FORCED REFLECTIONAL SYMMETRY-BREAKING OF AN O(2)-SYMMETRICAL HOMOCLINIC CYCLE

It has been known for several years that differential systems with O(2) symmetry can possess heteroclinic cycles between two equilibria which belong to the same group orbit. We call these objects 'homoclinic cycles' because they realize a homoclinic orbit from a group orbit of equilibria to itself. In this paper we show that under perturbations which break the reflectional symmetry in O(2), the homoclinic cycle generically bifurcates to a quasi-periodic flow on a 2-torus. The techniques applied to this problem are (i) the reduction of the system to the orbit space and (ii) a generalization of Melnikov's method by Lin for the study of perturbations of heteroclinic chains.