critical Points 98
4.7 Apparent Contour of a Surface in R3 100
4.8 Maps from R2 into R2 104
4.9 Envelopes of Plane Curves 108
4.10 Caustics 109
4.11 Genericity and Stability Ill
5. Catastrophe Theory 115
5.1 Introduction 115
5.2 The Language of Germs 117
5.3 r sufficient Jets; r determined Germs 119
5.4 The Jacobian Ideal 120
5.5 The Theorem on Sufficiency of Jets 123
5.6 Deformations of a Singularity 126
5.7 The Principles of Catastrophe Theory 130
5.8 Catastrophes of Cusp Type 133
5.9 A Cusp Example 135
5.10 Liquid Vapour Equilibrium 138
5.11 The Elementary Catastrophes 140
5.12 Catastrophes and Controversies 143
6. Vector Fields 147
6.1 Introduction 147
6.2 Examples of Vector Fields (Rn Case) 149
6.3 First Integrals 151
6.4 Vector Fields on Submanifolds 155
6.5 The Uniqueness Theorem and Maximal Integral Curves ... 157
6.6 Vector Fields and Line Fields. Elimination of the Time .. . 159
6.7 One parameter Groups of Diffeomorphisms 161
6.8 The Existence Theorem (Local Case) 164
6.9 The Existence Theorem (Global Case) 168
6.10 The Integral Flow of a Vector Field 170
6.11 The Main Features of a Phase Portrait 172
6.12 Discrete Flows and Continuous Flows 175
Table of Contents VII
7. Linear Vector Fields 179
7.1 Introduction 179
7.2 The Spectrum of an Endomorphism 181
7.3 Space Decomposition Corresponding to Partition of the
Spectrum 185
7.4 Norm and Eigenvalues 188
7.5 Contracting, Expanding and Hyperbolic Endomorphisms .. 191
7.6 The Exponential of an Endomorphism 193
7.7 One parameter Groups of Linear Transformations 196
7.8 The Image of the Exponential 200
7.9 Contracting, Expanding and Hyperbolic Exponential
Flows 203
7.10 Topological Classification of Linear Vector Fields 206
7.11 Topological Classification of Automorphisms 211
7.12 Classification of Linear Flows in Dimension 2 213
8. Singular Points of Vector Fields 219
8.1 Introduction 219
8.2 The Classification Problem 220
8.3 Linearization of a Vector Field in the Neighbourhood of a
Singular Point 223
8.4 Difficulties with Linearization 226
8.5 Singularities with Attracting Linearization 228
8.6 Lyapunov Theory 230
8.7 The Theorems of Grobman and Hartman 233
8.8 Stable and Unstable Manifolds of a Hyperbolic Singularity 234
8.9 Differentable Linearization: Statement of the Problem .... 238
8.10 Differentiate Linearization: Resonances 239
8.11 Differentiable Linearization: the Theorems of Sternberg
and Hartman 242
8.12 Linearization in Dimension 2 243
8.13 Some Historical Landmarks 247
9. Closed Orbits Structural Stability 249
9.1 Introduction 249
9.2 The Poincare Map 250
9.3 Characteristic Multipliers of a Closed Orbit 252
9.4 Attracting Closed Orbits 254
9.5 Classification of Closed Orbits and Classification of
Diffeomorphisms 256
9.6 Hyperbolic Closed Orbits 258
9.7 Local Structural Stability 260
VIII Table of Contents
9.8 The Kupka Smale Theorem 264
9.9 Morse Smale Fields 266
9.10 Structural Stability Through the Ages 268
10. Bifurcations of Phase Portraits 269
10.1 Introduction 269
10.2 What Do We Mean by Bifurcation? 270
10.3 The Centre Manifold Theorem 273
10.4 The Saddle Node Bifurcation 275
10.5 The Hopf Bifurcation 277
10.6 Local Bifurcations of a Closed Orbit 280
10.7 Saddle node Bifurcation for a Closed Orbit 282
10.8 Period doubling Bifurcation 282
10.9 Hopf Bifurcation for a Closed Orbit 284
10.10 An Example of a Codimension 2 Bifurcation 287
10.11 An Example of Non local Bifurcation 289
References 293
Index 295
Notation 303
|
| any_adam_object | 1 |
| author | Demazure, Michel 1937- |
| author_GND | (DE-588)121340201 |
| author_facet | Demazure, Michel 1937- |
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| classification_rvk | SK 350 SK 370 |
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| ctrlnum | (OCoLC)439822280 (DE-599)BVBBV012782152 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV012782152 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:33:35Z |
| institution | BVB |
| isbn | 3540521186 |
| language | German English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008693493 |
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| physical | VIII, 301 S. graph. Darst |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Springer |
| record_format | marc |
| series2 | Universitext |
| spelling | Demazure, Michel 1937- Verfasser (DE-588)121340201 aut Géométrie - catastrophes et bifurcations Bifurcations and catastrophes geometry of solutions to nonlinear problems Michel Demazure Berlin [u.a.] Springer 2000 VIII, 301 S. graph. Darst txt rdacontent n rdamedia nc rdacarrier Universitext Verzweigung Mathematik (DE-588)4078889-1 gnd rswk-swf Katastrophentheorie (DE-588)4029930-2 gnd rswk-swf Verzweigung Mathematik (DE-588)4078889-1 s Katastrophentheorie (DE-588)4029930-2 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008693493&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Demazure, Michel 1937- Bifurcations and catastrophes geometry of solutions to nonlinear problems Verzweigung Mathematik (DE-588)4078889-1 gnd Katastrophentheorie (DE-588)4029930-2 gnd |
| subject_GND | (DE-588)4078889-1 (DE-588)4029930-2 |
| title | Bifurcations and catastrophes geometry of solutions to nonlinear problems |
| title_alt | Géométrie - catastrophes et bifurcations |
| title_auth | Bifurcations and catastrophes geometry of solutions to nonlinear problems |
| title_exact_search | Bifurcations and catastrophes geometry of solutions to nonlinear problems |
| title_full | Bifurcations and catastrophes geometry of solutions to nonlinear problems Michel Demazure |
| title_fullStr | Bifurcations and catastrophes geometry of solutions to nonlinear problems Michel Demazure |
| title_full_unstemmed | Bifurcations and catastrophes geometry of solutions to nonlinear problems Michel Demazure |
| title_short | Bifurcations and catastrophes |
| title_sort | bifurcations and catastrophes geometry of solutions to nonlinear problems |
| title_sub | geometry of solutions to nonlinear problems |
| topic | Verzweigung Mathematik (DE-588)4078889-1 gnd Katastrophentheorie (DE-588)4029930-2 gnd |
| topic_facet | Verzweigung Mathematik Katastrophentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008693493&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT demazuremichel geometriecatastrophesetbifurcations AT demazuremichel bifurcationsandcatastrophesgeometryofsolutionstononlinearproblems |