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.