Bifurcations in Hamiltonian systems: computing singularities by Gröbner bases
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| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2003
|
| Series: | Lecture notes in mathematics
1806 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 169 S. Ill., graph. Darst. |
| ISBN: | 3540004033 |
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| 245 | 1 | 0 | |a Bifurcations in Hamiltonian systems |b computing singularities by Gröbner bases |c Henk Broer ... |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2003 | |
| 300 | |a XIII, 169 S. |b Ill., 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 1806 | |
| 650 | 7 | |a Bifurcatie |2 gtt | |
| 650 | 4 | |a Bifurcation, Théorie de la | |
| 650 | 4 | |a Gröbner, Bases de | |
| 650 | 7 | |a Hamilton-vergelijkingen |2 gtt | |
| 650 | 7 | |a SINGULARIDADES (TOPOLOGIA DIFERENCIAL) |2 larpcal | |
| 650 | 4 | |a Singularités (Mathématiques) | |
| 650 | 4 | |a Systèmes hamiltoniens | |
| 650 | 7 | |a TEORIA DA BIFURCAÇÃO (SISTEMAS DINÂMICOS) |2 larpcal | |
| 650 | 4 | |a Bifurcation theory | |
| 650 | 4 | |a Gröbner bases | |
| 650 | 4 | |a Hamiltonian systems | |
| 650 | 4 | |a Singularities (Mathematics) | |
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Record in the Search Index
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| adam_text | Table of Contents
Preface V
1 Introduction 1
1.1 A further setting of the problem 2
1.1.1 The planar reduction method 2
1.1.2 The energy momentum map 5
1.1.3 Standard bases 7
1.2 Sketch of the results 8
1.2.1 Reduction methods 8
1.2.2 The planar reduction method 9
1.2.3 The energy momentum map method 12
1.2.4 Standard bases 14
1.3 Discussion 15
1.3.1 Discussion reduction methods 15
1.3.2 Discussion standard bases 16
1.4 Outline 17
1 Applications
2 Method I: Planar reduction 21
2.1 Introduction 21
2.1.1 BCKV restricted morphisms 23
2.2 Details of the planar reduction method 23
2.2.1 Overview 24
2.2.2 Some notation 25
2.2.3 Birkhoff normalization 26
2.2.4 Reduction to planar 1 degree of freedom system 28
2.2.5 Reduction to the central singularity 30
2.2.6 Inducing the system from a versal deformation 30
2.2.7 BCKV normal form 31
2.3 Spring pendulum in l:2 resonance 32
2.3.1 The system 32
2.3.2 Reduction 33
2.3.3 Dynamics and bifurcations 39
XII Table of Contents
3 Method II: The energy momentum map 45
3.1 Introduction 45
3.2 Description of the method 47
3.2.1 Birkhoff normalization 47
3.2.2 Circle equivariant vector fields 47
3.2.3 The energy momentum map 49
3.2.4 Removing the ^ dependence 51
3.3 Application to several resonances 53
3.3.1 The 1:2 resonance 54
3.3.2 The 1:3 resonance 55
3.3.3 The 1:4 resonance 56
3.4 Spring pendulum in 1:2 resonance 57
3.4.1 Bifurcation analysis of the l:2 resonant normal form 57
3.4.2 Pictures 60
3.4.3 Inducing the system from the model 61
II Theory
4 Birkhoff normalization 71
4.1 Introduction 71
4.2 Introduction to Hamiltonian mechanics 72
4.3 Birkhoff normal form theorem 74
4.3.1 Semisimple quadratic part, and resonance 76
4.3.2 Second normalization 78
4.4 Algorithms for the Birkhoff normal form 79
4.4.1 Another formulation of Birkhoff s result 80
4.4.2 Deprit s algorithm 80
4.4.3 The exponential map algorithm 82
5 Singularity theory 85
5.1 Overview 85
5.2 Introduction: The finite dimensional case 86
5.2.1 Deformations 87
5.3 Functions and right transformations 88
5.3.1 Equivalence and versal deformations 88
5.3.2 Applications 90
5.3.3 BCKV normal form 91
5.4 Maps and left right transformations 94
5.4.1 The tangent space 95
Table of Contents XIII
6 Grobner bases and Standard bases 97
6.1 Introduction 97
6.1.1 Algorithms and real numbers 99
6.2 Motivation: Grobner bases 99
6.2.1 Term orders for Grobner bases 100
6.2.2 Basic question 101
6.2.3 Rephrasing the basic question 101
6.2.4 A criterion for Grobner bases 102
6.3 Standard bases 104
6.3.1 Overview 104
6.3.2 Definitions 104
6.3.3 Setup 106
6.3.4 Normal form property 107
6.3.5 The standard map theorem 108
6.3.6 Normal form algorithm 110
6.3.7 Reduced normal forms Ill
6.4 Instances of standard bases 112
6.4.1 Grobner bases 112
6.4.2 Standard bases for submodules 114
6.4.3 Standard bases for subalgebras 115
6.4.4 Standard bases for modules over subalgebras 119
6.4.5 Left Right tangent space 122
6.5 The ring of formal power series 128
6.5.1 Three approaches 129
6.5.2 Existence of a normal form for formal power series 129
6.5.3 Truncated formal power series 131
