Multifractality of the Feigenbaum Attractor and Fractional Derivatives View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2005-12

AUTHORS

U. Frisch, K. Khanin, T. Matsumoto

ABSTRACT

It is shown that fractional derivatives of the (integrated) invariant measure of the Feigenbaum map at the onset of chaos have power-law tails in their cumulative distributions, whose exponents can be related to the spectrum of singularities (f(α). This is a new way of characterizing multifractality in dynamical systems, so far applied only to multifractal random functions [Frisch and Matsumoto, J. Stat. Phys. 108:1181, 2002]. The relation between the thermodynamic approach [Vul, Sinai and Khanin, Russian Math. Surveys39:1, 1984] and that based on singularities of the invariant measures is also examined. The theory for fractional derivatives is developed from a heuristic point view and tested by very accurate simulations. More... »

PAGES

671-695

References to SciGraph publications

  • 1978-07. Quantitative universality for a class of nonlinear transformations in JOURNAL OF STATISTICAL PHYSICS
  • 2002-09. On Multifractality and Fractional Derivatives in JOURNAL OF STATISTICAL PHYSICS
  • 1987-03. Some characterizations of strange sets in JOURNAL OF STATISTICAL PHYSICS
  • 1987-03. Scaling spectra and return times of dynamical systems in JOURNAL OF STATISTICAL PHYSICS
  • 1973-12. Random difference equations and Renewal theory for products of random matrices in ACTA MATHEMATICA
  • 1987-06. The dimension spectrum of some dynamical systems in JOURNAL OF STATISTICAL PHYSICS
  • 1980-02. The transition to aperiodic behavior in turbulent systems in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s10955-005-7011-4

    DOI

    http://dx.doi.org/10.1007/s10955-005-7011-4

    DIMENSIONS

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    133 schema:name Dept. of Mathematics, Heriot-Watt University, EH14 4AS, Edinburgh, U.K
    134 Issac Newton Institute for Mathematical Sciences, 20 Clarkson Road, CB3 0EH, Cambridge, U.K
    135 Landau Institute for Theoretical Physics, Kosygina Str. 2, 117332, Moscow, Russia
    136 Observatoire de la Côte d’Azur, B.P. 4229, 06304, Nice Cedex 4, France
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    139 schema:name Observatoire de la Côte d’Azur, B.P. 4229, 06304, Nice Cedex 4, France
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