The Symmetry Perspective

Abstract

1. Steady-State Bifurcation.- 1.1. Two Examples.- 1.2. Symmetries of Differential Equations.- 1.3. Liapunov-Schmidt Reduction.- 1.4. The Equivariant Branching Lemma.- 1.5. Application to Speciation.- 1.6. Observational Evidence.- 1.7. Modeling Issues: Imperfect Symmetry.- 1.8. Generalization to Partial Differential Equations.- 2. Linear Stability.- 2.1. Symmetry of the Jacobian.- 2.2. Isotypic Components.- 2.3. General Comments on Stability of Equilibria.- 2.4. Hilbert Bases and Equivariant Mappings.- 2.5. Model-Independent Results for D3Steady-State Bifurcation.- 2.6. Invariant Theory for SN.- 2.7. Cubic Terms in the Speciation Model.- 2.8. Steady-State Bifurcations in Reaction-Diffusion Systems.- 3. Time Periodicity and Spatio-Temporal Symmetry.- 3.1. Animal Gaits and Space-Time Symmetries.- 3.2. Symmetries of Periodic Solutions.- 3.3. A Characterization of Possible Spatio-Temporal Symmetries.- 3.4. Rings of Cells.- 3.5. An Eight-Cell Locomotor CPG Model.- 3.6. Multifrequency Oscillations.- 3.7. A General Definition of a Coupled Cell Network.- 4. Hopf Bifurcation with Symmetry.- 4.1. Linear Analysis.- 4.2. The Equivariant Hopf Theorem.- 4.3. Poincare-Birkhoff Normal Form.- 4.4. ?(2) Phase-Amplitude Equations.- 4.5. Traveling Waves and Standing Waves.- 4.6. Spiral Waves and Target Patterns.- 4.7. ?(2) Hopf Bifurcation in Reaction-Diffusion Equations.- 4.8. Hopf Bifurcation in Coupled Cell Networks.- 4.9. Dynamic Symmetries Associated to Bifurcation.- 5. Steady-State Bifurcations in Euclidean Equivariant Systems.- 5.1. Translation Symmetry, Rotation Symmetry, and Dispersion Curves.- 5.2. Lattices, Dual Lattices, and Fourier Series.- 5.3. Actions on Kernels and Axial Subgroups.- 5.4. Reaction-Diffusion Systems.- 5.5. Pseudoscalar Equations.- 5.6. The Primary Visual Cortex.- 5.7. The Planar Benard Experiment.- 5.8. Liquid Crystals.- 5.9. Pattern Selection: Stability of Planforms.- 6. Bifurcation From Group Orbits.- 6.1. The Couette-Taylor Experiment.- 6.2. Bifurcations From Group Orbits of Equilibria.- 6.3. Relative Periodic Orbits.- 6.4. Hopf Bifurcation from Rotating Waves to Quasiperiodic Motion.- 6.5. Modulated Waves in Circular Domains.- 6.6. Spatial Patterns.- 6.7. Meandering of Spiral Waves.- 7. Hidden Symmetry and Genericity.- 7.1. The Faraday Experiment.- 7.2. Hidden Symmetry in PDEs.- 7.3. The Faraday Experiment Revisited.- 7.4. Mode Interactions and Higher-Dimensional Domains.- 7.5. Lapwood Convection.- 7.6. Hemispherical Domains.- 8. Heteroclinic Cycles.- 8.1. The Guckenheimer-Holmes Example.- 8.2. Heteroclinic Cycles by Group Theory.- 8.3. Pipe Systems and Bursting.- 8.4. Cycling Chaos.- 9. Symmetric Chaos.- 9.1. Admissible Subgroups.- 9.2. Invariant Measures and Ergodic Theory.- 9.3. Detectives.- 9.4. Instantaneous and Average Symmetries, and Patterns on Average.- 9.5. Synchrony of Chaotic Oscillations and Bubbling Bifurcations.- 10. Periodic Solutions of Symmetric Hamiltonian Systems.- 10.1. The Equivariant Moser-Weinstein Theorem.- 10.2. Many-Body Problems.- 10.3. Spatio-Temporal Symmetries in Hamiltonian Systems.- 10.4. Poincare-Birkhoff Normal Form.- 10.5. Linear Stability.- 10.6. Molecular Vibrations.