Spatio-Temporal Phase Patterns Near a Hopf Bifurcation in 2D Systems

  • D. Walgraef
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 41)

Abstract

The spontaneous nucleation of spatio-temporal patterns in systems driven far from thermal equilibrium by uniform constraints remains the subject of intensive theoretical and experimental research. Despite the complexity of the dynamics which gives rise to this phenomenon, great progress has been achieved in the understanding of pattern formation and stability near instability points where the reduction of the dynamics leads to amplitude equations for the patterns. Furthermore, since most of these structures appear via continuous symmetry breaking effects, long range fluctuations are expected to develop spontaneously in the ordered regime. The corresponding long wavelength modes which play the role of Goldstone modes in driven systems may be described by the appropriate phase dynamics. The case of translational symmetry-breaking has been widely investigated in the case of nonlinear reaction-diffusion equations, Rayleigh-Bénard, Taylor-Couette, convective or hydrodynamical instabilities in normal fluids or liquid crystals, …/1–3/. In the case of oscillations of the limit cycle type associated with a Hopf bifurcation, temporal symmetry breaking occurs and the phase dynamics leads to various kinds of spatiotemporal behaviors. Among them, concentric or spiral chemical waves and turbulent structures associated with the 1d Kuramoto- Sivashinsky equation have been widely investigated /4/.

Keywords

Convection Hexagonal 

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References

  1. 1.
    See for example : a) Cellular Structures in Instabilities, eds. J.E. Wesfreid and S. Zaleski, Springer Verlag, New York, 1984;Google Scholar
  2. b).
    Patterns, Defects and Microstructures in Nonequilibrium Systems, ed. D. Walgraef, Martinus Nijhoff, Dordrecht, 1987.Google Scholar
  3. 2.
    P. Coullet and S. Fauve, Phys. Rev. Lett., 55, 2857 (1985).CrossRefADSGoogle Scholar
  4. 3.
    K. Kawasaki and H.R. Brand, Annals of Physics, 160, 2, 420 (1985).CrossRefADSMathSciNetGoogle Scholar
  5. 4.
    Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer Series in Synergetics 19, 1984.CrossRefMATHGoogle Scholar
  6. 5.
    M. Lowe and J.P. Gollub, Phys. Rev. A31, 3893 (1985).ADSGoogle Scholar
  7. 6.
    See e.g. Non-Equilibrium Dynamics in Chemical Systems, eds. C. Vidal and A. Pacault, Springer Verlag, Berlin, 1984.MATHGoogle Scholar
  8. 7.
    P. Möckel, Natuwissenschaften, 64, 224 (1977);CrossRefADSGoogle Scholar
  9. M. Kagan, A. Levi, D. Avnir, ibid. 69, 548 (1982);Google Scholar
  10. M. Gimenez, J.C. Micheau, ibid. 70, 90 (1983).Google Scholar
  11. 8.
    K.I. Agladaze, V.I. Krinsky and A.M. Pertsov, Nature, 308, 834 (1984);CrossRefADSGoogle Scholar
  12. S. Müller in ref.lb.Google Scholar
  13. 9.
    N. Kopell and L.N. Howard, Adv. Appl. Math., 2, 417 (1981)CrossRefMATHMathSciNetGoogle Scholar
  14. P. Hagan, SIAM J. Appl. Math., 42, 762 (1982).CrossRefMATHMathSciNetGoogle Scholar
  15. 10.
    S. Chandrasekhar, Hydrodynamic and Hydrodynamic Stability, Oxford University Press (1961).MATHGoogle Scholar
  16. 11.
    L.R. Keefe, Stud. Appl. Math., 73, 91 (1985).MATHADSMathSciNetGoogle Scholar
  17. 12.
    G. Dewel, P. Borckmans and D. Walgraef in Chemical Instabilities, eds. G. Nicolis and F. Baras, Reidel, NATO ASI Series C120, Dordrecht, 1984.Google Scholar
  18. 13.
    H. Greenside and M.C. Cross, Phys. Rev. A31, 2492 (1985).ADSGoogle Scholar
  19. 14.
    H.R. Brand, P.S. Lomdhal and A.C. Newell, Physica D23, 345 (1985).ADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • D. Walgraef
    • 1
  1. 1.Service de Chimie-Physique IIUniversité Libre de BruxellesBrusselsBelgium

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