Hopf bifurcation with zero frequency and imperfect SO(2) symmetry

Rotating waves are periodic solutions in SO(2) equivariant dynamical sys- tems. Their precession frequency changes with parameters, and it may change sign, passing through zero. When this happens, the dynamical system is very sensitive to imperfections that break the SO(2) symmetry, and the waves may become trapped by the imperfections, resulting in steady solutions that exist in a nite region in parameter space, the so-called pinning phenomenon. In this study we analyze the breaking of the SO(2) symmetry in a dynamical sys- tem close to a Hopf bifurcation whose frequency changes sign along a curve in parameter space. The problem is more complex than expected, and the com- plete unfolding is of codimension six. A detailed analysis of dierent types of imperfections indicates that a pinning region surrounded by innite-period bi- furcation curves appears in all cases. Complex bifurcational processes, strongly dependent on the specics of how the symmetry is broken, appear very close to the intersection of the Hopf bifurcation and the pinning region. Scaling laws of the pinning region width, and partial breaking of SO(2) to Zm, are also con- sidered. Previous experimental and numerical studies of pinned rotating waves are reviewed in light of the new theoretical results.