2003/05/28 by P. C. Matthews, Matthews, P. C.
Computer Science · Engineering · Mathematics · #37G40 #Advanced Differential Equations and Dynamical Systems #Advanced Mathematical Modeling in Engineering #Dynamical Systems (math.DS) #Elasticity and Wave Propagation #FOS: Mathematics #math.DS #msc:37G40
paper · pdf · doi:10.48550/arxiv.math/0305392
10 pages, 1 figure
arxiv created 2003/05/28 · openalex publication_date 2003/05/28 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.