MEANDERING OF THE SPIRAL TIP - AN ALTERNATIVE APPROACH

Meandering of a one-armed spiral tip has been noted in chemical reacti ons and numerical simulations. Barkley, Kness, and Tuckerman show that meandering can begin by Hopf bifurcation from a rigidly rotating spir al wave (a point that is verified in a B-Z reaction by Li, Ouyang, Pet rov, and Swinney). At the codimension-two point where (in an appropria te sense) the frequency at Hopf bifurcation equals the frequency of th e spiral wave, Barkley notes that spiral tip meandering can turn to li nearly translating spiral tip motion. Barkley also presents a model sh owing that the linear motion of the spiral tip is a resonance phenomen on, and this point is verified experimentally by Li et al. and proved rigorously by Wulff. In this paper we suggest an alternative developme nt of Barkley's model extending the center bundle constructions of Kru pa from compact groups to noncompact groups and from finite dimensions to function spaces. Our reduction works only under certain simplifyin g assumptions which are not valid for Euclidean group actions. Recent work of Sandstede, Scheel, and Wulff shows how to overcome these diffi culties. This approach allows us to consider various bifurcations from a rotating wave. In particular, we analyze the codimension-two Barkle y bifurcation and the codimension-two Takens-Bogdanov bifurcation from a rotating wave. We also discuss Hopf bifurcation from a many-armed s piral showing that meandering and resonant linear motion of the spiral tip do nor always occur.