Institute for Mathematics and Its Applications
Ian Stewart, University of Warwick
It is well known from numerical simulations and other approaches that reaction-diffusion equations possess solutions of `target pattern' and `spiral' form, similar to phenomena observed in the BZ reaction. The talk will outline joint work in progress with Marty Golubitsky and Edgar Knobloch, which -- in a particular abstraction -- view such patterns as examples of symmetric Hopf bifurcation. For this purpose we work in a circular domain. In order to prove the existence of such solutions, it turns out that boundary conditions are crucial. With the usual Neumann or Dirichlet boundary conditions, for example, spirals do not occur. Robin boundary conditions, in contrast, lead to spirals and target patterns as the generic Hopf bifurcations. The geometry is closely related to Bessel functions evaluated along radial lines in the complex plane that do not coincide with either the real or imaginary axis.
The talk will also indicate the current limitations of this approach. In particular the occurrence of such bifurcations in realistic model equations remains to be demonstrated, and several other important issues remain unresolved.