, Volume 7, Issue 6, pp 557586
First online:
Meandering of the Spiral Tip: An Alternative Approach
 M. GolubitskyAffiliated withDepartment of Mathematics, University of Houston
 , V. G. LeBlancAffiliated withDepartment of Mathematics, University of Ottawa
 , I. MelbourneAffiliated withDepartment of Mathematics, University of Houston
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessSummary
Meandering of a onearmed 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 BZ reaction by Li, Ouyang, Petrov, and Swinney). At the codimensiontwo 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 codimensiontwo Barkley bifurcation and the codimensiontwoTakensBogdanovbifurcation froma rotatingwave.We alsodiscussHopf bifurcation from a manyarmed spiral showing that meandering and resonant linear motion of the spiral tip do not always occur.
Key words
spiral waves Euclidean symmetry meandering center bundleMSC numbers
58F14 58F39 35K57 Title
 Meandering of the Spiral Tip: An Alternative Approach
 Journal

Journal of Nonlinear Science
Volume 7, Issue 6 , pp 557586
 Cover Date
 199706
 DOI
 10.1007/s003329900040
 Print ISSN
 09388974
 Online ISSN
 14321467
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 spiral waves
 Euclidean symmetry
 meandering center bundle
 58F14
 58F39
 35K57
 Authors

 M. Golubitsky ^{(1)}
 V. G. LeBlanc ^{(2)}
 I. Melbourne ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of Houston, Houston, TX, 772043476, USA
 2. Department of Mathematics, University of Ottawa, Ottawa, ON, K1N 6N5, Canada