According to the Equivariant Hopf Theorem, a consequence of Hopf bifurcation in systems with dihedral Formula: see text-symmetry, where Formula: see text is the group of symmetries of an Formula: see text-gon, is for these systems to be able to exhibit, up to conjugacy, three types of responses: a traveling wave form and two standing wave forms. From related work, it is known that, generically, when Formula: see text, these are the only types of solutions that bifurcate from the trivial solution via a double Hopf bifurcation point. In this paper, we prove, analytically and computationally, that when the rotation symmetry is broken, this double Hopf bifurcation point splits into two standard, back-to-back Hopf bifurcation points, corresponding to the two types of standing wave solutions, but with fixed phase differences. Traveling waves are no longer found to be possible. The results are motivated by the analysis of network systems, in which rotation (or reflection) symmetry can be broken due to the presence of disorder or heterogeneities in system parameters; that is, due to forced symmetry-breaking bifurcations. A brief discussion of ongoing experimental work is also included. This work is expected to have relevance to a number of physical systems.
Sahoo et al. (Wed,) studied this question.