Symmetry and Perturbation Theory in Nonlinear Dynamics

by Cicogna, Giampaolo; Beiglbock, W.; Beig, R.; Ehlers, J.; Frisch, U.; Hepp, K.; Jaffe, R. L.

• ISBN13: 9783540659044
• ISBN10: 3540659048
• Format: Hardcover
• Copyright: 1999-06-01
• Publisher: Springer Verlag

Summary

This book deals with the theory of Poincarè--Birkhoff normal forms, studying symmetric systems in particular. Attention is focused on general Lie point symmetries, and not just on symmetries acting linearly. Some results on the simultaneous normalization of a vector field describing a dynamical system and vector fields describing its symmetry are presented and a perturbative approach is also used. Attention is given to the problem of convergence of the normalizing transformation in the presence of symmetry, with some other extensions of the theory. The results are discussed for the general case of dynamical systems and also for the specific Hamiltonian setting.

Table of Contents

Introduction	1	(3)
Perturbation Theory	3	(3)
Other Tools to Study Nonlinear Systems and Symmetry	6	(2)
Why Nonlinear Symmetries?	8	(2)
Why Normal Forms?	10	(4)
Normal Forms and Higher-Level Perturbation Theory	14	(2)
Direct Applications of Normal Forms Theory in Physics	16	(3)
Symmetry and Differential Equations	19	(22)
Geometrical Setting	21	(3)
Invariance of Equations and of Solutions	24	(5)
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	(3)
Appendix. The Prolongation Formula	39	(2)
Dynamical Systems	41	(26)
Dynamical Systems and Flows	41	(2)
Singular Points and Invariant Manifolds	43	(4)
Conserved Quantities	47	(1)
Perturbative Expansion	48	(3)
Poincare-Dulac Normal Forms	51	(5)
Birkhoff-Gustavson Normal Forms	56	(5)
Bifurcation Theory	61	(6)
Appendix. Lie Transforms	64	(3)
Symmetries of Dynamical Systems	67	(16)
Symmetries of Dynamical Systems	68	(1)
Lie-Point Time-Independent Symmetries	69	(2)
Constants of Motion and the Module Structure of the Symmetry Algebra	71	(1)
Symmetry and Topology of Trajectories	72	(3)
Time-Dependent Symmetries	75	(2)
Orbital Symmetries	77	(1)
Approximate Symmetries	78	(5)
Appendix. On the Module Structure	81	(2)
Normal Forms and Symmetries for Dynamical Systems	83	(32)
Perturbative Expansion of Determining Equations	83	(2)
Recursive Determination of Symmetries	85	(3)
Approximate Symmetries	88	(1)
Symmetry Characterization of Poincare-Dulac Normal Forms	89	(2)
Nonlinear Symmetries and Normal Forms	91	(5)
Linear Symmetries and Normal Forms	96	(1)
On Linear and Nonlinear Symmetries	97	(2)
Symmetry for Systems in Normal Form	99	(2)
Reduction to Normal Form of a Nilpotent Lie Algebra	101	(3)
Non-semisimple Normal Forms	104	(4)
The Linearization of a Dynamical System	108	(1)
Partial Joint Normal Form for Non-nilpotent Algebras	109	(6)
Appendix. Some Results on Matrices and Lie Algebras	112	(3)
Normal Forms and Symmetries for Hamiltonian Systems	115	(20)
Birkhoff-Gustavson Normal Forms	115	(3)
Birkhoff-Gustavson Normal Forms with Symmetries	118	(6)
The Case of d-Dimensional Algebras of Symmetries	124	(2)
Perturbative Construction of Symmetries	126	(2)
The Non-normal Case	128	(4)
The Normality of the Homological Operator	132	(3)
Convergence of the Normalizing Transformations	135	(16)
Conditions Ensuring Convergence	135	(4)
Normalizing Transformations in the Presence of Symmetries	139	(2)
Convergence and Symmetries: A General Result	141	(3)
Convergence and Symmetries: A Special Case	144	(4)
Convergence in the Case of Hamiltonian Problems	148	(3)
Invariant Manifolds	151	(12)
Some Preliminary Results on Flow Invariant Manifolds	151	(3)
Reduction to a Center Manifold	154	(1)
Normal Forms	155	(1)
Shoshitaishvili Theorem and Center Manifolds	156	(2)
Some Examples	158	(5)
Further Normalization	163	(20)
Higher-Order Terms in Poincare Transformations	164	(2)
The Homological Operators	166	(1)
Non-uniqueness of Poincare Normal Forms	166	(1)
Poincare Renormalization	167	(4)
Renormalization by Iterated Normalizations	171	(2)
Examples: Planar Vector Fields	173	(4)
The Hamiltonian Case	177	(2)
Renormalized Forms in the Presence of Symmetry	179	(4)
Asymptotic Symmetries	183	(10)
Notation and Basic Sets	184	(1)
Induced Actions on Functions and Equations	185	(2)
Symmetries and Asymptotic Symmetries	187	(2)
Asymptotic Symmetries and Space-Time Asymptotic Properties	189	(4)
References	193