Mixing pp 343-359 | Cite as

Renormalization Group Method in Chaotic Mixing

  • George M. Zaslavsky
Part of the NATO ASI Series book series (NSSB, volume 373)


There exist special values of a control parameter for which a self-similar hierarchy of islands occurs in the phase space of a system with chaotic dynamics. This set of islands impose an anomalous diffusion of particles which can be described by a fractional kinetic equation with fractional derivatives in space and time. The corresponding exponents of the derivatives can be related to the self-similarity properties of the islands using the renormalization group approach. Examples are given for the web-map and standard map.


Phase Space Renormalization Group Chaotic Dynamic Anomalous Diffusion Renormalization Group Method 
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]
    N.S. Krylov, Papers on the Foundation of Statistical Physics (Princeton Univ., Princeton, NJ, 1979).Google Scholar
  2. [2]
    M.F. Shlesinger, G.M. Zaslavsky, and J. Klafter, Nature 363, 31 (1993).ADSCrossRefGoogle Scholar
  3. [3]
    J.P. Bouchaud and A. Georges, Phys. Rep. 195, 127 (1990).MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    J. Klafter, M.F. Shlesinger, G. Zumofen, Physics Today 49, 33 (1996).CrossRefGoogle Scholar
  5. [5]
    “Lévy Flights and Related Topics in Physics”, Eds. M. Shlesinger, G. Zaslavsky, and U. Frisch (Springer, Heidelberg, 1995).zbMATHGoogle Scholar
  6. [6]
    V.V. Afanas’ev, R.Z. Sagdeev and G.M. Zaslavsky, Chaos 1, 143 (1991).MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    G.M. Zaslavsky, in: “Topological Aspects of the Dynamics of Fluids and Plasmas”, Eds. H.K. Moffatt, G.M. Zaslavsky, P. Comte, and M. Tabor (Kluwer, Boston, 1992) p. 481.Google Scholar
  8. [8]
    G.M. Zaslavsky, D. Stevens, H. Weitzner, Phys. Rev. E 48, 1683 (1993).MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    G.M. Zaslavsky, Chaos 4 25 (1994).MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. [10]
    G.M. Zaslavsky, Physica D 76 110 (1994).MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. [11]
    W. Young, A. Pumir, and Y. Pomeau, Phys. Fluids A 1, 462 (1989).MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. [12]
    G.M. Zaslavsky, M. Yu. Zakharov, R.Z. Sagdeev, D.A. Usikov and A.A. Chernikov, Sov. Phys.-JETP 64 294 (1986).MathSciNetGoogle Scholar
  13. [13]
    G.M. Zaslavsky, B.A. Niyazov, Phys. Reports (to appear).Google Scholar
  14. [14]
    J.D. Meiss, Phys. Rev. A 34, 2375 (1986).ADSCrossRefGoogle Scholar
  15. [15]
    J.D. Meiss, Rev. Mod. Phys. 64, 795 (1992).MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. [16]
    G.M. Zaslavsky, M. Edelman, and B. Nyazov, Chaos 7, No. 1 (1997).Google Scholar
  17. [17]
    B.V. Chirikov, Phys. Reports 52, 264 (1979).MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    J. Klafter, G. Zumofen, and M.F. Shlesinger, in Lévy Flights and Related Topics in Physics, Eds. M.F. Shlesinger, G.M. Zaslavsky, U. Frisch, Springer, 1995, p. 196.Google Scholar
  19. [19]
    S. Benkadda, S. Kassibrakis, R.B. White, and G.M. Zaslavsky, (not published).Google Scholar
  20. [20]
    T.H. Solomon, E.R. Weeks and H.L. Swinney, Phys. Rev. Lett. 71, 3975 (1993)ADSCrossRefGoogle Scholar
  21. [21]
    O. Cardoso, B. Gluckmann, O. Parcollet, and P. Tabeling, Phys. Fluids 8, 209 (1996).ADSCrossRefGoogle Scholar
  22. [22]
    P. Lévy, Theorie de l’Addition des Variables Aletoires (Gauthier-Villiers, Paris, 1937).Google Scholar
  23. [23]
    E.W. Montroll and M.F. Shlesinger, in “Studies in Statistical Mechanics”, Eds. J. Lebowitz and E. Montroll (North-Holland, Amsterdam, 1984), vol. 11, p. 1.Google Scholar
  24. [24]
    D. Escande, Phys. Rep. 121 (1985) 163.MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    G. Zaslavsky, Chaos in Dynamic Systems, (Harwood Acad. Publ., NY, 1995).Google Scholar
  26. [26]
    S.G. Samko, a.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives and Their Applications, Nauka i Tekhnika, Minsk, 1987. (Translation by Harwood Academic Publishers.)Google Scholar
  27. [27]
    K.S. Miller and B. Ross, An Introduction to the Fractional Differential Equations (John Wiley & Sons, NY, 1993).zbMATHGoogle Scholar
  28. [28]
    R. Ishizaki, T. Horito, T. Kobayashi, and M. Mori, Progr. Theor. Phys. 85, 1013 (1991).ADSCrossRefGoogle Scholar
  29. [29]
    B.V. Chirikov, D.L. Shepeliansky, Physica D 13, 394 (1984).ADSCrossRefGoogle Scholar
  30. [30]
    G.M. Zaslavsky and M.K. Tippett, Phys. Rev. Lett. 67 3251 (1991).ADSCrossRefGoogle Scholar
  31. [31]
    S. Benkadda, Y. Elskens, and B. Ragot, Phys. Rev. Lett. 72 2859 (1994).ADSCrossRefGoogle Scholar
  32. [32]
    G.M. Zaslavsky, Chaos 5 653 (1995).MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • George M. Zaslavsky
    • 1
    • 2
  1. 1.Department of PhysicsNew York UniversityNew YorkUSA
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

Personalised recommendations