Heteroclinic cycles in bifurcation problems with O(3) symmetry and the spherical Benard problem

It has been known since a paper of Armbruster and Chossat ([AC91]) that robust heteroclinic cycles between equilibria can bifurcate in differential systems which are invariant under the action of the group O(3) defined as the sum of its ''natural'' irreducible representations of degrees 1 and 2 (i.e., of dimensions 3 and 5). Moreover, these cycles can be seen numerically in the simulation of the amplitude equations resulting from a center manifold reduction of the Benard problem in a nonrotating spherical shell with suitable aspect ratio ([FH86]). In the present work we first generalize the results of [AC91] to the interactions of irreducible representations of degrees l and l + 1 for any l > 0. Heteroclinic cycles of various types are shown to exist under certain ''generic'' conditions and are classified. We show in particular that these conditions are satisfied in most cases when the differential system proceeds from a l, l + 1 mode interaction bifurcation in the spherical Benard problem.