M. Golubitsky, E. Knobloch, and I. Stewart
Target Patterns and Spirals in Planar Reaction-Diffusion Systems
J. Nonlinear Sci. . 10 (2000) 333-354.
Solutions of reaction-diffusion equations on a circular domain are
considered. With Robin boundary conditions, the primary instability may
be a Hopf bifurcation 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 conditions are
Neumann or Dirichlet, then
the eigenfunctions are defined by Bessel functions of real argument,
and take the form of body modes filling the interior of
the domain. Body modes typically do not exhibit spiral structure.
We argue that the wall modes are important for understanding the
formation process of spirals, even in extended systems. Specifically,
we conjecture that wall modes describe the core of the spiral; the
constant-amplitude
spiral visible outside the core is the result of strong
nonlinearities which enter almost immediately above threshold as a
consequence of the exponential radial growth of the wall modes.