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Resonant Pattern Formation in a Spatially Extended Chemical System

  • Anna L. Lin
  • Valery Petrov
  • Harry L. Swinney
  • Alexandre Ardelea
  • Graham F. Carey
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 115)

Abstract

When an oscillatory nonlinear system is driven by a periodic external stimulus, the system can lock at rational multiples p : q of the driving frequency. The frequency range of this resonant locking at a given p : q depends on the amplitude of the stimulus; the frequency width of locking increases from zero as the stimulus amplitude increases from zero, generating an “Arnol’d tongue” in a graph of stimulus amplitude vs stimulus frequency. Physical systems that exhibit frequency locking include electronic circuits [1, 2], Josephson junctions [3], chemical reactions [4], fields of fireflies [5, 6], and forced cardiac systems [7, 8]. Most studies of frequency locking have concerned either maps or systems of a few coupled ODEs. The Arnol’d tongue structure of the sine circle map has been extensively studied, and the theory of periodically driven ODE systems has been well developed [9], but there has been very little analysis of frequency locking phenomena in PDEs, except for a few studies of the parametrically excited Mathieu equation with diffusion and damping [10, 11, 12] and the parametrically excited complex Ginzburg-Landau equation [13, 14]. Our interest here is in the effect of periodic forcing on pattern forming systems such as convecting fluids, liquid crystals, granular media, and reaction-diffusion systems. Such systems are often subject to periodic forcing (e.g., circadian forcing of biological systems), but the effect of forcing on the bifurcations to patterns has not been examined in experiments or analyzed in PDE models of these systems.

Keywords

Spiral Wave Force Period Arnold Tongue Oscillatory Nonlinear System Couple ODEs 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    L. PIVKA, A. L. ZHELEZNYAK, L. O. CHUA, Int. J. of Bif. and Chaos, 4, 1743, (1994).MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    M. ITOH, H. MURAKAMI, L. O. CHUA, Int. J. of Bif. and Chaos, 4, 1721, (1994).zbMATHCrossRefGoogle Scholar
  3. [3]
    T. BOHR, P. BAK, M. H. JENSEN, Phys. Rev. A, 30, 1970, (1984).MathSciNetCrossRefGoogle Scholar
  4. [4]
    V. PETROV, Q. OUYANG AND H. L. SWINNEY, Nature, 388, 655, (1997).CrossRefGoogle Scholar
  5. [5]
    J. BUCK AND E. BUCK, Sci. Am., 234, 74, (1976).CrossRefGoogle Scholar
  6. [6]
    R. E. MIROLLO, S. H. STROGATZ, SIAM J. Appl. Math., 50, 1645, (1990).MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    M. R. GUEVARA, L. GLASS, J. Math. Biology, 14, 1, (1982).MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    L. GLASS, Physics Today, 49, 40, (1996), A. M. KUNYSZ, A. SHRIER, L. GLASS, Am. J. Physiol., 273, (Cell Physiol., 42), 331, (1997).CrossRefGoogle Scholar
  9. [9]
    N. E. SANCHEZ, AND A. H. NAYFEH, Journal of Sound and Vibration, 207, 137, (1997).MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    R. H. RAND, Mechanics Research Communications, 23, 283, (1996).MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    R. H. RAND, R. ZOUNES AND R. HASTINGS, Nonlinear Dynamics, The Richard Rand 50th Anniversary Volume, 203, (1997).Google Scholar
  12. [12]
    R. H. RAND, B. C. DENARDO, W. I. NEWMAN AND A. L. NEWMAN, Design Engineering Technical Conferences, DE-Vol. 84-1, 3, part A. ASME, (1995).Google Scholar
  13. [13]
    P. COULLET AND K. EMILSSON, Physica D, 61, 119, (1992).zbMATHCrossRefGoogle Scholar
  14. [14]
    D. WALGRAEF, Spatiotemporal Pattern Formation, Springer, New York, (1997).CrossRefGoogle Scholar
  15. [15]
    D. RAND, S. OSTLUND, J. SETHNA, AND E. D. SIGGIA, Phys. Rev. Lett., 49, 387, (1982).MathSciNetCrossRefGoogle Scholar
  16. [16]
    R. E. ECKE, J. D. FARMER AND D. K. UMBERGER, Nonlinearity, 2, 175, (1989).MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    L. GLASS AND J. SUN, Phys. Rev. E, 50, 5077, (1994).CrossRefGoogle Scholar
  18. [18]
    A. M. DAVIE, Nonlinearity, 9, 421, (1996).MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    M. KRUPA AND M. ROBERTS, Physica D, 57, 417, (1992).MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    J. A. GLAZIER AND A. LIBCHABER, IEEE Transactions on Circuits and Systems, 35, 790, (1988).MathSciNetCrossRefGoogle Scholar
  21. [21]
    P. BAK, Physics Today, 39, 38, (1986).CrossRefGoogle Scholar
  22. [22]
    T. BOHR, P. BAK AND M. H. JENSEN, Phys. Rev. A, 30, 1970, (1984).Google Scholar
  23. [23]
    H. G. E. HENTSCHEL AND I. PROCACCIA, Physica D, 8, 435, (1983).MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    A. CHIFFAUDEL AND S. FAUVE, Phys. Rev., A35, 4004, (1987).Google Scholar
  25. [25]
    C. ELPHICK, A. HAGBERG AND E. MERON, Phys. Rev. Lett., 80, 5007, (1998).CrossRefGoogle Scholar
  26. [26]
    P. COULLET, T. FRISCH, AND G. SONNINO, Phys. Rev. E, 49, 2087, (1994).CrossRefGoogle Scholar
  27. [27]
    I. PRIGOGINE AND R. LEFEVER, J. Chem. Phys., 48, 1695, (1968).CrossRefGoogle Scholar
  28. [28]
    T. KAI AND K. TOMITA, Progr. Theor. Physics, 61, 54, (1979).CrossRefGoogle Scholar
  29. [29]
    S. KADAR, T. AMEMIYA AND K. SHOWALTER, J. Phys. Chem. A, 101, 8200, (1997).CrossRefGoogle Scholar
  30. [30]
    M. JINGUJI, M. ISHIHARA AND T. NAKAZAWA, J. Phys. Chem., 96, 4279, (1992).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Anna L. Lin
    • 1
  • Valery Petrov
    • 1
  • Harry L. Swinney
    • 1
  • Alexandre Ardelea
    • 2
  • Graham F. Carey
    • 2
  1. 1.Center for Nonlinear DynamicsThe University of Texas at AustinAustinUSA
  2. 2.CFD Lab ASE/EM DepartmentThe University of Texas at AustinAustinUSA

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