Dynamics of bifurcations for variational problems with \(O(3)\) equivariance: A Conley index approach
type: Item
instance of: scholarly article
title: Dynamics of bifurcations for variational problems with \(O(3)\) equivariance: A Conley index approach
zbMATH Open document ID: 0756.49003
DOI: 10.1007/BF00375120
author: B. Fiedler
author: K. Mischaikow
published in: Archive for Rational Mechanics and Analysis
publication date: 1993-01-17T00:00:00Z
review text: The paper investigates the nonlinear dynamics of symmetry breaking with orthogonal symmetry group \(O(3)\). A motivating example is the scalar reaction diffusion equation \(u_ t=\Delta u+\lambda f(u)\) on the 2- sphere; \(f(0)=0\), \(f'(0)=1\). Consider stationary bifurcation from the trivial solution, involving the seven-dimensional irreducible representation of \(SO(3)\) on spherical harmonics of order 3. As input information, use the known homology of the coset spaces \(O(3)/K\), where \(K\) denotes the isotropy subgroups of the three bifurcating stationary solution branches. Under nondegeneracy assumptions, the bifurcation directions then determine the unstable dimensions (Morse indices) of the bifurcating stationary solutions. Moreover, the heteroclinic connections between these equilibria are found. Going beyond the model example, the result is relevant even for non- variational \(O(3)\)-equivariant problems, since equivariant lowest order terms are automatically of variational type. The result is obtained via Conley index (generalized Morse theory) and the connection matrix. Therefore the result is persistent, even when breaking the equivariance of the system slightly.
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 58E07
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 96946
zbMATH Keywords: heteroclinic orbits
zbMATH Keywords: nonlinear dynamics of symmetry breaking
zbMATH Keywords: orthogonal symmetry group \(O(3)\)
zbMATH Keywords: scalar reaction diffusion equation
zbMATH Keywords: stationary bifurcation
zbMATH Keywords: Morse indices
zbMATH Keywords: Conley index
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 96946