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Phase Dynamics and Spatial Patterns in Oscillating and Excitable Media

  • P. Hanusse
  • V. Pérez-Muñuzuri
  • C. Vidal
Chapter
Part of the NATO ASI Series book series (NSSB, volume 244)

Abstract

As far as pattern formation, front propagation and phase dynamics are concerned, a lot has been achieved in various, yet related, specific fields. Let us mention just some of these: in hydrodynamics, the equations of the Rayleigh-Benard convection by Newell & Whitehead [1] and the instability of convective flows [2], the phase instability of oscillating systems by Kuramoto [3], the propagation of fronts in combustion by Sivashinski [4], the Eckhaus instability of steady structures [5], the topological turbulence in the Ginzburg-Landau equation [6]. Finally, to return to topics more directly related to this work, and with no pretention to exhaustivity, we should cite the works of Winfree [7], Tyson [8], Fife [9], Keener [10], Mikhailov & Krinskii [11] Meron & Pelce [12]. These mostly deal with theory. Of course, much has been done in experiments, which we should only cite indirectly, through review articles or collective works [13, 14, 15].

Keywords

Hopf Bifurcation Cellular Automaton Phase Dynamic Bifurcation Theory Global Bifurcation 
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]
    Newell, A.C. & Whitehead, J.C. (1969). J. Fluid Mech. 38, 279.ADSCrossRefzbMATHGoogle Scholar
  2. [2]
    Pomeau, Y. & Manneville, P. (1979). J. Phys. Lett. 40, L610.MathSciNetGoogle Scholar
  3. [3]
    Kuramoto, Y. (1984). Chemical Oscillations, Waves and Turbulence. Springer.CrossRefzbMATHGoogle Scholar
  4. [4]
    Sivashinski, G.I. (1977). Acta Astron. 4, 1177.CrossRefGoogle Scholar
  5. [5]
    Eckhaus, W. (1965). Studies in Nonlinear Stability Theory. Springer.CrossRefGoogle Scholar
  6. [6]
    Coullet, P., Gil, L. & Lega, J. (1989). Phys. Rev. Lett. 62(14), 1619.ADSCrossRefGoogle Scholar
  7. [7]
    Winfree, A.T. (1980). The geometry of biological time. Springer.zbMATHGoogle Scholar
  8. [8]
    Tyson, J. (1987). J. Chim. Phys. 84, 1359–1365.MathSciNetGoogle Scholar
  9. [9]
    Fife, P.C. (1984). In Non-equilibrium Dynamics in Chemical systems, Vidal, C. & Pacault, A. (eds.), pp. 76–88. Springer.CrossRefGoogle Scholar
  10. [10]
    Keener, J. (1989). Physica D34, 378–390.zbMATHMathSciNetGoogle Scholar
  11. [11]
    Mikhailov, A.C.& Krinskii, V.I. (1983). Physica D9, 346–371.MathSciNetGoogle Scholar
  12. [12]
    Meron, E. & Pelcé, P. (1988). Phys. Rev. Lett. 60, 1880–1883.ADSCrossRefMathSciNetGoogle Scholar
  13. [13]
    Vidal, C. & Hanusse, P. (1986). Int. rev. Phys. Chem. 5, 1–155.CrossRefGoogle Scholar
  14. [14]
    Field, R.J. & Burger, M. (eds.) (1985). Oscillations and travelling waves in chemical systems. Wiley: New York.Google Scholar
  15. [15]
    Tyson, J. & Keener, J. (1988). Physica D32, 327–361.zbMATHMathSciNetGoogle Scholar
  16. [16]
    Hanusse, P. (1987). J. Chim. Phys. 84, 1315–1327.Google Scholar
  17. [17]
    Hanusse, P. & Guillataud, P. Object detection and identification by hierarchical segmentation, to be published.Google Scholar
  18. [18]
    Marsden, J.E. & McCracken, M. (1976). The Hopf bifurcation and its applications. Springer.CrossRefzbMATHGoogle Scholar
  19. [19]
    Guckenheimer, J. & Holmes, P. (1983). Nonlinear oscillations, Dynamical sytems, Bifurcations of vector fields, Vol. I. Springer.Google Scholar
  20. [20]
    Golubitsky, M. & Schaeffer, D.G. (1985). Singularities and groups in bifurcation theory. Springer.CrossRefzbMATHGoogle Scholar
  21. [21]
    de Kepper, P., Pacault, A. & Rossi, A. (1976). C.R. Acad. Sci. Paris 282C, 199.Google Scholar
  22. [22]
    FitzHugh, R. (1960). J. Gen. Physiol. 43, 867–896.CrossRefGoogle Scholar
  23. [23]
    Kuramoto, Y. (1976). Prog. Theor. Phys. 56, 724–740.ADSCrossRefMathSciNetGoogle Scholar
  24. [24]
    Boissonade, J. & de Kepper, P. (1981). J. Chem. Phys. 75, 189–195.ADSCrossRefGoogle Scholar
  25. [25]
    Hanusse, P. & Perez-Muñuzuri, V., to be published.Google Scholar
  26. [26]
    Winfree, A.T., Winfree, E.M. & Seifert, H. (1985). Physica D17, 109.zbMATHMathSciNetGoogle Scholar
  27. [27]
    Wolfram, S. (1986). Theory and applications of Cellular Automata. World Scientific: Singapore.zbMATHGoogle Scholar
  28. [28]
    Markus, M., see contribution in this volume.Google Scholar
  29. [29]
    Hanusse, P. & Pérez-Muñuzuri, V., in preparation.Google Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • P. Hanusse
    • 1
  • V. Pérez-Muñuzuri
    • 1
    • 2
  • C. Vidal
    • 1
  1. 1.Centre de Recherche Paul Pascal/CNRSUniversité de Bordeaux IPessacFrance
  2. 2.Dept. of Física de la Materia CondensadaUniversity of Santiago de CompostelaSpain

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