Target patterns and spirals in planar reaction-diffusion systems

Solutions of reaction-diffusion equations on a circular domain are consider ed. With Robin boundary conditions, the primary instability may be a Hopf b ifurcation with eigenfunctions exhibiting prominent spiral features. These eigenfunctions, defined by Bessel functions of complex argument, peak near the boundary and are called wall modes. In contrast, if the boundary condit ions are Neumann or Dirichlet, then the eigenfunctions are defined by Besse l functions of real argument, and take the form of body modes filling the i nterior of the domain. Body modes typically do not exhibit pronounced spira l structure. We argue that the wall modes are important for understanding t he formation process of spirals, even in extended systems. Specifically, we conjecture that wall modes describe the core of the spiral; the constant-a mplitude spiral visible outside the core is the result of strong nonlineari ties which enter almost immediately above threshold as a consequence of the exponential radial growth of the wall modes.