M. Dellnitz, M. Golubitsky, A. Hohmann and I. Stewart
Spirals in scalar reaction diffusion equations
Intern. J. Bifur. & Chaos. 5 (6) (1995) 1487-1501.
Spiral patterns have been observed experimentally, numerically and theoretically in a variety of systems. It is often believed that these spiral wave patterns can occur only in systems of reaction-diffusion equations. We show, both theoretically (using Hopf bifurcation techniques) and numerically (using both direct simulation and continuation of rotating waves) that spiral wave patterns can appear in a single reaction-diffusion equation (in u(x,t)) on a disk, if one assumes `spiral' boundary conditions (u_r=mu_theta). Spiral boundary conditions are motivated by assuming that a solution is infinitesimally an Archemedian spiral near the boundary. It follows from a bifurcation analysis that for this form of spirals there are no singularities in the spiral pattern (technically there is no spiral tip) and that at bifurcation there is a steep gradient between the `red' and `blue' arms of the spiral.