P. Chossat et F. Guyard, HETEROCLINIC CYCLES IN BIFURCATION PROBLEMS WITH O(3) SYMMETRY AND THE SPHERICAL BENARD-PROBLEM, Journal of nonlinear science, 6(3), 1996, pp. 201-238
It has been known since a paper of Armbruster and Chossat ([AC91]) tha
t robust heteroclinic cycles between equilibria can bifurcate in diffe
rential 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 r
esulting from a center manifold reduction of the Benard problem in a n
onrotating spherical shell with suitable aspect ratio ([FH86]). In the
present work we first generalize the results of [AC91] to the interac
tions of irreducible representations of degrees l and l + 1 for any l
> 0. Heteroclinic cycles of various types are shown to exist under cer
tain ''generic'' conditions and are classified. We show in particular
that these conditions are satisfied in most cases when the differentia
l system proceeds from a l, l + 1 mode interaction bifurcation in the
spherical Benard problem.