E-grāmata: Singularities and Groups in Bifurcation Theory: Volume II

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 Bénard 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.
Introduction.- §1. The Recognition Problem.- §2.* Proof of Theorem 1.3.- §3.
Sample Computations of RT(h, ?).- §4. Sample Recognition Problems.- §5.
Linearized Stability and ?-equivalence.- §6. Intrinsic Ideals and Intrinsic
Submodules.- §7. Higher Order Terms.- XV Equivariant Unfolding Theory.- §0.
Introduction.- §1. Basic Definitions.- §2. The Equivariant Universal
Unfolding Theorem.- §3. Sample Universal ?-unfoldings.- §4. Bifurcation with
D3 Symmetry.- §5. The Spherical Bénard Problem.- §6. Spherical Harmonics of
Order 2.- §7.* Proof of the Equivariant Universal Unfolding Theorem.- §8.*
The Equivariant PreparationTheorem.- Case Study 5 The Traction Problem for
Mooney-Rivlin Material.- §0. Introduction.- §1. Reduction to D3 Symmetry in
the Plane.- §2. Taylor Coefficients in the Bifurcation Equation.- §3.
Bifurcations of the Rivlin Cube.- XVI Symmetry-Breaking in Hopf Bifurcation.-
§0. Introduction.- §1. Conditions for Imaginary Eigenvalues.- §2. A Simple
Hopf Theorem with Symmetry.- §3. The Circle Group Action.- §4. The Hopf
Theorem with Symmetry.- §5. Birkhoff Normal Form and Symmetry.- §6. Floquet
Theory and Asymptotic Stability.- §7. Isotropy Subgroups of ? × S1.- §8.*
Dimensions of Fixed-Point Subspaces.- §9. Invariant Theory for ? × S1.-
10.
Relationship Between Liapunov-Schmidt Reduction and Birkhoff Normal Form.-
§11.* Stability in Truncated Birkhoff Normal Form.- XVII Hopf Bifurcation
with O(2) Symmetry.- §0. Introduction.- §1. The Action of O(2) × S1.- §2.
Invariant Theory for O(2) × S1.- §3. The Branching Equations.- §4. Amplitude
Equations, D4 Symmetry, and Stability.- §5. Hopf Bifurcation with O(n)
Symmetry.- §6. Bifurcation with D4 Symmetry.- §7. The Bifurcation Diagrams.-
§8. Rotating Waves and SO(2) or Zn
Symmetry.- XVIII Further Examples of Hopf Bifurcation with Symmetry.- §0.
Introduction.- §1. The Action of Dn × S1.- §2. Invariant Theory for Dn × S1.-
§3. Branching and Stability for Dn.- §4. Oscillations of Identical Cells
Coupled in a Ring.- §5. Hopf Bifurcation with O(3) Symmetry.- §6. Hopf
Bifurcation on the Hexagonal Lattice.- XIX Mode Interactions.- §0.
Introduction.- §
1. Hopf/Steady-State Interaction.- §2. Bifurcation Problems
with Z2 Symmetry.- §3. Bifurcation Diagrams with Z2 Symmetry.- §4. Hopf/Hopf
Interaction.- XX Mode Interactions with O(2) Symmetry.- §0. Introduction.-
§l.Steady-State Mode Interaction.- §2. Hopf/Steady-State Mode Interaction.-
§3. Hopf/Hopf Mode Interaction.- Case Study 6 The Taylor-Couette System.-
§0. Introduction.- §1. Detailed Overview.- §2. The Bifurcation Theory
Analysis.- §3. Finite Length Effects.