Symmetry breaking and symmetry locking in equivariant circle maps

Circle maps which commute with actions of a finite cyclic or dihedral group, G, are considered as models for symmetric dynamical systems which undergo "symmetry breaking" and "symmetry increasing" bifurcations. The possible symmetry breaking bifurcations of periodic orbits are classified. Two notions of "symmetry locking" are defined. Full symmetry locking occurs when the symmetry groups of (the closures of) all trajectories of a map are contained in some proper subgroup of G. Partial symmetry locking occurs when the symmetry groups of all trajectories in some invariant open subset of the circle are contained in a proper subgroup of G. Full symmetry locking is essentially equivalent to frequency locking. Partial symmetry locking occurs in "unimodal windows" in parameter space and in certain regions of the Arnold tongues of the maps. A brief discussion of "symmetry restoration" via attractor "crises", which occurs on the boundaries of these regions, is included.