Journal of Nonlinear Science

, Volume 7, Issue 6, pp 557–586

Meandering of the Spiral Tip: An Alternative Approach

  • M. Golubitsky
  • V. G. LeBlanc
  • I. Melbourne
Article

DOI: 10.1007/s003329900040

Cite this article as:
Golubitsky, M., LeBlanc, V.G. & Melbourne, I. J. Nonlinear Sci. (1997) 7: 557. doi:10.1007/s003329900040

Summary

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-twoTakens-Bogdanovbifurcation froma rotatingwave.We alsodiscussHopf bifurcation from a many-armed spiral showing that meandering and resonant linear motion of the spiral tip do not always occur.

Key words

spiral wavesEuclidean symmetrymeandering center bundle

MSC numbers

58F1458F3935K57

Copyright information

© Springer-Verlag New York Inc. 1997

Authors and Affiliations

  • M. Golubitsky
    • 1
  • V. G. LeBlanc
    • 2
  • I. Melbourne
    • 1
  1. 1.Department of MathematicsUniversity of HoustonHoustonUSA
  2. 2.Department of MathematicsUniversity of OttawaOttawaCanada