Drift bifurcations of relative equilibria and transitions of spiral waves

We consider dynamical systems that are equivariant under a noncompact Lie g roup of symmetries and the drift of relative equilibria in such systems. In particular, we investigate how the drift for a parametrized family of norm ally hyperbolic relative equilibria can change character at what we call a 'drift bifurcation'. To do this, we use results of Arnold to analyse parame trized families of elements in the Lie algebra of the symmetry group. We examine effects in physical space of such drift bifurcations for planar reaction-diffusion systems and note that these effects can explain certain aspects of the transition from rigidly rotating spirals to rigidly propagat ing 'retracting waves'. This is a bifurcation observed in numerical simulat ions of excitable media where the rotation rate of a family of spirals slow s down and gives way to a semi-infinite translating wavefront.