MODE INTERACTIONS WITH SPHERICAL SYMMETRY

Abstract

We study mode interaction steady-state bifurcation problems with spherical symmetry. Using the representation of O(3) in terms of spherical harmonics, we study the interactions of the modes of dimension 1, 3 and 5. When studying mode interactions involving the 3- and the 5-dimensional modes, we come across a very natural representation, which turns out to be that of SO(3) instead of O(3). Given that the study of problems with this latter symmetry has already been done, we then study problems with SO(3) symmetry. For all these problems, we stress the existence of secondary bifurcations giving rise to the existence of limit cycles and the occurrence of heteroclinic connections between equilibria.