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.
Bibliographical noteFunding Information:
PC thanks Ignacio Bosch-Vivancos both for helpful discussions and for performing preliminary numerical simulations of the model equations. IM acknowledges the hospitality of the INLN where part of this work was carried out. All four authors acknowledge travel support of the European Bifurcation Theory Group. AS acknowledge travel support of the DFG Schwerpunkt 'Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme' and the Technical University of Vienna. The research of KM was supported in part by Fonds zur F6rderung der wissenschaftlichen Forschung of Austria under the grant P 101115-MAT. The research of IM was supported in part by NSF Grant DMS-9403624, by ONR Grant N00014-94-1-0317 and by the CNRS.