Stability of the Scenarios Towards Chaos

  • P. Coullet
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 24)


For the last ten years, the study of chaotic behaviors generated by deterministic dynamical systems has received an increasing interest among Physicists and Mathematicians. A fascinating aspect of this field of research is its interdisciplinarity. The deep reason for this universality lies in the very nature of the subject: The Mathematics of Time [1]. The study of simple dynamical systems, such as the iterations of mappings or the ordinary differential equations thus appears as a study of these Mathematics of time. Two kinds of problems can be easily identified. The first one deals with the transition towards complex behaviors. The second one is concerned by the chaotic behaviors themselves. In this paper, we will only discuss some general aspects of the first class of problems. In fact, these problems are in some sense simpler to study. They generally lead to universal scenarios [2] by which a periodic or quasiperiodic signal progressively looses its regularity, as an external constraint varies. These problems of transition bear a striking analogy with critical phenomena. The common analytical tool is the renormalization group [3].


Chaotic Behavior Critical Phenomenon Stable Manifold Period Doubling Invariant Torus 
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.
    S. Smale: The Mathematics of Time (Springer Verlag, 1980)zbMATHCrossRefGoogle Scholar
  2. 2.
    J.P. Eckmann: Rev. Mod. Phys. 53, 643 (1981)MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 3.
    K.G. Wilson: Rev. Mod. Phys. 55, 583 (1983)ADSCrossRefGoogle Scholar
  4. 4.
    M.J. Feigenbaum: J. Stat. Phys. 19, 25 (1978) M.J. Feigenbaum: J. Stat. Phys. 21, 669 (1979)MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    J.P. Crutchfield, N. Nauenberg and J. Rudnick: Phys. Rev. Lett. 46, 933 (1981)MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    B. Shraiman, C.E. Wayne and P.C. Martin: Phys. Rev. Lett. 46, 935 (1981)MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    A. Arneodo, P. Coullet and A.E. Spiegel: Phys. Lett. 94A, 1 (1983)MathSciNetCrossRefGoogle Scholar
  8. 8.
    V. Franceschini: Physica 6D, 285 (1983)MathSciNetzbMATHGoogle Scholar
  9. 9.
    K. Kaneko: “Doubling of Torus” preprint (1983)Google Scholar
  10. 10.
    A. Arneodo, P. Coullet and C. Tresser: Phys. Lett. 81A, 197 (1981)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Y. Kuramoto and S. Koga: Phys. Lett. 92A, 1 (1982)MathSciNetCrossRefGoogle Scholar
  12. 12.
    P. Coullet and C. Tresser: J. de Physique, colloque 39, C.-25 (1978) C. Tresser and P. Coullet: C.R.Acad. Sc. Paris 287, 577 (1978)MathSciNetzbMATHGoogle Scholar
  13. 13.
    J.P. Eckmann “Routes to chaos with special emphasis on period doubling” Les Houches Summer school 1981 Chaotic Behavior in Deterministic Dynamical Systems, Ed. G. IOOSS, HELLEMAN, STORA (North Holland 1983 )Google Scholar
  14. 14.
    P. Collet and J.P. Eckmann: Iterated maps on an interval as dynamical systems (Birkhauser, Boston 1980 )Google Scholar
  15. 15.
    S.K. Ma: Modern theory of critical phenomena (Benjamin, Reading, Mass. 1976 )Google Scholar
  16. 16.
    B.A. Huberman and J. Rudnick: Phys. Rev. Lett. 45, 154 (1980)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    A. Chenciner and G. Iooss: Arch. Rational Mech. Anal. 69, 109 (1979)MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. 18.
    Bifurcation de diffgomorphismes de ℝ2 au voisinage d’un point fixe elliptique” in Les Houches Summer school 1981 Chaotic Behavior in Deterministic Dynamical Systems, Ed. IOOSS, HELLEMAN, STORA (North Holland 1983)Google Scholar
  19. 19.
    P. Coullet:“Chaotic behaviors in the unfolding of singular vector fields” to be published in the Proceedings of the “Workshop on Common Trends in Particle and Condensed Matter Physics” at Les Houches (1983)Google Scholar
  20. 20.
    E.N. Lorenz: J. Atmos. Sci. 20, 130 (1963)ADSCrossRefGoogle Scholar
  21. 21.
    J. Guckenheimer and R.F. Williams: Publ. Math. IHES 50, 307 (1979)Google Scholar
  22. 22.
    C. Tresser: C.R. Acad. Sei. Paris 296, 729 (1983)MathSciNetzbMATHGoogle Scholar
  23. 23.
    P. Manneville and Y. Pomeau: Phys. Lett. 75 A, 1 (1979); Commun Math. Phys. 74, 74 (1980; Physica ID, 219 (1980)MathSciNetADSGoogle Scholar
  24. 24.
    J.E. Hirsch, N. Nauenberg and D.J. Scalapino: Phys. Lett. 87A, 391 (1982)MathSciNetADSGoogle Scholar
  25. 25.
    B. Hu, and J. Rudnick: Phys. Rev. Lett. 48, 1645 (1982)MathSciNetADSCrossRefGoogle Scholar
  26. 26.
    H. Daido: Prog. Theor. Phys. 68, 1935 (1982)MathSciNetADSzbMATHCrossRefGoogle Scholar
  27. 27.
    M.J. Feigenbaum, L.P. Kadanoff and S.J. Shenker: Physica 5D, 370 (1982)MathSciNetGoogle Scholar
  28. 28.
    D. Rand, S. Ostlund, J.P. Setna and E.D. Siggia: Phys. Rev. Lett. 49, 132 (1982)MathSciNetADSCrossRefGoogle Scholar
  29. 29.
    J.P. Setna and E.D. Siggia “Universal transition in a dynamical system forced at two incommensurate frequencies” Preprint 1983Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • P. Coullet
    • 1
  1. 1.Mécanique StatistiqueLaboratoire de Physique de la Matière CondenséeNice CedexFrance

Personalised recommendations