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The Theory of Chaotic Attractors pp 170–189Cite as

Measuring the Strangeness of Strange Attractors

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Abstract

We study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise. The exponent v is closely related to the fractal dimension and the information dimension, but its computation is considerably easier. Its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed. Algorithms for extracting v from the time series of a single variable are proposed. The relations between the various measures of strange attractors and between them and the Lyapunov exponents are discussed. It is shown that the conjecture of Kaplan and Yorke for the dimension gives an upper bound for v. Various examples of finite and infinite dimensional systems are treated, both numerically and analytically.

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Author information

Authors and Affiliations

  1. Department of Chemical Physics, Weizmann Institute of Science, Rehovot, 76100, Israel

    Peter Grassberger & Itamar Procaccia

  2. Department of Physics, University of Wuppertal, W. Germany

    Peter Grassberger

Authors
  1. Peter Grassberger
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  2. Itamar Procaccia
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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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Grassberger, P., Procaccia, I. (2004). Measuring the Strangeness of Strange 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_12

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  • DOI: https://doi.org/10.1007/978-0-387-21830-4_12

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4419-2330-1

  • Online ISBN: 978-0-387-21830-4

  • eBook Packages: Springer Book Archive

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