Multifractals: Formalism and Experiments

  • Mogens H. Jensen
Part of the NATO ASI Series book series (volume 167)


We review briefly the formalism for studying multifractal scaling properties. The scaling structure is conveniently described by means of an f-α spectrum. For the onset of chaos via quasiperiodicity and period doubling we obtain universal spectra. These spectra are compared with spectra obtained from a forced Rayleigh-Benard experiment and very good agreement is found between theory and experiment. Finally, we show that the experimental spectra can be inverted and give information about the underlying dynamical process.


Partition Function Period Doubling Legendre Transformation Infinite Slope Concentrate Part 
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.
    B.B. Mandelbrot, J.Fluid.Mech., 62: 331 (1974).ADSMATHCrossRefGoogle Scholar
  2. 2.
    U. Frisch and G. Parisi, “Varanna School LXXXXVIII”, M. Ghil, R. Benzi, and G. Parisi, eds., North-Holland, New York (1985), p.84; R. Benzi, G. Paladin, G. Parisi, and A. Vulpiani, J.Phys.A, 17:352 (1984).Google Scholar
  3. 3.
    T.C. Halsey, P. Meakin, and I. Procaccia, Phys.Rev.Lett. 56: 854 (1986).ADSCrossRefGoogle Scholar
  4. 4.
    T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia, and B.I. Shraiman, Phys.Rev.A, 33: 1141 (1986).MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    L. de Arcangelis, S. Redner, and A. Coniglio, Phys.Rev.B, 31: 4725 (1985).ADSGoogle Scholar
  6. 6.
    M.H. Jensen, L.P. Kadanoff, A. Libchaber, I. Procaccia, and J. Stavans, Phys.Rev.Lett., 55: 2798 (1985).ADSCrossRefGoogle Scholar
  7. 7.
    M.J. Feigenbaum, J.Stat.Phys. (to be published).Google Scholar
  8. 8.
    M.E. Cates and T.A. Witten, Phys.Rev.Lett., 56: 2497 (1986).ADSCrossRefGoogle Scholar
  9. 9.
    R. Blumenfeld, Y. Meir, A.B. Harris, and A. Aharony, J.Phys.A, 19: L791 (1986).ADSCrossRefGoogle Scholar
  10. 10.
    T. Bohr and D. Rand, Physica, 25D: 387 (1987).MathSciNetMATHGoogle Scholar
  11. 11.
    L. Pietronero and A.P. Siebesma, Phys.Rev.Lett., 57: 1098 (1986).ADSCrossRefGoogle Scholar
  12. 12.
    R. Badii and A. Politi, Phys.Rev.Lett., 52: 1661 (1984)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    C. Amitrano, A. Coniglio, and F. di Liberto, Phys.Rev.Lett., 57: 1016 (1986).ADSCrossRefGoogle Scholar
  14. 14.
    A. Renyi, “Probability Theory”, North-Holland, Amsterdam (1970).Google Scholar
  15. 15.
    H.G.E. Hentschel and I. Procaccia, Physica, 8D:435 (1983); see also P. Grassberger, Phys.Lett., 107A: 101 (1985).CrossRefGoogle Scholar
  16. 16.
    J.A. Glazier, M.H. Jensen, A. Libchaber, and J. Stavans, Phys.Rev.A, 34: 1621 (1986).ADSCrossRefGoogle Scholar
  17. 17.
    M.J. Feigenbaum, J.Stat.Phys., 19: 25 (1978);MathSciNetADSMATHCrossRefGoogle Scholar
  18. M.J. Feigenbaum, J.Stat.Phys. 21: 669 (1979).MathSciNetADSMATHCrossRefGoogle Scholar
  19. 18.
    M.J. Feigenbaum, Comm.Math.Phys., 77: 65 (1980).MathSciNetADSMATHCrossRefGoogle Scholar
  20. 19.
    D. Bensimon, M.H. Jensen, and L.P. Kadanoff, Phys.Rev.A, 33: 3622 (1986).ADSCrossRefGoogle Scholar
  21. 20.
    E. Aurell, Phys.Rev.A, 34: 5135 (1986).ADSCrossRefGoogle Scholar
  22. 21.
    Scott J. Shenker, Physica, 5D: 405 (1982);MathSciNetGoogle Scholar
  23. M.J. Feigenbaum, L.P. Kadanoff, and Scott J. Shenker, Physica, 5D: 370 (1982);Google Scholar
  24. S. Ost-lund, D. Rand, J.P. Sethna, and E.D. Siggia, Physica, 8D: 303 (1983).MathSciNetMATHGoogle Scholar
  25. 22.
    V.I. Arnold, Am.Math.Soc.Trans., Ser. 2, 46: 213 (1965).Google Scholar
  26. 23.
    M.H. Jensen, P. Bak, and T. Bohr, Phys.Rev.Lett., 50: 1637 (1983);ADSCrossRefGoogle Scholar
  27. M.H. Jensen, P. Bak, and T. Bohr, Phys.Rev.A, 30: 1960 (1984).MathSciNetADSCrossRefGoogle Scholar
  28. 24.
    A. Libchaber, C. Laroche, and S. Fauve, Physica, 7D: 73 (1983);Google Scholar
  29. A. Libchaber, C. Laroche, and S. Fauve, J.Phys (Paris) Lett., 43: L211 (1982).CrossRefGoogle Scholar
  30. 25.
    J. Stavans, F. Heslot, and A. Libchaber, Phys.Rev.Lett., 55: 596 (1985).ADSCrossRefGoogle Scholar
  31. 26.
    E.G. Gwinn and R.M. Westervelt, Phys.Rev.Lett.Google Scholar
  32. 27.
    D. Ruelle, “Statistical Mechanics, Thermodynamic Formalism”, Addison-Wesley, Reading (1978).Google Scholar
  33. 28.
    E.B. Vul, Ya.G. Sinai, and K.M. Khanin, Usp.Mat.Nauk,39:3 (1984) [Russ.Mat.Surveys,39:1 (1984)].Google Scholar
  34. 29.
    M.J. Feigenbaum, M.H. Jensen, and I. Procaccia, Phys.Rev.Lett., 56: 1503 (1986).MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1987

Authors and Affiliations

  • Mogens H. Jensen
    • 1
  1. 1.NorditaCopenhagenDenmark

Personalised recommendations