Field M.J. Dynamics and Symmetry

Imperial College Press, 2007. — 492 p. — ISBN: 1860948286.This book contains the first systematic exposition of the global and local theory of dynamics equivariant with respect to a (compact) Lie group. Aside from general genericity and normal form theorems on equivariant bifurcation, it describes many general families of examples of equivariant bifurcation and includes a number of novel geometric techniques, in particular, equivariant transversality.Groups
Definition of a group and examples
Homomorphisms, subgroups and quotient groups
Constructions
Topological groups
Lie groups
HaarmeasureGroup Actions and RepresentationsGroups and G-spaces
Orbit spaces and actions
Twisted products
Isotropy type and stratification by isotropy type
Representations
Irreducible representations and the isotypic decomposition
Orbit structure for representations
Slices
Invariant and equivariant mapsSmooth G-manifolds
Proper G-manifolds
G-vector bundles
Infinitesimal theory
Riemannianmanifolds
The differentiable slice theorem
Equivariant isotopy extension theorem
Orbit structure for G-manifolds
The stratification of M by normal isotropy type
Stratified sets
Invariant Riemannian metrics on a compact Lie groupEquivariant Bifurcation Theory: Steady State Bifurcation
Introduction and preliminaries
Solution branches and the branching pattern
Symmetry breaking—the MISC
Determinacy
The hyperoctahedral family
Phase vector field and maps of hyperbolic type
Transforming to generalized spherical polar coordinates
d(V,G)-determinacy, d(V,G) = ,
Counting branches and finding their location
The symmetric and alternating groups
The groups S_{k+ 1} Z2 and A_k+1 * Z₂
Appendix: Proof of theorem on hyperbolic elements
Notes on ChapterEquivariant Bifurcation Theory: Dynamics
The invariant sphere theorem
The examples of dos Reis and Guckenheimer & Holmes
Steady state bifurcation to limit cycles
Bifurcation to complex dynamics in dimension four
The converse to theMISC
Hopf bifurcation and the invariant sphere theorem
Notes on Chapter
Equivariant TransversalityC⁰⁰ -topologies on function spaces
Transversality
Stratumwise transversality and stability
Reduction to a problem about solving equations
Invariants and equivariants
The universal variety
Stratifications and semialgebraic sets
Canonical stratification of the universal variety?
Stratifying Z_T and U
Equivariant coordinate changes on V * R^s and W
Symmetries of the stratification A
Openness of equivariant transversality
Global definitions and results
Solutions with specific isotropy type
Notes on ChapterApplications of G-transversality to Bifurcation Theory I
Weak stability and determinacy
Jet transversality
Equivariant jet transversality for families
Stability and determinacy
Higher order versions of G-transversality
Extensions to the case of non-finite G
Notes on ChapterEquivariant Dynamics
Invariant G-orbits
Stabilities and normal hyperbolicity
Relative fixed and periodic sets for diffeomorphisms
Genericity theorems for equivariant diffeomorphisms
Equivariant vector fields
Genericity theorems for equivariant vector fields
Notes on ChapterDynamical Systems on G-manifolds
Skew products
Gradient dynamics
G-subshifts of finite type
Suspensions
The inverse limit: turning maps into homeomorphisms
Solenoidal attractors
Equivariant Anosov diffeomorphisms
Notes on ChapterApplications of G-transversality to Bifurcation Theory II
Technical preliminaries and basic notations
The universal variety for relative equilibria
Weak stability and determinacy
Weak stability and determinacy for reversible systems
Stability and determinacy
Bifurcation from relative equilibria
Stability and determinacy for maps
Strong determinacy
Normal form theorems
Relative periodic orbits
Notes for ChapterBibliography
Index of Notational Conventions
Index