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AuthorGolubitsky, Martin. author
TitleSingularities and Groups in Bifurcation Theory [electronic resource] : Volume II / by Martin Golubitsky, Ian Stewart, David G. Schaeffer
ImprintNew York, NY : Springer New York, 1988
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Descript XVI, 536 p. online resource


Bifurcation theory studies how the structure of solutions to equations changes as parameters are varied. The nature of these changes depends both on the number of parameters and on the symmetries of the equations. Volume I discusses how singularity-theoretic techniques aid the understanding of transitions in multiparameter systems. This volume focuses on bifurcation problems with symmetry and shows how group-theoretic techniques aid the understanding of transitions in symmetric systems. Four broad topics are covered: group theory and steady-state bifurcation, equicariant singularity theory, Hopf bifurcation with symmetry, and mode interactions. The opening chapter provides an introduction to these subjects and motivates the study of systems with symmetry. Detailed case studies illustrate how group-theoretic methods can be used to analyze specific problems arising in applications


of Volume II -- XI Introduction -- ยง0. Introduction -- ยง1. Equations with Symmetry -- ยง2. Techniques -- ยง3. Mode Interactions -- ยง4. Overview -- XII Group-Theoretic Preliminaries -- ยง0. Introduction -- ยง1. Group Theory -- ยง2. Irreducibility -- ยง3. Commuting Linear Mappings and Absolute Irreducibility -- ยง4. Invariant Functions -- ยง5. Nonlinear Commuting Mappings -- ยง6.* Proofs of Theorems in ยงยง4 and 5 -- ยง7.* Tori -- XIII Symmetry-Breaking in Steady-State Bifurcation -- ยง0. Introduction -- ยง1. Orbits and Isotropy Subgroups -- ยง2. Fixed-Point Subspaces and the Trace Formula -- ยง3. The Equivariant Branching Lemma -- ยง4. Orbital Asymptotic Stability -- ยง5. Bifurcation Diagrams and DnSymmetry -- ยง6.โ{128}  Subgroups of SO(3) -- ยง7.โ{128}  Representations of SO(3) and O(3): Spherical Harmonics -- ยง8.โ{128}  Symmetry-Breaking from SO(3) -- ยง9.โ{128}  Symmetry-Breaking from O(3) -- ยง10.* Generic Spontaneous Symmetry-Breaking -- Case Study 4 The Planar Bรฉnard Problem -- ยง0. Introduction -- ยง1. Discussion of the PDE -- ยง2. One-Dimensional Fixed-Point Subspaces -- ยง3. Bifurcation Diagrams and Asymptotic Stability -- XIV Equivariant Normal Forms -- ยง0. Introduction -- ยง1. The Recognition Problem -- ยง2.* Proof of Theorem 1.3 -- ยง3. Sample Computations of RT(h, ?) -- ยง4. Sample Recognition Problems -- ยง5. Linearized Stability and ?-equivalence -- ยง6. Intrinsic Ideals and Intrinsic Submodules -- ยง7. Higher Order Terms -- XV Equivariant Unfolding Theory -- ยง0. Introduction -- ยง1. Basic Definitions -- ยง2. The Equivariant Universal Unfolding Theorem -- ยง3. Sample Universal ?-unfoldings -- ยง4. Bifurcation with D3 Symmetry -- ยง5.โ{128}  The Spherical Bรฉnard Problem -- ยง6.โ{128}  Spherical Harmonics of Order 2 -- ยง7.* Proof of the Equivariant Universal Unfolding Theorem -- ยง8.* The Equivariant Preparation Theorem -- Case Study 5 The Traction Problem for Mooney-Rivlin Material -- ยง0. Introduction -- ยง1. Reduction to D3 Symmetry in the Plane -- ยง2. Taylor Coefficients in the Bifurcation Equation -- ยง3. Bifurcations of the Rivlin Cube -- XVI Symmetry-Breaking in Hopf Bifurcation -- ยง0. Introduction -- ยง1. Conditions for Imaginary Eigenvalues -- ยง2. A Simple Hopf Theorem with Symmetry -- ยง3. The Circle Group Action -- ยง4. The Hopf Theorem with Symmetry -- ยง5. Birkhoff Normal Form and Symmetry -- ยง6. Floquet Theory and Asymptotic Stability -- ยง7. Isotropy Subgroups of ? ร{151} S1 -- ยง8.* Dimensions of Fixed-Point Subspaces -- ยง9. Invariant Theory for ? ร{151} S1 -- 10. Relationship Between Liapunov-Schmidt Reduction and Birkhoff Normal Form -- ยง11.* Stability in Truncated Birkhoff Normal Form -- XVII Hopf Bifurcation with O(2) Symmetry -- ยง0. Introduction -- ยง1. The Action of O(2) ร{151} S1 -- ยง2. Invariant Theory for O(2) ร{151} S1 -- ยง3. The Branching Equations -- ยง4. Amplitude Equations, D4 Symmetry, and Stability -- ยง5.โ{128}  Hopf Bifurcation with O(n) Symmetry -- ยง6.โ{128}  Bifurcation with D4 Symmetry -- ยง7. The Bifurcation Diagrams -- ยง8.โ{128}  Rotating Waves and SO(2) or Zn Symmetry -- XVIII Further Examples of Hopf Bifurcation with Symmetry -- ยง0. Introduction -- ยง1. The Action of Dn ร{151} S1 -- ยง2. Invariant Theory for Dn ร{151} S1 -- ยง3. Branching and Stability for Dn -- ยง4. Oscillations of Identical Cells Coupled in a Ring -- ยง5.โ{128}  Hopf Bifurcation with O(3) Symmetry -- ยง6.โ{128}  Hopf Bifurcation on the Hexagonal Lattice -- XIX Mode Interactions -- ยง0. Introduction -- ยง 1. Hopf/Steady-State Interaction -- ยง2. Bifurcation Problems with Z2 Symmetry -- ยง3. Bifurcation Diagrams with Z2 Symmetry -- ยง4. Hopf/Hopf Interaction -- XX Mode Interactions with O(2) Symmetry -- ยง0. Introduction -- ยงl.โ{128}  Steady-State Mode Interaction -- ยง2. Hopf/Steady-State Mode Interaction -- ยง3.โ{128}  Hopf/Hopf Mode Interaction -- Case Study 6 The Taylor-Couette System -- ยง0. Introduction -- ยง1. Detailed Overview -- ยง2. The Bifurcation Theory Analysis -- ยง3. Finite Length Effects

Mathematics Group theory Mathematics Group Theory and Generalizations


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