Bifurcations in Hamiltonian Systems: Computing Singularities by Gröbner Bases

Henk Broer, Igor Hoveijn, Gerton Lunter, Gert Vegter9783540004035, 3540004033

The authors consider applications of singularity theory and computer algebra to bifurcations of Hamiltonian dynamical systems. They restrict themselves to the case were the following simplification is possible. Near the equilibrium or (quasi-) periodic solution under consideration the linear part allows approximation by a normalized Hamiltonian system with a torus symmetry. It is assumed that reduction by this symmetry leads to a system with one degree of freedom. The volume focuses on two such reduction methods, the planar reduction (or polar coordinates) method and the reduction by the energy momentum mapping. The one-degree-of-freedom system then is tackled by singularity theory, where computer algebra, in particular, Gröbner basis techniques, are applied. The readership addressed consists of advanced graduate students and researchers in dynamical systems.

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Table of contents :
Springer LNM1806; Bifurcations in Hamiltonian Systems……Page 1
Preface……Page 2
Table of Contents……Page 7
2.1 Introduction……Page 10
2.2 Details of the planar reduction method……Page 12
2.2.1 Overview……Page 13
2.2.2 Some notation……Page 14
2.2.3 Birkhoff normalization……Page 15
2.2.4 Reduction to planar 1 degree-of-freedom system……Page 17
2.2.6 Inducing the system from a versal deformation……Page 19
2.2.7 BCKV normal form……Page 20
2.3.1 The system……Page 21
2.3.2 Reduction……Page 22
2.3.3 Dynamics and bifurcations……Page 28
3.1 Introduction……Page 34
3.2.2 Circle-equivariant vector fields……Page 36
3.2.3 The energy–momentum map……Page 38
3.2.4 Removing the $chi$-dependence……Page 40
3.3 Application to several resonances……Page 42
3.3.1 The 1:2 resonance……Page 43
3.3.2 The 1:3 resonance……Page 44
3.3.3 The 1:4 resonance……Page 45
3.4.1 Bifurcation analysis of the 1:2-resonant normal form……Page 46
3.4.2 Pictures……Page 49
3.4.3 Inducing the system from the model……Page 50
4.1 Introduction……Page 58
4.2 Introduction to Hamiltonian mechanics……Page 59
4.3 Birkhoff normal form theorem……Page 61
4.3.1 Semisimple quadratic part, and resonance……Page 63
4.3.2 Second normalization……Page 65
4.4 Algorithms for the Birkhoff normal form……Page 66
4.4.2 Deprit’s algorithm……Page 67
4.4.3 The exponential-map algorithm……Page 69
5.1 Overview……Page 72
5.2 Introduction: The finite dimensional case……Page 73
5.2.1 Deformations……Page 74
5.3.1 Equivalence and versal deformations……Page 75
5.3.2 Applications……Page 77
5.3.3 BCKV normal form……Page 78
5.4 Maps and left-right transformations……Page 81
5.4.1 The tangent space……Page 82
6.1 Introduction……Page 84
6.2 Motivation: Gröbner bases……Page 86
6.2.1 Term orders for Gröbner bases……Page 87
6.2.3 Rephrasing the basic question……Page 88
6.2.4 A criterion for Gröbner bases……Page 89
6.3.2 Definitions……Page 91
6.3.3 Setup……Page 93
6.3.4 Normal form property……Page 94
6.3.5 The standard map theorem……Page 95
6.3.6 Normal form algorithm……Page 97
6.3.7 Reduced normal forms……Page 98
6.4.1 Gröbner bases……Page 99
6.4.2 Standard bases for submodules……Page 101
6.4.3 Standard bases for subalgebras……Page 102
6.4.4 Standard bases for modules over subalgebras……Page 106
6.4.5 Left-Right tangent space……Page 109
6.5 The ring of formal power series……Page 115
6.5.2 Existence of a normal form for formal power series……Page 116
6.5.3 Truncated formal power series……Page 118
7.1 Introduction……Page 120
7.2 Deformations of functions……Page 121
7.2.1 Finding a universal deformation……Page 122
7.2.2 The algorithm of Kas and Schlessinger……Page 123
7.2.3 Solving the infinitesimal stability equation……Page 124
7.2.4 Application: The hyperbolic umbilic……Page 126
7.3.1 Adaptation of Kas and Schlessinger’s algorithm……Page 127
7.3.2 Solving the infinitesimal stability equation……Page 131
7.3.3 Example of a LR-tangent space calculation……Page 133
A.1 Classification of term orders……Page 139
A.2 Proof of Proposition 5.8……Page 141