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Rotating waves are periodic solutions in SO(2) equivariant dynamical systems.Their precession frequency changes with parameters and it may change sign,passing through zero. When this happens, the dynamical system is very sensitiveto imperfections that break the SO(2) symmetry and the waves may become trappedby the imperfections, resulting in steady solutions that exist in a finiteregion in parameter space. This is the so-called pinning phenomenon. In thisstudy, we analyze the breaking of the SO(2) symmetry in a dynamical systemclose to a Hopf bifurcation whose frequency changes sign along a curve inparameter space. The problem is very complex, as it involves the completeunfolding of high codimension. A detailed analysis of different types ofimperfections indicates that a pinning region surrounded by infinite-periodbifurcation curves appears in all cases. Complex bifurcational processes,strongly dependent on the specifics of the symmetry breaking, appear very closeto the intersection of the Hopf bifurcation and the pinning region. Scalinglaws of the pinning region width, and partial breaking of SO(2) to Zm, are alsoconsidered. Previous and new experimental and numerical studies of pinnedrotating waves are reviewed in light of the new theoretical results.

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