Author | Golubitsky, Martin. author |
---|---|
Title | Singularities and Groups in Bifurcation Theory [electronic resource] : Volume II / by Martin Golubitsky, Ian Stewart, David G. Schaeffer |
Imprint | New York, NY : Springer New York, 1988 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-4574-2 |
Descript | XVI, 536 p. online resource |
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 Subgroups of SO(3) -- ยง7.โ Representations of SO(3) and O(3): Spherical Harmonics -- ยง8.โ Symmetry-Breaking from SO(3) -- ยง9.โ 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 The Spherical Bรฉnard Problem -- ยง6.โ 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 S1 -- ยง8.* Dimensions of Fixed-Point Subspaces -- ยง9. Invariant Theory for ? ร 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 S1 -- ยง2. Invariant Theory for O(2) ร S1 -- ยง3. The Branching Equations -- ยง4. Amplitude Equations, D4 Symmetry, and Stability -- ยง5.โ Hopf Bifurcation with O(n) Symmetry -- ยง6.โ Bifurcation with D4 Symmetry -- ยง7. The Bifurcation Diagrams -- ยง8.โ Rotating Waves and SO(2) or Zn Symmetry -- XVIII Further Examples of Hopf Bifurcation with Symmetry -- ยง0. Introduction -- ยง1. The Action of Dn ร S1 -- ยง2. Invariant Theory for Dn ร S1 -- ยง3. Branching and Stability for Dn -- ยง4. Oscillations of Identical Cells Coupled in a Ring -- ยง5.โ Hopf Bifurcation with O(3) Symmetry -- ยง6.โ 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 Steady-State Mode Interaction -- ยง2. Hopf/Steady-State Mode Interaction -- ยง3.โ 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