Transition to Deterministic Chaos in a Hydrodynamic System

  • M. Giglio
  • S. Musazzi
  • U. Perini
Conference paper
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 47)


Nonlinear hydrodynamic systems often exhibit transition to chaotic behaviour when a properly defined stress parameter is increased. According to an idea due to Landau (1) the route to chaos has been conjectured to consist of a sequence of bifurcation points at which many incommensurate frequencies are gradually introduced. At variance with this, Lorenz (2) and Ruelle and Takens (3) have sugqested that fluid turbulence is associated with the existence of a strange attractor in phase space. Therefore chaotic behaviour is actually caused by finite dimensional deterministic dynamics and is a consequence of the sensitivity of the solutions to the initial conditions.


Lyapunov Exponent Rayleigh Number Bifurcation Point Chaotic Behaviour Strange Attractor 
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  1. 1.
    L.D. Landau and E.M. Lifshitz: Fluid Mechanics (Pergamon, Oxford, 1959).Google Scholar
  2. 2.
    E.N. Lorenz, J. Atmos. Sci. 20, 130 (1963).ADSCrossRefGoogle Scholar
  3. 3.
    D. Ruelle and F. Takens, Commun. Math. Phys. 20, 167 (1971).MathSciNetADSMATHCrossRefGoogle Scholar
  4. 4a.
    J. Maurer and A. Libchaber, J. Phys. (Paris) Lett. 41, L515 (1980);CrossRefGoogle Scholar
  5. 4b.
    A. Libchaber and J. Maurer, J. Phys, (Paris) Coll. C3 41, C3 51 (1980);Google Scholar
  6. 4c.
    A. Libchaber and J. Maurer, Nonlinear Phenomena at Phase Transitions, ed. T. Riste (Plenum, 1982).Google Scholar
  7. 5.
    M. Giglio, S. Musazzi and U. Perini, Phys.Rev.Lett. 47, 243 (1981).ADSCrossRefGoogle Scholar
  8. 6.
    A. Libchaber, C. Laroche and S. Fauve, J. Phys. (Paris) Lett. 43, L211 (1982).CrossRefGoogle Scholar
  9. 7.
    M.J. Feigenbaum, J. Stat. Phys. 19, 25 (1978), Phys. Lett. 74A, 375MathSciNetADSMATHCrossRefGoogle Scholar
  10. (1979).Commun. Math. Phys. 77, 65 (1980).Google Scholar
  11. 8.
    At the NATO Advanced Study Workshop Testing Non-Linear Dynamics, Haverf. College, Pa., June 83, we learned of studies of this type by three groups: P. Berge and M. Dubois; J. Guckenheimer, G. Buzyna and R. Pfeffer; A. Brandstater, J. Swift, H. Swinney, A. Wolf, J.D. Farmer, E. Jen and J.P. Crutchfield.Google Scholar
  12. 9.
    S. Chandrasekkar, Hydrodynamic and Hydromagnetic Stability, (Clarendon, Oxford, 1961).Google Scholar
  13. 10.
    F.H. Busse, Rep. Progr. Physics 41, 1929 (1978).ADSCrossRefGoogle Scholar
  14. 11.
    M. Giglio, S. Musazzi and U. Perini, Evolution of Order and Chaos, ed. H. Haken (Springer-Verlag, Berlin Heidelberg New York, 1982).Google Scholar
  15. 12a.
    For further reading on this subject see, for exampleGoogle Scholar
  16. E. Ott, Rev. Mod. Phys. 53, 655 (1981)MathSciNetADSMATHCrossRefGoogle Scholar
  17. 12b.
    R. Shaw, Z. Naturforsch. 36 a, 80 (1981)MathSciNetADSMATHGoogle Scholar
  18. 12c.
    J.D. Farmer, Physica 4D, 366 (1982).MathSciNetADSGoogle Scholar
  19. 13.
    B. Mandelbrot, Fractals-Form, Chance and Dimension (Freeman, San Francisco, 1977).MATHGoogle Scholar
  20. 14.
    J.L. Kaplan and J.A. Yorke, Functional Differential Equations and Approximations of Fixed Points, eds. H.O. Walther, Lecture Notes in Math. 730 (Springer, 1979).Google Scholar
  21. 15.
    N.H. Packard, J.P. Crutchfield, J.D. Farmer and K.S. Shaw, Phys. Rev. Lett. 45, 712 (1980).ADSCrossRefGoogle Scholar
  22. 16.
    J. Guckenheimer, Nature 298, 358 (1982).ADSCrossRefGoogle Scholar
  23. 17.
    P. Grassberger and J. Procaccia, Phys. Rev. Lett. 50, 346 (1983).MathSciNetADSCrossRefGoogle Scholar
  24. 18.
    A. Ben-Mizrachi, I. Procaccia and P. Granberger, Phys. Rev. A, to appear.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

  • M. Giglio
    • 1
  • S. Musazzi
    • 1
  • U. Perini
    • 1
  1. 1.CISE SpaMilanoItaly

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