The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space

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.