Singularities and Groups in Bifurcation Theory: Volume II [Hardcover]

Bifurcation theory studies how the structure of solutions to equations changes as parameters are varied. The nature of these changes depends both on the number of parameters and on the symmetries of the equations. Volume I discusses how singularity-theoretic techniques aid the understanding of transitions in multiparameter systems. This volume focuses on bifurcation problems with symmetry and shows how group-theoretic techniques aid the understanding of transitions in symmetric systems. Four broad topics are covered: group theory and steady-state bifurcation, equicariant singularity theory, Hopf bifurcation with symmetry, and mode interactions. The opening chapter provides an introduction to these subjects and motivates the study of systems with symmetry. Detailed case studies illustrate how group-theoretic methods can be used to analyze specific problems arising in applications.of Volume II.- XI Introduction.- ?0. Introduction.- ?1. Equations with Symmetry.- ?2. Techniques.- ?3. Mode Interactions.- ?4. Overview.- XII Group-Theoretic Preliminaries.- ?0. Introduction.- ?1. Group Theory.- ?2. Irreducibility.- ?3. Commuting Linear Mappings and Absolute Irreducibility.- ?4. Invariant Functions.- ?5. Nonlinear Commuting Mappings.- ?6.* Proofs of Theorems in ??4 and 5.- ?7.* Tori.- XIII Symmetry-Breaking in Steady-State Bifurcation.- ?0. Introduction.- ?1. Orbits and Isotropy Subgroups.- ?2. Fixed-Point Subspaces and the Trace Formula.- ?3. The Equivariant Branching Lemma.- ?4. Orbital Asymptotic Stability.- ?5. Bifurcation Diagrams and DnSymmetry.- ?6. Subgroups of SO(3).- ?7. Representations of SO(3) and O(3): Spherical Harmonics.- ?8. Symmetry-Breaking from SO(3).- ?9. Symmetry-Breaking from O(3).- ?10.* Generic Spontaneous Symmetry-Breaking.- Case Study 4 The Planar B?nard Problem.- ?0. Introduction.- ?1. Discussion of the PDE.- ?2. One-Dimensional Fixed-Point Subspaces.- ?3. Bifurcation Diagrams and Asymptotic Stability.- XIV Equivariant Normal Forms.- ?0. Introdlck