7 Computing normalizing transformations 133
7.1 Introduction 133
7.2 Deformations of functions 134
7.2.1 Finding a universal deformation 135
7.2.2 The algorithm of Kas and Schlessinger 136
7.2.3 Solving the infinitesimal stability equation 137
7.2.4 Application: The hyperbolic umbilic 139
7.3 Deformations of maps 140
7.3.1 Adaptation of Kas and Schlessinger s algorithm 140
7.3.2 Solving the infinitesimal stability equation 144
7.3.3 Example of a LR tangent space calculation 146
A Appendix 153
A.I Classification of term orders 153
A.2 Proof of Proposition 5.8 155
References 159
Index 167
|
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| indexdate | 2024-07-09T19:10:43Z |
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| physical | XIII, 169 S. Ill., graph. Darst. |
| publishDate | 2003 |
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| spelling | Bifurcations in Hamiltonian systems computing singularities by Gröbner bases Henk Broer ... Berlin [u.a.] Springer 2003 XIII, 169 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1806 Bifurcatie gtt Bifurcation, Théorie de la Gröbner, Bases de Hamilton-vergelijkingen gtt SINGULARIDADES (TOPOLOGIA DIFERENCIAL) larpcal Singularités (Mathématiques) Systèmes hamiltoniens TEORIA DA BIFURCAÇÃO (SISTEMAS DINÂMICOS) larpcal Bifurcation theory Gröbner bases Hamiltonian systems Singularities (Mathematics) Singularität Mathematik (DE-588)4077459-4 gnd rswk-swf Gröbner-Basis (DE-588)4276378-2 gnd rswk-swf Hamiltonsches System (DE-588)4139943-2 gnd rswk-swf Verzweigung Mathematik (DE-588)4078889-1 gnd rswk-swf Hamiltonsches System (DE-588)4139943-2 s Verzweigung Mathematik (DE-588)4078889-1 s Singularität Mathematik (DE-588)4077459-4 s Gröbner-Basis (DE-588)4276378-2 s DE-604 Broer, Hendrik W. 1950- Sonstige (DE-588)124653782 oth Lecture notes in mathematics 1806 (DE-604)BV000676446 1806 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010175727&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bifurcations in Hamiltonian systems computing singularities by Gröbner bases Lecture notes in mathematics Bifurcatie gtt Bifurcation, Théorie de la Gröbner, Bases de Hamilton-vergelijkingen gtt SINGULARIDADES (TOPOLOGIA DIFERENCIAL) larpcal Singularités (Mathématiques) Systèmes hamiltoniens TEORIA DA BIFURCAÇÃO (SISTEMAS DINÂMICOS) larpcal Bifurcation theory Gröbner bases Hamiltonian systems Singularities (Mathematics) Singularität Mathematik (DE-588)4077459-4 gnd Gröbner-Basis (DE-588)4276378-2 gnd Hamiltonsches System (DE-588)4139943-2 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd |
| subject_GND | (DE-588)4077459-4 (DE-588)4276378-2 (DE-588)4139943-2 (DE-588)4078889-1 |
| title | Bifurcations in Hamiltonian systems computing singularities by Gröbner bases |
| title_auth | Bifurcations in Hamiltonian systems computing singularities by Gröbner bases |
| title_exact_search | Bifurcations in Hamiltonian systems computing singularities by Gröbner bases |
| title_full | Bifurcations in Hamiltonian systems computing singularities by Gröbner bases Henk Broer ... |
| title_fullStr | Bifurcations in Hamiltonian systems computing singularities by Gröbner bases Henk Broer ... |
| title_full_unstemmed | Bifurcations in Hamiltonian systems computing singularities by Gröbner bases Henk Broer ... |
| title_short | Bifurcations in Hamiltonian systems |
| title_sort | bifurcations in hamiltonian systems computing singularities by grobner bases |
| title_sub | computing singularities by Gröbner bases |
| topic | Bifurcatie gtt Bifurcation, Théorie de la Gröbner, Bases de Hamilton-vergelijkingen gtt SINGULARIDADES (TOPOLOGIA DIFERENCIAL) larpcal Singularités (Mathématiques) Systèmes hamiltoniens TEORIA DA BIFURCAÇÃO (SISTEMAS DINÂMICOS) larpcal Bifurcation theory Gröbner bases Hamiltonian systems Singularities (Mathematics) Singularität Mathematik (DE-588)4077459-4 gnd Gröbner-Basis (DE-588)4276378-2 gnd Hamiltonsches System (DE-588)4139943-2 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd |
| topic_facet | Bifurcatie Bifurcation, Théorie de la Gröbner, Bases de Hamilton-vergelijkingen SINGULARIDADES (TOPOLOGIA DIFERENCIAL) Singularités (Mathématiques) Systèmes hamiltoniens TEORIA DA BIFURCAÇÃO (SISTEMAS DINÂMICOS) Bifurcation theory Gröbner bases Hamiltonian systems Singularities (Mathematics) Singularität Mathematik Gröbner-Basis Hamiltonsches System Verzweigung Mathematik |
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