Continuation of spiral waves

We describe a new numerical method of computing rigidly rotating spiral waves, which is based on solving the Neumann boundary value problem for the radius-dependent angular Fourier modes. Utilizing the established continuation engine AUTO, our method is simple in implementation and can be easily modified to suit a particular reaction-diffusion system. Since the method does not involve direct simulations of the reaction-diffusion system, unstable branches of rigidly rotating spiral waves can be computed as well. We illustrate our method by computing single- and multi-armed spirals in the Barkley model. Continuation of single-armed spirals displays nearly identical results with the Barkley's continuation code STEADY. The dependence of spiral waves on the geometry of the medium reproduces the results of numerical simulations reported before in [A.M. Pertsov, E.A. Ermakova, AN. Panfilov, Rotating spiral waves in a modified Fitz-Hugh-Nagumo model, Physica D 14 (1) (1984) 117-124], revealing, however, some subtle details like non-monotonous dependence of the rotation frequency on the disc radius and the existence of an unstable rotating solution that separates coexisting free and pinned spirals. We demonstrate that on bounded discs, spiral waves are accompanied by boundary spots - slowly rotating solutions which are localized near the outer boundary of the disc. Boundary spots are shown to be closely related to one- and two-dimensional unstable critical solutions, such as unstable pulses in one dimension and critical fingers in two dimensions, which separate spiral waves from shrinking wave segments. (C) 2007 Elsevier B.V. All rights reserved.