Complexity and Chaos in Thermal Convection

  • S. Ciliberto
  • M. Caponeri
Part of the NATO ASI Series book series (NSSB, volume 222)


The stability of spatial patterns,that may appear in chemical, physical and biological systems is certainly a complex problem that has received a lot interest, both from an experimental1–3 and theoretical point of view4–12. General methods that allow a complete characterization of the transition to spatiotemporal chaos are not jet available, because this phenomenon presents many different features depending on the system under study. In fluid dynamics the analysis of this transition is very useful in order to understand the relationship between low dimensional chaos13,14 and the turbulent regimes, in which the fluid motion presents a chaotic evolution both in space and time. Indeed in spatially extended systems the transition to low dimensional chaos is associated with relevant spatial effects, such as mode competition, travelling waves, localized oscillations14, however the unpredictable time evolution does not influence the spatial order and the correlation length is comparable with the size of the system. To give more insight into the problem of the transition to turbulence it is very important to study the role of the spatial degrees of freedom in temporal chaotic regimes and the mechanisms that reduce the spatial coherence. The simplest mathematical models, in which the features of the transition to spatiotemporal chaos may be analysed, are systems of coupled maps5–7, one dimensional partial differential equations4,7,8 and cellular automata10. These models have a physical relevance because many of the features, that they present, are similar to those observed in experiments on boundary layer flow15, thermal convection2,3, liquid crystal and surface wavesl.


