Radially Symmetric Patterns of Reaction-Diffusion Systems

In this paper, bifurcations of stationary and time-periodic solutions to reactiondiffusion systems are studied. We develop a center-manifold and normal form theory for radial dynamics which allows for a complete description of radially symmetric patterns. In particular, we show the existence of localized pulses near saddle-nodes, critical Gibbs kernels in the cusp, focus patterns in Turing instabilities, and active or passive target patterns in oscillatory instabilities. Received by the editor February 15, 2001. 1991 Mathematics Subject Classification. Primary 35K57, 35B32, 37L10, 37L15, 34C37 ; Secondary 35J60, 35B40, 37G40.