Dynamic Instabilities Observed in the Belousov-Zhabotinsky System

  • C. Vidal
Part of the Springer Series in Synergetics book series (SSSYN, volume 11)


Since the work of RUELLE and TAKENS [l], which established that turbulence may occur in systems with few degrees of freedom, a lot of effort has been devoted to study the so-called weak turbulence. Theory and experimentation (numerical as well as bench experiments) have both contributed to a new insight into the onset of turbulence. Let me briefly recall some salient results of the experiments performed on real hydrodynamical systems. Two geometries have been more thoroughly studied than any others, namely the circular Couette flow [2] and the Rayleigh-Benard instability. The more comprehensive results belong to the latter, having given rise to a great number of observations. For large aspect ratios, turbulence occurs at, or very near to, the threshold of convective instability [3.a]. On the other hand, several bifurcations lead to turbulence when cells with a low aspect ratio are used. Amongst other factors, the detailed sequence of bifurcations depends on the Prandtl number of the fluid, i.e. on the ratio of its kinematic viscosity to its thermal diffusivity. Although the instabilities involved are not the same, there are strong similarities in the behaviour of liquid helium [3c], silicon oil [3b] and water [4]. On the different routes leading to turbulence, three phenomena may be encountered: a cascade of period doubling bifurcations (sometimes named the Feigenbaum cascade), a quasi-periodic regime involving 2 or even 3 independent frequencies which, eventually, “lock in”, and an intermittency phenomenon, that is to say, bursts of noise emerging from time to time in a coherent regime. In agreement with the basic prediction of RUELLE and TAKENS, it has been observed in all cases that the transition to chaotic flow always takes place through a small number of bifurcations. Other experiments on the circular Couette flow, and numerical integration of sets of differential equations, such as the celebrated Lorenz model, have also led to the same general conclusion.


Fourier Spectrum Bifurcation Parameter Periodic Regime Bench Experiment Coherent Regime 
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  1. 1.
    D. Ruelle and F. Takens, Comm. Math. Phys. 20, 167 - 192 (1971).CrossRefGoogle Scholar
  2. 2.
    P.R. Fenstermacher, H.L. Swinney and J.P. Gollub, J. Fluid. Mech. 94 103–128 (1979).CrossRefGoogle Scholar
  3. 3.
    a G. Ahlhers, this volume, b P. Bergé, this volume. c A. Libchaber, this volume.Google Scholar
  4. 4.
    J.P. Gollub and S.V. Benson, J. Fluid. Mech. 100, 449–470 (1980).CrossRefGoogle Scholar
  5. 5.
    D. Ruelle, Trans. N.Y. Acad. Sci. 35, 66–71 (1973).Google Scholar
  6. 6.
    O.E. Rossler and K. Wegmann, Nature, 271, 5460–5461 (1978). K. Wegmann and O.E. Rossler, Z. Naturforsch. 33a, 1179–1183 (1978).CrossRefGoogle Scholar
  7. 7.
    R.A. Schmitz, K.R. Graziani and J.L. Hudson, J. Chem. Phys. 67, 3040–3044 (1977).CrossRefGoogle Scholar
  8. 8.
    J.L. Hudson, M. Hart and D. Marinko, J. Chem. Phys. 71, 1601–1606 (1979).CrossRefGoogle Scholar
  9. 9.
    C. Vidal, J.C. Roux, A. Rossi and S. Bachelart, C.R. Acad. Sci. Paris, serie C, 289, 73–76 (1979). C. Vidal, J.C. Roux, S. Bachelart and A. Rossi, Ann. N.Y. Acad. Sci. 357, 377–396 (1980).Google Scholar
  10. 10.
    J.C. Roux, A. Rossi, S. Bachelart and C. Vidal, Phys. Lett. 77A, 391–393 (1980).CrossRefGoogle Scholar
  11. 11.
    J.C. Roux, A. Rossi, S. Bachelart and C. Vidal, Physica D, in press.Google Scholar
  12. 12.
    R.J. Field and R.M. Noyes, J. Chem. Phys. 60, 1877–1884 (1974). R.J. Field and R.M. Noyes, J. Am. Chem. Soc., 96, 2001–2006 (1974). R.J. Field, J. Chem. Phys., 63, 2284–2296 (1975).CrossRefGoogle Scholar
  13. 13.
    R.J. Field, E. Koros and R.M. Noyes, J. Am. Chem. Soc., 94, 8649–8664 (1972).CrossRefGoogle Scholar
  14. 14.
    B. Belousov, “Sb. Ref. Radiat. Med.”, 145–147, Medgiz, Moscow (1959).Google Scholar
  15. 15.
    C. Yidal, J.C. Roux and A. Rossi, J. Am, Chem. Soc., 102, 1241–1245 (1980).CrossRefGoogle Scholar
  16. 16.
    R.K. Otnes and L. Enochson, “Digital time series analysis” Wiley New-York (1972).Google Scholar
  17. 17.
    J.P. Crutchfield, J.D. Farmer, N.H. Packard, R.S. Shaw, G. Jones and R.J. Donnelly, Phys. Lett. 76A, 1–4 (1980).CrossRefGoogle Scholar
  18. 18.
    M.J. Feigenbaum, Phys. Lett. 74A, 375–378 (1979); Comm. Math. Pfiys. 77, 65–86 (1980).Google Scholar
  19. 19.
    D. Ruelle, private communication.Google Scholar
  20. 20.
    N.H. Packard, J.P. Crutchfield, J.D. Farmer and R.S. Shaw, Phys. Rev. Lett., 45, 712–716 (1980).CrossRefGoogle Scholar
  21. 21.
    Y. Pomeau, J.C. Roux, S. Bachelart, A. Rossi and C. Vidal, submitted to J. Phys. Lett. C. Vidal, S. Bachelart and A. Rossi, to be published.Google Scholar
  22. 22.
    K. Tomita and I. Tsuda, Phys. Lett., 71A, 489–492 (1979).CrossRefGoogle Scholar
  23. 23.
    Y. Pomeau and P. Manneville, Commun. Math. Phys., 74, 189–197 (1980).CrossRefGoogle Scholar
  24. 24.
    P. de Kepper, A. Rossi and A. Pacault, C.R. Acad. Sci. Paris, Ser. C, 283, 371–376 (1976).Google Scholar
  25. 25.
    R.M. May, Nature, 261, 459–467 (1976).PubMedCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1981

Authors and Affiliations

  • C. Vidal
    • 1
  1. 1.Centre de Recherches Paul PascalTalenceFrance

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