Symmetry breaking and symmetry locking in equivariant circle maps
The families of \(Z_ k\)-equivariant circle maps, typically given by \[ f_ k(x,\omega,\mu)=x+\omega-(\mu/2\pi k)\sin 2\pi kx\bmod 1, \] which commute with actions of finite cyclic or dihedral groups \(G\), are considered as models for symmetric dynamical systems which undergo symmetry breaking and symmetry increasing bifurcations. A classification of generic bifurcations of periodic orbits for maps with \(D_ k\) and \(Z_ k\) symmetry is given. The authors analyze the phenomena of symmetry locking and symmetry restoration. So, two notions of symmetry locking are defined --- full and partial. Full symmetry locking occurs when the symmetry groups of the closure 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 \(G\). It is proved that for a ``nice class'' of maps, the full symmetry locking is essentially equivalent to frequency locking, and that partial symmetry locking exists within periodic windows, that is regions of the parameter space where an iterate of the map has an invariant subinterval on which it has the structure of a unimodal map. Also, some results that suggest that symmetry restoration occurs at the end of periodic windows are given.
scientific article; zbMATH DE number 94098
scientific article