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The Dimension of Chaotic Attractors

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The Theory of Chaotic Attractors

Abstract

Dimension is perhaps the most basic property of an attractor. In this paper we discuss a variety of different definitions of dimension, compute their values for a typical example, and review previous work on the dimension of chaotic attractors. The relevant definitions of dimension are of two general types, those that depend only on metric properties, and those that depend on the frequency with which a typical trajectory visits different regions of the attractor. Both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value, which we call the “dimension of the natural measure”, and all of the metric dimensions take on a common value, which we call the “fractal dimension”. Furthermore, the dimension of the natural measure is typically equal to the Lyapunov dimension, which is defined in terms of Lyapunov numbers, and thus is usually far easier to calculate than any other definition. Because it is computable and more physically relevant, we feel that the dimension of the natural measure is more important than the fractal dimension.

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Author information

Authors and Affiliations

  1. Center for Nonlinear Studies and Theoretical Division, MS B258, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA

    J. Doyne Farmer

  2. Laboratory of Plasma and Fusion Energy Studies, University of Maryland, College Park, Maryland, USA

    Edward Ott

  3. Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, Maryland, USA

    James A. Yorke

Authors
  1. J. Doyne Farmer
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  2. Edward Ott
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  3. James A. Yorke
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Editor information

Editors and Affiliations

  1. Institute for Physical Science and Technology, University of Maryland, 20742, College Park, MD, USA

    Brian R. Hunt

  2. Department of Mathematics, Michigan State University, 48824, East Lansing, Michigan, USA

    Tien-Yien Li

  3. Department of Mathematics, University of Delaware, 19716, Newark, Delaware, USA

    Judy A. Kennedy

  4. Department of Econometrics, University of Groningen, NL-9700 AV, Groningen, The Netherlands

    Helena E. Nusse

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© 1983 Springer Science+Business Media New York

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Farmer, J.D., Ott, E., Yorke, J.A. (1983). The Dimension of Chaotic Attractors. In: Hunt, B.R., Li, TY., Kennedy, J.A., Nusse, H.E. (eds) The Theory of Chaotic Attractors. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21830-4_11

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  • DOI: https://doi.org/10.1007/978-0-387-21830-4_11

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4419-2330-1

  • Online ISBN: 978-0-387-21830-4

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