  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 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 finite
region in parameter space. This is the so-called pinning phenomenon. In this
study, we analyze the breaking of the SO(2) symmetry in a dynamical system
close to a Hopf bifurcation whose frequency changes sign along a curve in
parameter space. The problem is very complex, as it involves the complete
unfolding of high codimension. A detailed analysis of different types of
imperfections indicates that a pinning region surrounded by infinite-period
bifurcation curves appears in all cases. Complex bifurcational processes,
strongly dependent on the specifics of the symmetry breaking, 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
considered. Previous and new experimental and numerical studies of pinned
rotating waves are reviewed in light of the new theoretical results.
