Meandering of the spiral tip: an alternative approach

J. Nonlin. Sci. 7 (1997) 557-586

Marty Golubitsky, Victor G. LeBlanc and Ian Melbourne


Meandering of a one-armed spiral tip has been noted in chemical reactions and numerical simulations. Barkley, Kness and Tuckerman show that meandering can begin by Hopf bifurcation from a rigidly rotating spiral wave (a point that is verified in a B-Z reaction by Li, Ouyang, Petrov and Swinney). At the codimension two point where (in an appropriate sense) the frequency at Hopf bifurcation equals the frequency of the spiral wave, Barkley notes that spiral tip meandering can turn to linearly translating spiral tip motion.

Barkley also presents a model showing that the linear motion of the spiral tip is a resonance phenomenon, and this point is verified experimentally by Li et al and proved rigorously by Wulff. In this paper we suggest an alternative development of Barkley's model extending the center bundle constructions of Krupa from compact groups to noncompact groups and from finite dimensions to function spaces. This approach allows us to consider various bifurcations from a rotating wave. In particular, we can analyze in a straightforward manner the codimension two Barkley bifurcation and the codimension two Takens-Bogdanov bifurcation from a rotating wave. We also discuss Hopf bifurcation from a many armed spiral showing that meandering and resonant linear motion of the spiral tip do not always occur.

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