Author | Golubitsky, Martin. author |
---|---|
Title | The Symmetry Perspective [electronic resource] : From Equilibrium to Chaos in Phase Space and Physical Space / by Martin Golubitsky, Ian Stewart |
Imprint | Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 2002 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-8167-8 |
Descript | XVII, 325 p. online resource |
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. Poincarรฉ-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 Bรฉnard 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. Poincarรฉ-Birkhoff Normal Form -- 10.5. Linear Stability -- 10.6. Molecular Vibrations