等变化分叉和动力系统中的方法及应用METHODS IN EQUIVARIANT BIFURCATION AND DYNAMICAL SYSTEMS, AND APPLICATIONS
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作者： Pascal Chossat著
出 版 社： 东南大学出版社
出版时间： 2000-12-1字数：版次： 1页数： 404印刷时间： 1999/12/01开本：印次： 1纸张： 胶版纸I S B N ： 9789810238285包装： 精装内容简介
This invaluable book presents a comprehensive introduction to bifurcation theory in the presence of symmetry, an applied mathematical topic which has developed considerably over the past twenty years and has been very successful in analysing and predicting pattern formation and other critical phenomena in most areas of science where nonlinear models are involved, like fluid flow instabilities, chemical waves, elasticity and population dynamics.
The book has two aims. One is to expound the mathematical methods of equivariant bifurcation theory. Beyond the classical bifurcation tools, such as center manifold and normal form reductions, the presence of symmetry requires the introduction of the algebraic and geometric formalism of Lie group theory and transformation group methods. For the first time, all these methods in equivariant bifurcations are presented in a coherent and self-consistent way in a book.
The other aim is to present the most recent ideas and results in this theory, in relation to applications. This includes bifurcations of relative equilibria and relative periodic orbits for compact and noncompact group actions, heteroclinic cycles and forced symmetry-breaking perturbations. Although not all recent contributions could be included and a choice had to be made, a rather complete description of these new developments is provided. At the end of every chapter, exercises are offered to the reader.
目录
Preface
Chapter 1 Symmetries in ODE's and PDE's
1.1 Euclidean symmetries : the basic notions
1.1.1 The Euclidean group
1.1.2 The closed subgroups of 0(2) and 0(3)
1.1.3 Lattice groups and lattice symmetries
1.2 Differential systems of physics and their symmetries
1.2.1 Examples of ODEs with symmetry : coupled oscillators
1.2.2 Elasticity : buckling problems
1.2.3 Reaction-diffusion equations
1.2.4 Hydrodynamical models
1.2.5 The symmetry of classical differential operators
1.3 Exercises
Chapter 2 Equivariant bifurcations, a first look
2.1 Group actions on Banach spaces
2.2 The equivariant Lyapunov-Schmidt decomposition
2.3 The equivariant branching lemmas
2.3.1 The steady-state equivariant branching lemma
2.3.2 The equivariant branching lemma for symmetry groups acting in R2 and R3
2.4 The equivariant Hopf bifurcation
2.4.1 Hopf bifurcation as a symmetry-breaking bifurcation problem
2.4.2 The equivariant Hopf branching lemma
2.5 Exercises
Chapter 3 Invariant manifolds and normal forms
3.1 Invariant manifolds for autonomous ODE's
3.2 The normal form reduction
3.3 Center manifolds and normal forms in bifurcation problems
3.3.1 Center manifold and normal form for a parameter dependent ODE
3.3.2 Effective computation of the center manifold and normal form
3.4 Center manifolds for partial differential equations
3.4.1 Evolution equations in Banach spaces and center manifolds
3.4.2 An example: the Swift-Hohenberg equation on the sphere
3.5 Exercises
Chapter 4 Linear Lie Group Actions
4.1 Introduction
4.2 Lie groups
4.3 Induced actions
4.4 Representations
4.5 Characters
4.6 Representations of some continuous groups
4.6.1 The group SO(2)
4.6.2 The group 0(2)
4.6.3 The group 0(3)
4.6.4 The group Dm ** T2
4.7 A remark on non-compact groups
4.8 Infinite dimensional representations
4.9 Generic one parameter families of equivariant linear maps . . .
4.10 Geometry of representations
4.11 The equivariant Whitney embedding theorem
4.12 Exercises
Chapter 5The Equivariant Structure of Bifurcation Equations
Chapter 6Reduction Techniques for Equivariant Systems
Chapter 7Relative Equilibria and Relative Periodic Orbits
Chapter 8Bifurcations in Equivariant Systems
Chapter 9Heteroclinic Cycles
Chapter 10Perturbation of Equivariant Systems
Appendix A Miscellanea on the Group SO(3)
Appendix B Translation table for the subgroups of O(3)
Bibliography
Index
