Journal of Nonlinear Science

, Volume 7, Issue 6, pp 557–586 | Cite as

Meandering of the Spiral Tip: An Alternative Approach

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

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 waves Euclidean symmetry meandering center bundle 

MSC numbers

58F14 58F39 35K57 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D. Barkley. A model for fast computer-simulation of waves in excitable media. Physica D 49(1991) 61–70.CrossRefGoogle Scholar
  2. [2]
    D. Barkley. Linear stability analysis of rotating spiral waves in excitable media. Phys. Rev. Lett. 68(1992) 2090–2093.CrossRefGoogle Scholar
  3. [3]
    D. Barkley. Euclidean symmetry and the dynamics of rotating spiral waves. Phys. Rev. Lett. 72(1994) 165–167.Google Scholar
  4. [4]
    D. Barkley, M. Kness, and L. S. Tuckerman. Spiral-wave dynamics in a simple model of excitable media: The transition from simple to compound rotation. Phys. Rev. A 42(1990) 2489–2492.CrossRefMathSciNetGoogle Scholar
  5. [5]
    V. N. Biktashev, A. V. Holden, and E. V. Nikolaev. Spiral wave meander and symmetry in the plane. Int. J. Bifur. & Chaos 6 No. 12 (1996).Google Scholar
  6. [6]
    B. Fiedler, B. Sandstede, A. Scheel, and C. Wulff. Bifurcation from relative equilibria of noncompact group actions: Skew products, meanders, and drifts. Documenta Math. 1(1996) 479–505.Google Scholar
  7. [7]
    M. J. Field. Equivariant dynamical systems. Trans. Amer. Math. Soc. 259(1980) 185–205.CrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    M. Golubitsky and I.N. Stewart. Hopf bifurcation in the presence of symmetry. Arch. Ratl. Mech. & Anal. 87 No. 2 (1985) 107–165.CrossRefGoogle Scholar
  9. [9]
    J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Appl. Math Sci. 42, Springer-Verlag, New York, 1983.Google Scholar
  10. [10]
    D. Henry. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math. 840, Springer-Verlag, Berlin, 1981.Google Scholar
  11. [11]
    R. Kapral and K. Showalter. Chemical Waves and Patterns. Kluwer Academic Publishers, Amsterdam, 1995.Google Scholar
  12. [12]
    M. Krupa. Bifurcations from relative equilibria. SIAM J. Math. Anal. 21(1990) 1453–1486.CrossRefMathSciNetGoogle Scholar
  13. [13]
    W. Jahnke, W. E. Skaggs, and A. T. Winfree. Chemical vortex dynamics in the Belousov-Zhabotinsky reaction and in the two-variable Oregonator model. J. Phys. Chem. 93(1989) 740–749.CrossRefGoogle Scholar
  14. [14]
    G. Li, Q. Ouyang, V. Petrov, and H. L. Swinney. Transition from simple rotating chemical spirals to meandering and traveling spirals. Phys. Rev. Lett. 77(1996) 2105–2108.CrossRefGoogle Scholar
  15. [15]
    D. Rand. Dynamics and symmetry. Predictions for modulated waves in rotating fluids, Arch. Ratl. Mech. & Anal. 79(1982) 1–38.CrossRefGoogle Scholar
  16. [16]
    M. Renardy. Bifurcation from rotating waves. Arch. Ratl. Mech. & Anal. 79(1982) 49–84.Google Scholar
  17. [17]
    B. Sandstede, A. Scheel, and C. Wulff. Center-manifold reduction for spiral waves. C. R. Acad. Sci. To appear.Google Scholar
  18. [18]
    B. Sandstede, A. Scheel, and C. Wulff. Dynamics of spiral waves in unbounded domains using center-manifold reductions. Preprint.Google Scholar
  19. [19]
    A. Vanderbauwhede, M. Krupa, and M. Golubitsky. Secondary bifurcations in symmetric systems. In Proc. Equadiff Conference, 1987, (C. M. Dafermos, G. Ladas, and G. Papanicolaou, Eds.) Lect. Notes in Pure & Appl. Math. 118, Marcel Dekker, New York, 1989, 709–716.Google Scholar
  20. [20]
    A.T. Winfree. Scroll-shaped waves of chemical activity in three dimensions, Science 181 (1973) 937–939.CrossRefGoogle Scholar
  21. [21]
    C. Wulff. Theory of Meandering and Drifting Spiral Waves in Reaction-Diffusion Systems. Freie Universitä t Berlin, Thesis, 1996.Google Scholar

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

Personalised recommendations