The framework of ‘symmetry’ provides an important route between the abstract theory and experimental observations. The book applies symmetry methods to dynamical systems, focusing on bifurcation and chaos theory. Its exposition is organized around a wide variety of relevant applications.
From the reviews:
"[ The] rich collection of examples makes the book...extremely useful for motivation and for spreading the ideas to a large Community."--MATHEMATICAL REVIEWS
This book applies symmetry methods to dynamical systems, focusing on bifurcation and chaos theory. Award-winning monograph of the Ferran Sunyer i Balaguer Prize 2001. From the reviews:"This excellent book reflects the authors' experience in [ the] exploration of the role of symmetry in...pattern formation in nonlinear dynamics over the past fifteen years. The selection of the material, logical structure of the monograph and presentation are perfect." --ZENTRALBLATT MATH
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From the reviews:
"This book was awarded the Ferran Sunyier i Balaguer Prize for 2001, and I am sure that it will be a very useful resource not only for researchers in this area but also for those who want to obtain the benefits of using this approach in applications." --Bulletin of the American Mathematical Society (Review of hardcover edition)
[ The] rich collection of examples makes the book extremely useful for motivation and for spreading the ideas to a large Community. This [ review] is far from complete and cannot reflect the authors unique way of presenting examples, asking questions, giving answers or forming an intuition.(MATHEMATICAL REVIEWS)
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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.
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