Muitifractals in Convection and Aggregation View Full Text


Ontology type: schema:Chapter     


Chapter Info

DATE

1988

AUTHORS

Mogens H. Jensen

ABSTRACT

Patterns found in convection and aggregation are often multifractals and can be characterized by a continuum of exponents. This differs from critical phenomena where typically a finite number of relevant exponents is needed. The set of exponents for a multifractal is conveniently presented as an f(α) spectrum, i.e., a spectrum of pointwise dimensions. We shall illustrate the formalism by a Rayleigh-Benard convection experiment developed by Libchaber and co-workers. The convec-tive state exhibits an unstable mode. This mode is coupled to an external oscillation and it is possible to drive the system towards the onset of chaos via quasiperiod-icity, by tuning the ratio between the two frequencies to an irrational number. At the critical point an attractor is extracted from a stroboscopic temperature signal and the corresponding f(α) spectrum is computed. A similar scenario is found in the theory of dynamical systems with two characteristic frequencies. The simplest such systems are discrete mappings expressed in terms of angular variables, circle maps. For circle maps one can tune the ratio of the frequencies to the same irrational value as in the convection experiment. At the critical point where chaos sets in, an attractor is extracted. The associated f(α) spectrum is calculated and a renormalization group analysis establishes it to be universal. This is a theory of no adjustable parameters which can be compared to the experiment. Good agreement is found providing a strong evidence that the convection experiment and circle maps belong to the same universality class. More... »

PAGES

292-309

References to SciGraph publications

  • 1987-03. Scaling spectra and return times of dynamical systems in JOURNAL OF STATISTICAL PHYSICS
  • 1987-03. Some characterizations of strange sets in JOURNAL OF STATISTICAL PHYSICS
  • 1975. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms in NONE
  • 1978-07. Quantitative universality for a class of nonlinear transformations in JOURNAL OF STATISTICAL PHYSICS
  • 1986-05. Renormalization group analysis of the global properties of a strange attractor in JOURNAL OF STATISTICAL PHYSICS
  • 1979-12. The universal metric properties of nonlinear transformations in JOURNAL OF STATISTICAL PHYSICS
  • 1987. Dynamics and Structure of Viscous Fingers in Porous Media in TIME-DEPENDENT EFFECTS IN DISORDERED MATERIALS
  • Book

    TITLE

    Random Fluctuations and Pattern Growth: Experiments and Models

    ISBN

    978-0-7923-0073-1
    978-94-009-2653-0

    Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/978-94-009-2653-0_41

    DOI

    http://dx.doi.org/10.1007/978-94-009-2653-0_41

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1041210152


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    40 schema:description Patterns found in convection and aggregation are often multifractals and can be characterized by a continuum of exponents. This differs from critical phenomena where typically a finite number of relevant exponents is needed. The set of exponents for a multifractal is conveniently presented as an f(α) spectrum, i.e., a spectrum of pointwise dimensions. We shall illustrate the formalism by a Rayleigh-Benard convection experiment developed by Libchaber and co-workers. The convec-tive state exhibits an unstable mode. This mode is coupled to an external oscillation and it is possible to drive the system towards the onset of chaos via quasiperiod-icity, by tuning the ratio between the two frequencies to an irrational number. At the critical point an attractor is extracted from a stroboscopic temperature signal and the corresponding f(α) spectrum is computed. A similar scenario is found in the theory of dynamical systems with two characteristic frequencies. The simplest such systems are discrete mappings expressed in terms of angular variables, circle maps. For circle maps one can tune the ratio of the frequencies to the same irrational value as in the convection experiment. At the critical point where chaos sets in, an attractor is extracted. The associated f(α) spectrum is calculated and a renormalization group analysis establishes it to be universal. This is a theory of no adjustable parameters which can be compared to the experiment. Good agreement is found providing a strong evidence that the convection experiment and circle maps belong to the same universality class.
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