Bifurcations and Dynamics of Spiral Waves

Spiral waves have been observed in several chemical systems, for instance, in the Belousov-Zhabotinsky reaction and in the catalysis on platinum surfaces. Seemingly key to the dynamics of spiral waves is the Euclidean symmetry group SE(N). In this article, an SE(N)-equivariant center-manifold reduction near meandering spirals and other waves is developed. The vector eld on the center manifold has a certain skew-product structure which is important for the understanding of bifurcations from spiral waves. The main diiculty is that the Euclidean symmetry group is not compact and its action is not diierentiable; in fact, the group may act discontinuously on the underlying function space. Using the center-manifold reduction, Hopf bifurcations and periodic forcing of spiral waves are then investigated. The results explain various experiments and numerical simulations.