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 ZnSymmetry. - 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. Language: English