Fourier Mode Cellular Automaton Model Energy Fluctuation Spatial Order Spectral Entropy 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    P. Kolodner, A. Passner, C. M. Surko, R. W. Walden, Phys. Rev. Lett. 56, 2621 (1986); A. Pocheau Jour. de Phys. 49, 1127 (1988)I; I. Rehberg, S. Rasenat, J. Finberg, L. de la Torre Juarez Phys. Rev. Lett. 61, 2449(1988); N. B. Trufillaro, R. Ramshankar, J. P. Gollub Phys. Rev. Lett. 62, 422 (1989); V. Croquette, H. Williams Physica 37 D, 300 (1989).Google Scholar
  2. 2.
    P. Berge’, in “ The Physics of Chaos and System Far From Equilibrium”, M.Duong-van and B. Nicolaenko, eds. (Nuclear Physics B, proceedings supplement 1988); F. Daviaud, M. Dubois, P.Berg, Europhys. Lett. 9,441 (1989).Google Scholar
  3. 3.
    S.Ciliberto,P.Bigazzi,Phys.Rev.Lett. 60, 286 (1988).Google Scholar
  4. 4.
    B.Nicolaenko, in “ The Physics of Chaos and Systems Far From Equilibrium”, M.Duong-Van and B.Nicolaenko, eds. ( Nuclear Physics B, proceedings supplement 1988 ).Google Scholar
  5. 5.
    G. L. Oppo, R. Kapral Phys. Rev. A 3, 4219 (1986).CrossRefGoogle Scholar
  6. 6.
    K. Kaneko, Prog. Theor. Phys. 74, 1033 (1985); J. Crutchfield K. Kaneko in “Direction in Chaos”, B. L. Hao (World Scientific Singapore 1987 ); R. Lima, Bunimovich preprint.Google Scholar
  7. 7.
    H. Chate’, P. Manneville, Phys. Rev. Lett. 54, 112 (1987); Europhysics Letters 6,591(1988);Physica D 32, 409 (1988)MathSciNetGoogle Scholar
  8. 8.
    H. Chate’, B. Nicolaenko, to be published in the proceedings of the conference: “New trends in nonlinear dynamics and pattern forming phenomena”, Cargese 1988;Google Scholar
  9. 9.
    Y. Pomeau, A. Pumir and P. Pelce’, J. Stat. Phys. 37, 39 (1984)MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    F. Bagnoli, S. Ciliberto, A. Francescato, R. Livi, S. Ruffo, in “Chaos and complexity”, M. Buiatti, S. Ciliberto, R. Livi, S. Ruffo eds., (World Scientific Singapore 1988 ); F. Bagnoli, S. Ciliberto, A. Francescato, R. Livi, S. Ruffo, in the proceedings of the school on Cellular Automata, Les Houches (1989).Google Scholar
  11. 11.
    a) P. Coullett;b) P. Procaccia;c) A. Politi. in the proceedings of the confer.Google Scholar
  12. 12.
    P.C.Hoemberg,B.Shraiman, Physica 37D, 109 (1989).MathSciNetGoogle Scholar
  13. 13.
    For a general review of low dimensional chaos see for example: J. P. Eckmann, D. Ruelle, Rev. Mod. Phys. 1987; P. Berge, Y. Pomeau, Ch. Vidal, L’Ordre dans le Chaos ( Hermann, Paris 1984 ).Google Scholar
  14. 14.
    A. Libchaber, C. Laroche, S. Fauve, J. Physique Lett. 43, 221, (1982); M. Giglio, S. Musazzi, U. Perini, Phys. Rev. Lett. 53, 2402 (1984); M. Dubois, M. Rubio, P. Berge’, Phys. Rev. Lett. 51, 1446 (1983); S. Ciliberto, M. A. Rubio, Phys. Rev. Lett. 58, 25 (1987); S. Ciliberto, J. P. Gollub, J. Fluid Mech. 158, 381 (1984); S. Ciliberto, Europhysics Letters 4, 685 (1987).CrossRefGoogle Scholar
  15. 15.
    M. Van Dyke, An Album of Fluid Motion (Parabolic Press, Stanford, 1982 ); D. J. Tritton, Physical Fluid Dynamics (Van Nostrand Reinold, New York, 1979), Chaps. 19–22Google Scholar
  16. 16.
    S. Chandrasekar, Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford 1961; F. H. Busse, Rep. Prog. Phys. 41 (1978) 1929; Ch. Normand, Y. Pomeau, M. Velarde Rev. Mod. Phys. 49, 581, (1977).CrossRefGoogle Scholar
  17. 17.
    S. Ciliberto in “ Dynamics and Stochastic Processes “ R. Vilela Mendes ed.(Springer 1989 ); S. Ciliberto in “ Quantitative Measures of Complex Dynamical System”, N. Abraham, A. Albano eds.(Plenum 1989 ).Google Scholar
  18. 18.
    W. Merzkirch, Flow Visualisation, Academic Press, New York 1974.Google Scholar
  19. 19.
    S. Ciliberto, F. Francini,F. Simonelli, Opt. Commun. 54, 38 (1985).CrossRefGoogle Scholar
  20. 20.
    S. Ciliberto, M. Caponeri, F. Bagnoli, to be published in Nuovo Cimento D.Google Scholar
  21. 21.
    S. Ciliberto, B. Nicolaenko submitted for publication; A. Newell, D.Rand, D.Russell, Physica 33 D, 281 (1988);, N. Aubry, P. Holmes J L Lumley, E. Stone, J. Fluid Mech. 192, 115 (1988).Google Scholar
  22. 22.
    H. Muller-Krumbhaar in ‘Monte Carlo Methods in Statistical Physics”, edited by K. Binder (Springer- Verlag,New York 1979); D. R. Nelson,’Phase transitions and critical phenomena’, edited by C. Domb and J.L. Lebowitz (Academic Press London 1983 )Google Scholar
  23. 23.
    R. Livi, M. Pettini, S.Ruffo, M.Sparpaglione, A.Vulpiani, Phys. Rev. A 31, 1039 (1985); M. A. Rubio, P. Bigazzi, L. Albavetti, S. Ciliberto J. Fluid Mech. to be published.Google Scholar
  24. 24.
    A. Pumir, J. de Physique 46, 511 (1985).MathSciNetCrossRefGoogle Scholar
  25. 25.
    S.Zalensky,Physica 34 D, 427 (1989).Google Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • S. Ciliberto
    • 1
  • M. Caponeri
    • 1
  1. 1.Istituto Nazionale OtticaFirenzeItaly

Personalised recommendations