Bifurcation to spiral waves in reaction-diffusion systems

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Abstract

For a large class of reaction-diffusion systems on the plane, we show rigorously that m-armed spiral waves bifurcate from a homogeneous equilibrium when the latter undergoes a Hopf bifurcation. In particular, we construct a finite-dimensional manifold which contains the set of small rotating waves close to the homogeneous equilibrium. Examining the flow on this center-manifold in a very general example, we find different types of spiral waves, distinguished by their speed of rotation and their asymptotic shape at large distances of the tip. The relation to the special class of λ-ω systems and the validity of these systems as an approximation is discussed.

Original languageEnglish (US)
Pages (from-to)1399-1418
Number of pages20
JournalSIAM Journal on Mathematical Analysis
Volume29
Issue number6
DOIs
StatePublished - Nov 1998

Keywords

  • Center-manifolds
  • Ginzburg-Landau equations
  • Spiral waves
  • λ-ω systems

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