Bifurcations and Dynamics of Spiral Waves

(N)

. In this article, it is shown that the dynamics near meandering spiral waves or other patterns is determined by a finite-dimensional vector field that has a certain skew-product structure over the group SESE(N) . This generalizes our earlier work on center-manifold theory near rigidly rotating spiral waves to meandering spirals. In particular, for meandering spirals, it is much more sophisticated to extract the aforementioned skew-product structure since spatio-temporal rather than only spatial symmetries have to be accounted for. Another difficulty is that the action of the Euclidean symmetry group on the underlying function space is not differentiable, and in fact may be discontinuous. Using this center-manifold reduction, Hopf bifurcations and periodic forcing of spiral waves are then investigated. The results explain the transitions to patterns with two or more temporal frequencies that have been observed in various experiments and numerical simulations.