Dynamics of bifurcations for variational problems with \(O(3)\) equivariance: A Conley index approach
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.
scientific article; zbMATH DE number 96946
scientific article