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.

Produktinformation

Utgivningsdatum2011-10-08
Mått155 x 235 x 30 mm
Vikt832 g
FormatHäftad
SpråkEngelska
SerieApplied Mathematical Sciences
Antal sidor536
FörlagSpringer-Verlag New York Inc.
ISBN9781461289296

Tillhör följande kategorier

Grupper och gruppteori inom Naturvetenskap och teknik

of Volume II.- XI Introduction.- §0. Introduction.- §1. Equations with Symmetry.- §2. Techniques.- §3. Mode Interactions.- §4. Overview.- XII Group-Theoretic Preliminaries.- §0. Introduction.- §1. Group Theory.- §2. Irreducibility.- §3. Commuting Linear Mappings and Absolute Irreducibility.- §4. Invariant Functions.- §5. Nonlinear Commuting Mappings.- §6.* Proofs of Theorems in §§4 and 5.- §7.* Tori.- XIII Symmetry-Breaking in Steady-State Bifurcation.- §0. Introduction.- §1. Orbits and Isotropy Subgroups.- §2. Fixed-Point Subspaces and the Trace Formula.- §3. The Equivariant Branching Lemma.- §4. Orbital Asymptotic Stability.- §5. Bifurcation Diagrams and DnSymmetry.- §6.† Subgroups of SO(3).- §7.† Representations of SO(3) and O(3): Spherical Harmonics.- §8.† Symmetry-Breaking from SO(3).- §9.† Symmetry-Breaking from O(3).- §10.* Generic Spontaneous Symmetry-Breaking.- Case Study 4 The Planar Bénard Problem.- §0. Introduction.- §1. Discussion of the PDE.- §2. One-Dimensional Fixed-Point Subspaces.- §3. Bifurcation Diagrams and Asymptotic Stability.- XIV Equivariant Normal Forms.- §0. Introduction.- §1. The Recognition Problem.- §2.* Proof of Theorem 1.3.- §3. Sample Computations of RT(h, ?).- §4. Sample Recognition Problems.- §5. Linearized Stability and ?-equivalence.- §6. Intrinsic Ideals and Intrinsic Submodules.- §7. Higher Order Terms.- XV Equivariant Unfolding Theory.- §0. Introduction.- §1. Basic Definitions.- §2. The Equivariant Universal Unfolding Theorem.- §3. Sample Universal ?-unfoldings.- §4. Bifurcation with D3 Symmetry.- §5.† The Spherical Bénard Problem.- §6.† Spherical Harmonics of Order 2.- §7.* Proof of the Equivariant Universal Unfolding Theorem.- §8.* The Equivariant PreparationTheorem.- Case Study 5 The Traction Problem for Mooney-Rivlin Material.- §0. Introduction.- §1. Reduction to D3 Symmetry in the Plane.- §2. Taylor Coefficients in the Bifurcation Equation.- §3. Bifurcations of the Rivlin Cube.- XVI Symmetry-Breaking in Hopf Bifurcation.- §0. Introduction.- §1. Conditions for Imaginary Eigenvalues.- §2. A Simple Hopf Theorem with Symmetry.- §3. The Circle Group Action.- §4. The Hopf Theorem with Symmetry.- §5. Birkhoff Normal Form and Symmetry.- §6. Floquet Theory and Asymptotic Stability.- §7. Isotropy Subgroups of ? × S1.- §8.* Dimensions of Fixed-Point Subspaces.- §9. Invariant Theory for ? × S1.- 10. Relationship Between Liapunov-Schmidt Reduction and Birkhoff Normal Form.- §11.* Stability in Truncated Birkhoff Normal Form.- XVII Hopf Bifurcation with O(2) Symmetry.- §0. Introduction.- §1. The Action of O(2) × S1.- §2. Invariant Theory for O(2) × S1.- §3. The Branching Equations.- §4. Amplitude Equations, D4 Symmetry, and Stability.- §5.† Hopf Bifurcation with O(n) Symmetry.- §6.† Bifurcation with D4 Symmetry.- §7. The Bifurcation Diagrams.- §8.† Rotating Waves and SO(2) or ZnSymmetry.- XVIII Further Examples of Hopf Bifurcation with Symmetry.- §0. Introduction.- §1. The Action of Dn × S1.- §2. Invariant Theory for Dn × S1.- §3. Branching and Stability for Dn.- §4. Oscillations of Identical Cells Coupled in a Ring.- §5.† Hopf Bifurcation with O(3) Symmetry.- §6.† Hopf Bifurcation on the Hexagonal Lattice.- XIX Mode Interactions.- §0. Introduction.- § 1. Hopf/Steady-State Interaction.- §2. Bifurcation Problems with Z2 Symmetry.- §3. Bifurcation Diagrams with Z2 Symmetry.- §4. Hopf/Hopf Interaction.- XX Mode Interactions with O(2) Symmetry.- §0. Introduction.- §l.†Steady-State Mode Interaction.- §2. Hopf/Steady-State Mode Interaction.- §3.† Hopf/Hopf Mode Interaction.- Case Study 6 The Taylor-Couette System.- §0. Introduction.- §1. Detailed Overview.- §2. The Bifurcation Theory Analysis.- §3. Finite Length Effects.

Mer från samma författare

Dynamics and Bifurcation in Networks

Martin Golubitsky, Ian Stewart

Inbunden, 2023

1 999 kr

Del 115

Pattern Formation in Continuous and Coupled Systems

Martin Golubitsky, Dan Luss, Steven H. Strogatz

Inbunden, 1999

719 kr

Symmetry in Chaos

Michael Field, Martin Golubitsky

Inbunden, 2009

1 129 kr

Del 115

Pattern Formation in Continuous and Coupled Systems

Martin Golubitsky, Dan Luss, Steven H. Strogatz

Häftad, 2012

719 kr

Mathematical Approaches to Challenges in Biology and Biomedicine

Martin Golubitsky, Stefano Boccaletti, Carla M.A. Pinto

Inbunden, 2026

3 149 kr

Del 200

Symmetry Perspective

Martin Golubitsky, Ian Stewart

Häftad, 2003

1 259 kr

Del 200

Symmetry Perspective

Martin Golubitsky, Ian Stewart

Inbunden, 2002

1 289 kr

Mer från samma serie

Del 125

Topological Methods in Hydrodynamics

Vladimir I. Arnold, Boris A. Khesin

Häftad, 2022

1 569 kr

Del 171

Shapes and Diffeomorphisms

Laurent Younes

Inbunden, 2019

1 329 kr

Energy Method, Stability, and Nonlinear Convection

Brian Straughan

Inbunden, 2003

1 439 kr

Del 159

Theory and Practice of Finite Elements

Alexandre Ern, Jean-Luc Guermond

Inbunden, 2004

1 859 kr

Nonlinear Problems of Elasticity

Stuart Antman

Inbunden, 2005

3 289 kr

Statistical and Computational Inverse Problems

Jari Kaipio, E. Somersalo

Inbunden, 2004

2 009 kr

Least-Squares Finite Element Methods

Pavel B. Bochev, Max D. Gunzburger

Inbunden, 2009

1 859 kr

Variational Methods in Imaging

Otmar Scherzer, Markus Grasmair, Harald Grossauer, Markus Haltmeier, Frank Lenzen

Inbunden, 2008

719 kr

Mathematical Problems in Image Processing

Gilles Aubert, Pierre Kornprobst

Inbunden, 2006

2 429 kr

Direct Methods in the Calculus of Variations

Bernard Dacorogna

Inbunden, 2007

2 429 kr

Du kanske också är intresserad av

Del 69