| 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 5 The Equivariant Structure of Bifurcation Equations Chapter 6 Reduction Techniques for Equivariant Systems Chapter 7 Relative Equilibria and Relative Periodic Orbits Chapter 8 Bifurcations in Equivariant Systems Chapter 9 Heteroclinic Cycles Chapter 10 Perturbation of Equivariant Systems Appendix A Miscellanea on the Group SO(3) Appendix B Translation table for the subgroups of O(3) Bibliography Index |
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