The framework of symmetry provides an important route between the abstract theory and experimental observations, especially for the study of pattern formation in physical systems. The book applies symmetry methods to dynamical systems, focusing on bifurcation and chaos theory. It s exposition is organized around a wide variety of applications ranging from theoretical physics to evolutionary biology.

The framework of 'symmetry' provides an important route between the abstract theory and experimental observations, especially for the study of pattern formation in physical systems. The book applies symmetry methods to dynamical systems, focusing on bifurcation and chaos theory. It's exposition is organized around a wide variety of applications ranging from theoretical physics to evolutionary biology.

Winner of the Ferran Sunyer i Balaguer Prize 2000 New revised softbound edition

Autorentext
Ian Stewart is Professor of Mathematics at Warwick University, and Director of the Mathematics Awareness Centre at Warwick. An active research mathematician, he is also a well-known popularizer of mathematics and related areas of science. In 1995 he was awarded the Royal Society's Michael Faraday Award for furthering the public understanding of science; his book 'Nature's Numbers' was shortlisted for the 1996 Rhone-Poulenc Prize for Science Books; and he delivered the 1997 Royal Institution Christmas Lectures, televised by the BBC. In 2001 he was elected a Fellow of the Royal Society.

Inhalt

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. Poincaré-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 VisualCortex.- 5.7. The Planar Bénard 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. Poincaré-Birkhoff Normal Form.- 10.5. Linear Stability.- 10.6. Molecular Vibrations.