Transverse bifurcations of homoclinic cycles

Homoclinic cycles exist robustly in dynamical systems with symmetry, and may undergo various bifurcations, not all of which have an analog in the absence of symmetry. We analyze such a bifurcation, the transverse bifurcation, and uncover a variety of phenomena that can be distinguished representation-theoretically. For example, exponentially flat branches of periodic solutions (a typical feature of bifurcation from homoclinic cycles) occur for some but not all representations of the symmetry group. Our study of transverse bifurcations is motivated by the problem of intermittent dynamos in rotating convection.