Advertisement

Dynamical Behavior of Patterns with Euclidean Symmetry

  • BjöRn Sandstede
  • Arnd Scheel
  • Claudia Wulff
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 115)

Abstract

Recent results on the dynamical behavior of patterns in two and three spatial dimensions are reviewed. Based upon spatio-temporal symmetries of patterns, it is shown that transitions to other patterns can be explained by analyzing low-dimensional model equations. Examples include the dynamics of periodically forced twisted scroll waves and transitions from rigidly-rotating spiral waves to meandering or drifting spirals.

Key words

Spiral waves twisted scroll waves Euclidean symmetry meandering, drifting. 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P. ASHWIN AND I. MELBOURNE, Noncompact drift for relative equilibria and relative periodic orbits, Nonlinearity, 10 (1997), pp. 595–616.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    D. BARKLEY, Linear stability analysis of rotating spiral waves in excitable media, Phys. Rev. Lett., 68 (1992), pp. 2090–2093.CrossRefGoogle Scholar
  3. [3]
    D. BARKLEY, Euclidean symmetry and the dynamics of rotating spiral waves, Phys. Rev. Lett., 72 (1994), pp. 164–167.CrossRefGoogle Scholar
  4. [4]
    D. BARKLEY, Spiral meandering, in Chemical waves and patterns, R. Kapral and K. Showalter (eds), Kluwer, Doordrecht, 1995, pp. 163–188.CrossRefGoogle Scholar
  5. [5]
    D. BARKLEY, EZ-spiral: a code for simulating spiral waves, available from ”http://www.ima.umn.edu/~barkley”.Google Scholar
  6. [6]
    V. N. BIKTASHEV, A. V. HOLDEN, AND E. V. NIKOLAEV, Spiral wave meander and symmetry of the plane, Int. J. Bifurcation Chaos, 6 (1996), pp. 2433–2440.MathSciNetCrossRefGoogle Scholar
  7. [7]
    V. N. BIKTASHEV, A. V. HOLDEN, AND H. ZHANG, Tension of organizing filaments of scroll waves, Phil. Trans. R. Soc. Lond. A, 347 (1994), pp. 611–630.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    M. BRAUNE AND H. ENGEL, Compound rotation of spiral waves in a light-sensitive Belousov-Zhabotinsky medium, Chem. Phys. Lett., 204 (1993), pp. 257–264.CrossRefGoogle Scholar
  9. [9]
    Focus Issue: Fibrillation in normal ventricular myocardium, Chaos, 8(1) (1998).Google Scholar
  10. [10]
    M. DOYLE, R.-M. MANTEL, AND D. BARKLEY, Fast simulation of waves in three-dimensional excitable media, Int. J. Bifurcation Chaos, 7 (1997), pp. 2529–2546.CrossRefGoogle Scholar
  11. [11]
    B. FIEDLER, B. SANDSTEDE, A. SCHEEL, AND C. WULFF, Bifurcation from relative equilibria to non-compact group actions: Skew products, meanders, and drifts, Doc. Math. J. DMV, 1 (1996), pp. 479–505.MathSciNetzbMATHGoogle Scholar
  12. [12]
    B. FIEDLER AND D. TURAEV, Normal forms, resonances, and meandering tip motions near relative equilibria of Euclidean group actions, Arch. Rat. Mech. Anal., to appear.Google Scholar
  13. [13]
    M. J. FIELD, Equivariant dynamical systems, Trans. Amer. Math. Soc., 259 (1980), pp. 185–205.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    M. GOLUBITSKY, V. LEBLANC, AND I. MELBOURNE, Meandering of the spiral tip — an alternative approach, J. Nonl. Sci., 7 (1997), pp. 557–586.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    M. GOLUBITSKY, V. LEBLANC, AND I. MELBOURNE, Hopf bifurcation from rotating waves and patterns in physical space, Preprint, 1998.Google Scholar
  16. [16]
    M. GOLUBITSKY, I. STEWART, AND D. G. SCHAEFFER, Singularities and groups in bifurcation theory II, Springer-Verlag, New York, 1988.zbMATHCrossRefGoogle Scholar
  17. [17]
    A. V. HOLDEN, Mathematics — The restless heart of a spiral, Nature, 387 (1997) pp. 655–656.CrossRefGoogle Scholar
  18. [18]
    W. JAHNKE, W. E. SKAGGS, AND A. T. WINFREE, Chemical vortex dynamics in the Belousov-Zhabotinskii reaction and in the two-variable Oregonator model, J. Chem. Phys., 93 (1989), pp. 740–749.CrossRefGoogle Scholar
  19. [19]
    J. P. KEENER, The dynamics of 3-dimensional scroll waves in excitable media, Physica D, 31 (1988), pp. 269–276.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    D. T. KIM, Y. KWAN, J. J. LEE, T. IKEDA, T. UCHIDA, K. KAMJOO, Y.-H. KIM, J. J. C. ONG, C. A. ATHILL, T.-J. WU, L. CZER, AND H. S. KARAGUEUZIAN, Patterns of spiral tip motion in cardiac tissues, Chaos, 8 (1998), pp. 137–148.CrossRefGoogle Scholar
