Forced reflectional symmetry breaking of an O(2)-symmetric homoclinic cycle

Abstract

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. The author calls these objects 'homoclinic cycles' because they realize a homoclinic orbit from a group orbit of equilibria to itself. The author shows 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 for the study of perturbations of heteroclinic chains.