Meandering of the Spiral Tip: An Alternative Approach
Golubitsky, M., LeBlanc, V. G. and Melbourne, Ian (1997) Meandering of the Spiral Tip: An Alternative Approach Journal of Nonlinear Science, 7 (6). pp. 557-586. ISSN 0938-8974
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. Our reduction works only under certain simplifying assumptions which are not valid for Euclidean group actions. Recent work of Sandstede, Scheel, and Wulff shows how to overcome these difficulties.
This approach allows us to consider various bifurcations from a rotating wave. In particular, we analyze 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.
|Additional Information:||This is a pre-press version of an article published in Journal of Nonlinear Science, 7, 557-586. Click here to access the published version. © 1997 Springer.|
|Uncontrolled Keywords:||spiral waves, Euclidean symmetry, meandering center bundle MSC numbers|
|Divisions:||Faculty of Engineering and Physical Sciences > Mathematics|
|Depositing User:||Mr Adam Field|
|Date Deposited:||27 May 2010 14:42|
|Last Modified:||23 Sep 2013 18:33|
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