Bifurcation in two-dimensional fixed point subspaces

TL;DR

This paper analyzes bifurcations with symmetry in two-dimensional fixed point subspaces, identifying possible solution branches, phase portraits, and stability conditions, with applications to spherical symmetry cases.

Contribution

It provides a comprehensive classification of bifurcation scenarios with two-dimensional fixed point subspaces and derives conditions for different phase portraits, extending understanding of symmetric bifurcations.

Findings

One or three solution branches typically occur.

Five distinct phase portraits are identified.

Conditions for phase portrait occurrence are established.

Abstract

Bifurcation with symmetry is considered in the case of an isotropy subgroup with a two-dimensional fixed point subspace and non-zero quadratic terms. In general, there are one or three branches of solutions, and five qualitatively different phase portraits, provided that two non-degeneracy conditions are satisfied. Conditions are also derived to determine which of the five possible phase portraits occurs, given the coefficients of the quadratic terms. The results are applied to the problem of bifurcation with spherical symmetry, where there are six irreducible representations for which the subspace of solutions with cubic symmetry is two-dimensional. In each case, the number of solutions and their stability is found.