Bifurcation to Spiral Waves in Reaction-diffusion Systems

For a large class of reaction-diiusion systems on the plane, we show rigorously that m-armed spiral waves bifurcate from a homogeneous equilibrium when the latter undergoes a Hopf bifurcation. In particular, we construct a nite-dimensional manifold which contains the set of small rotating waves close to the homogeneous equilibrium. Examining the ow on this center-manifold in a very general example, we nd diierent types of spiral waves, distinguished by their speed of rotation and their asymptotic shape at large distances of the tip. The relation to the special class of-! systems and the validity of these systems as an approximation is discussed.