ISBN: 3-540-65904-8
TITLE: Symmetry and Perturbation Theory in Nonlinear Dynamics
AUTHOR: Cicogna, Giampaolo; Gaeta, Giuseppe
TOC:

Introduction 1
1. Perturbation Theory 3
2. Other Tools to Study Nonlinear Systems and Symmetry 6
3. Why Nonlinear Symmetries? 8
4. Why Normal Forms? 10
5. Normal Forms and Higher-Level Perturbation Theory 14
6. Direct Applications of Normal Forms Theory in Physics 16
I. Symmetry and Differential Equations 19
1. Geometrical Setting 21
2. Invariance of Equations and of Solutions 24
3. Reduction and Solution of Ordinary Differential Equations 29
4. Reduction and Solution of Partial Differential Equations 33
5. On the Application of Symmetry Methods 38
Appendix. The Prolongation Formula 39
II. Dynamical Systems 41
1. Dynamical Systems and Flows 41
2. Singular Points and Invariant Manifolds 43
3. Conserved Quantities 47
4. Perturbative Expansion 48
5. Poincar-Dulac Normal Forms 51
6. Birkhoff-Gustavson Normal Forms 56
7. Bifurcation Theory 61
Appendix. Lie Transforms 64
III. Symmetries of Dynamical Systems 67
1. Symmetries of Dynamical Systems 68
2. Lie-Point Time-Independent Symmetries 69
3. Constants of Motion and the Module Structure of the Symmetry Algebra 71
4. Symmetry and Topology of Trajectories 72
5. Time-Dependent Symmetries 75
6. Orbital Symmetries 77
7. Approximate Symmetries 78
Appendix. On the Module Structure 81
IV. Normal Forms and Symmetries for Dynamical Systems 83
1. Perturbative Expansion of Determining Equations 83
2. Recursive Determination of Symmetries 85
3. Approximate Symmetries 88
4. Symmetry Characterization of Poincar-Dulac Normal Forms 89
5. Nonlinear Symmetries and Normal Forms 91
6. Linear Symmetries and Normal Forms 96
7. On Linear and Nonlinear Symmetries 97
8. Symmetry for Systems in Normal Form 99
9. Reduction to Normal Form of a Nilpotent Lie Algebra 101
10. Non-semisimple Normal Forms 104
11. The Linearization of a Dynamical System 108
12. Partial Joint Normal Form for Non-nilpotent Algebras 109
Appendix. Some Results on Matrices and Lie Algebras 112
V. Normal Forms and Symmetries for Hamiltonian Systems 115
1. Birkhoff-Gustavson Normal Forms 115
2. Birkhoff-Gustavson Normal Forms with Symmetries 118
3. The Case of d-Dimensional Algebras of Symmetries 124
4. Perturbative Construction of Symmetries 126
5. The Non-normal Case 128
6. The Normality of the Homological Operator 132
VI. Convergence of the Normalizing Transformations 135
1. Conditions Ensuring Convergence 135
2. Normalizing Transformations in the Presence of Symmetries 139
3. Convergence and Symmetries: A General Result 141
4. Convergence and Symmetries:-A Special Case 144
5. Convergence in the Case of Hamiltonian Problems 148
VII. Invariant Manifolds 151
1. Some Preliminary Results on Flow Invariant Manifolds 151
2. Reduction to a Center Manifold 154
3. Normal Forms 155
4. Shoshitaishvili Theorem and Center Manifolds 156
5. Some Examples 158
VIII. Further Normalization 163
1. Higher-Order Terms in Poincar Transformations 164
2. The Homological Operators 166
3. Non-uniqueness of Poincar Normal Forms 166
4. Poincar Renormalization 167
5. Renormalization by Iterated Normalizations 171
6. Examples: Planar Vector Fields 173
7. The Hamiltonian Case 177
8. Renormalized Forms in the Presence of Symmetry 179
IX. Asymptotic Symmetries 183
1. Notation and Basic Sets 184
2. Induced Actions on Functions and Equations 185
3. Symmetries and Asymptotic Symmetries 187
4. Asymptotic Symmetries and Space-Time Asymptotic Properties 189
References 193
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