  21. [21]
    M. KRUPA, Bifurcations of relative equilibria, SIAM J. Math. Anal., 21 (1990), pp. 1453–1486.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    G. Li, Q. OUYANG, V. PETROV, AND H. L. SWINNEY, Transition from simple rotating chemical spirals to meandering and travelling spirals, Phys. Rev. Lett., 77 (1996), pp. 2105–2108.CrossRefGoogle Scholar
  23. [23]
    R.-M. MANTEL AND D. BARKLEY, Periodic forcing of spiral waves in excitable media, Phys. Rev. E, 54 (1996), pp. 4791–4802.CrossRefGoogle Scholar
  24. [24]
    R.-M. MANTEL AND D. BARKLEY, Parametric forcing of scroll-wave patterns in three-dimensional excitable media, Preprint, 1998.Google Scholar
  25. [25]
    J. D. MURRAY, Mathematical biology, Springer-Verlag, Heidelberg, 1989.zbMATHGoogle Scholar
  26. [26]
    S. NETTESHEIM, A. VON OERTZEN, H. H. ROTERMUND, AND G. ERTL, Reaction diffusion patterns in the catalytic CO-oxidation on Pt(110) — front propagation and spiral waves, J. Chem. Phys., 98 (1993), pp. 9977–9985.CrossRefGoogle Scholar
  27. [27]
    A. V. PANFILOV, A. N. RUDENKO, AND A. M. PERTSOV, Twisted scroll waves in active 3-dimensional media, Dokl. Akad. Nauk. SSSR, 279 (1984), pp. 1000–1002.Google Scholar
  28. [28]
    A. V. PANFILOV AND A. T. WINFREE, Dynamical simulations of twisted scroll rings in three dimensional excitable media, Physica D, 17 (1985), pp. 323–330.MathSciNetCrossRefGoogle Scholar
  29. [29]
    B. B. PLAPP AND E. BODENSCHATZ, Core dynamics of multi-armed spirals in Rayleigh-Benard convection, Physica Scripta, T67 (1996), pp. 111–116.Google Scholar
  30. [30]
    T. PLESSER AND K.-H. MüLLER, Fourier analysis of the complex motion of spiral tips in excitable media, Int. J. Bifurcation Chaos, 5 (1995), pp. 1071–1084.zbMATHCrossRefGoogle Scholar
  31. [31]
    B. SANDSTEDE, A. SCHEEL, AND C. WULFF, Center-manifold reduction for spiral waves, C. R. Acad. Sci. Paris, Se I, Math., 324 (1997), pp. 153–158.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    B. SANDSTEDE, A. SCHEEL, AND C. WULFF, Dynamics of spiral waves on unbounded domains using center-manifold reductions, J. Diff. Eq., 141 (1997), pp. 122–149.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    B. SANDSTEDE, A. SCHEEL, AND C. WULFF, Bifurcations and dynamics of spiral waves,J. Nonl. Sci., to appear.Google Scholar
  34. [34]
    A. SCHRADER, M. BRAUNE, AND H. ENGEL, Dynamics of spiral waves in excitable media subjected to external periodic forcing, Phys. Rev. E, 52 (1995), pp. 98–108.CrossRefGoogle Scholar
  35. [35]
    G. S. SKINNER AND H. L. SWINNEY, Periodic to quasiperiodic transition of chemical spiral rotation, Physica D, 48 (1991), pp. 1–16.zbMATHCrossRefGoogle Scholar
  36. [36]
    J. J. TYSON AND J. P. KEENER, Singular perturbation theory of traveling waves in excitable media (a review), Physica D, 32 (1988), pp. 327–361.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    P. B. UMBANHOWAR, F. MELO, AND H. L. SWINNEY, Periodic, aperiodic, and transient patterns in vibrated granular layers, Physica A, 249 (1998), pp. 1–9.CrossRefGoogle Scholar
  38. [38]
    A. T. WINFREE, When time breaks down, Princeton University Press, Princeton, 1987.Google Scholar
  39. [39]
    A. T. WINFREE, Persistent tangles of vortex rings in excitable media, Physica D, 84 (1995), pp. 126–147.CrossRefGoogle Scholar
  40. [40]
    A. T. WINFREE AND S. H. STROGATZ, Singular filaments organize chemical waves in 3 dimensions: 2. twisted waves, Physica D, 9 (1983), pp. 65–80.MathSciNetCrossRefGoogle Scholar
  41. [41]
    A. T. WINFREE AND S. H. STROGATZ, Organizing centres for three-dimensional chemical waves, Nature, 311 (1984), pp. 611–615.CrossRefGoogle Scholar
  42. [42]
    C. WULFF, Theory of meandering and drifting spiral waves in reaction-diffusion systems, PhD thesis, FU Berlin, 1996.Google Scholar
  43. [43]
    V. ZYKOV, O. STEINBOCK, AND S. C. MüLLER, External forcing of spiral waves, Chaos, 4 (1994), pp. 509–518.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • BjöRn Sandstede
    • 1
  • Arnd Scheel
    • 2
  • Claudia Wulff
    • 2
  1. 1.Department of MathematicsOhio State UniversityColumbusUSA
  2. 2.Institut für Mathematik IFreie Universität BerlinBerlinGermany

Personalised recommendations