Dynamics and symmetry [electronic resource] / Michael J. Field.

Field, Mike.
London : Imperial College Press ; Singapore ; Hackensack, NJ : Distributed by World Scientific, c2007.
1 online resource (492 p.)
Imperial College Press advanced texts in mathematics ; v. 3.
ICP advanced texts in mathematics, 1753-657X ; v. 3

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Topological dynamics.
Lie groups.
Hamiltonian systems.
Bifurcation theory.
Symmetry (Mathematics).
Electronic books.
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. This important book forms a theoretical basis of future work on equivariant reversible and Hamiltonian systems. This book also provides a general and comprehensive introduction to codimension one equi
Contents; Preface; 1. Groups; 1.1 Definition of a group and examples; 1.2 Homomorphisms, subgroups and quotient groups; 1.2.1 Generators and relations for .nite groups; 1.3 Constructions; 1.4 Topological groups; 1.5 Lie groups; 1.5.1 The Lie bracket of vector fields; 1.5.2 The Lie algebra of G; 1.5.3 The exponential map of g; 1.5.4 Additional properties of brackets and exp; 1.5.5 Closed subgroups of a Lie group; 1.6 Haarmeasure; 2. Group Actions and Representations; 2.1 Introduction; 2.2 Groups and G-spaces; 2.2.1 Continuous actions and G-spaces; 2.3 Orbit spaces and actions
2.4 Twisted products2.4.1 Induced G-spaces; 2.5 Isotropy type and stratification by isotropy type; 2.6 Representations; 2.6.1 Averaging over G; 2.7 Irreducible representations and the isotypic decomposition; 2.7.1 C-representations; 2.7.2 Absolutely irreducible representations; 2.8 Orbit structure for representations; 2.9 Slices; 2.9.1 Slices for linear finite group actions; 2.10 Invariant and equivariant maps; 2.10.1 Smooth invariant and equivariant maps on representations; 2.10.2 Equivariant vector fields and flows; 3. Smooth G-manifolds; 3.1 Proper G-manifolds; 3.1.1 Proper free actions
3.2 G-vector bundles3.3 Infinitesimal theory; 3.4 Riemannianmanifolds; 3.4.1 Exponential map of a complete Riemannian manifold; 3.4.2 The tubular neighbourhood theorem; 3.4.3 Riemannian G-manifolds; 3.5 The differentiable slice theorem; 3.6 Equivariant isotopy extension theorem; 3.7 Orbit structure for G-manifolds; 3.7.1 Closed filtration of M by isotropy type; 3.8 The stratification of M by normal isotropy type; 3.9 Stratified sets; 3.9.1 Transversality to a Whitney stratification; 3.9.2 Regularity of stratification by normal isotropy type
3.10 Invariant Riemannian metrics on a compact Lie group3.10.1 The adjoint representations; 3.10.2 The exponential map; 3.10.3 Closed subgroups of a Lie group; 4. Equivariant Bifurcation Theory: Steady State Bifurcation; 4.1 Introduction and preliminaries; 4.1.1 Normalized families; 4.2 Solution branches and the branching pattern; 4.2.1 Stability of branching patterns; 4.3 Symmetry breaking-theMISC; 4.3.1 Symmetry breaking isotropy types; 4.3.2 Maximal isotropy subgroup conjecture; 4.4 Determinacy; 4.4.1 Polynomial maps; 4.4.2 Finite determinacy; 4.5 The hyperoctahedral family
4.5.1 The representations (Rk,Hk)4.5.2 Invariants and equivariants for Hk; 4.5.3 Cubic equivariants for Hk; 4.5.4 Bifurcation for cubic families; 4.5.5 Subgroups of Hk; 4.5.6 Some subgroups of the symmetric group; 4.5.7 A big family of counterexamples to the MISC; 4.5.8 Examples where P3G (Rk, Rk) = P3H k (Rk, Rk); 4.5.9 Stable solution branches of maximal index and trivial isotropy; 4.5.10 An example with applications to phase transitions; 4.6 Phase vector field and maps of hyperbolic type; 4.6.1 Cubic polynomial maps; 4.6.2 Phase vector field; 4.6.3 Normalized families
4.6.4 Maps of hyperbolic type
Description based upon print version of record.
Includes bibliographical references (p. 457-466) and indexes.