ESTIMATING CORRELATION DIMENSION FROM A CHAOTIC TIME-SERIES - WHEN DOES PLATEAU ONSET OCCUR

M Z DING, C GREBOGI, E OTT, T SAUER, J A YORKE

Research output: Contribution to journalArticle

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Abstract

Suppose that a dynamical system has a chaotic attractor A with a correlation dimension D-2. A common technique to probe the system is by measuring a single scalar function of the system state and reconstructing the dynamics in an m-dimensional space using the delay-coordinate technique. The estimated correlation dimension of the reconstructed attractor typically increases with m and reaches a plateau (on which the dimension estimate is relatively constant) for a range of large enough m values. The plateaued dimension value is then assumed to be an estimate of D-2 for the attractor in the original full phase space. In this paper we first present rigorous results which state that, for a long enough data string with low enough noise, the plateau onset occurs at m = Ceil(D-2), where Ceil(D-2), standing for ceiling of D-2, is the smallest integer greater than or equal to D-2. We then show numerical examples illustrating the theoretical prediction. In addition, we discuss new findings showing how practical factors such as a lack of data and observational noise can produce results that may seem to be inconsistent with the theoretically predicted plateau onset at m = Ceil(D-2).

Original languageEnglish
Pages (from-to)404-424
Number of pages21
JournalPhysica. D, Nonlinear Phenomena
Volume69
Issue number3-4
Publication statusPublished - 15 Dec 1993

Keywords

  • SMALL DATA SETS
  • STRANGE ATTRACTORS
  • SYSTEMS

Cite this

ESTIMATING CORRELATION DIMENSION FROM A CHAOTIC TIME-SERIES - WHEN DOES PLATEAU ONSET OCCUR. / DING, M Z ; GREBOGI, C ; OTT, E ; SAUER, T ; YORKE, J A .

In: Physica. D, Nonlinear Phenomena, Vol. 69, No. 3-4, 15.12.1993, p. 404-424.

Research output: Contribution to journalArticle

DING, M Z ; GREBOGI, C ; OTT, E ; SAUER, T ; YORKE, J A . / ESTIMATING CORRELATION DIMENSION FROM A CHAOTIC TIME-SERIES - WHEN DOES PLATEAU ONSET OCCUR. In: Physica. D, Nonlinear Phenomena. 1993 ; Vol. 69, No. 3-4. pp. 404-424.
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N2 - Suppose that a dynamical system has a chaotic attractor A with a correlation dimension D-2. A common technique to probe the system is by measuring a single scalar function of the system state and reconstructing the dynamics in an m-dimensional space using the delay-coordinate technique. The estimated correlation dimension of the reconstructed attractor typically increases with m and reaches a plateau (on which the dimension estimate is relatively constant) for a range of large enough m values. The plateaued dimension value is then assumed to be an estimate of D-2 for the attractor in the original full phase space. In this paper we first present rigorous results which state that, for a long enough data string with low enough noise, the plateau onset occurs at m = Ceil(D-2), where Ceil(D-2), standing for ceiling of D-2, is the smallest integer greater than or equal to D-2. We then show numerical examples illustrating the theoretical prediction. In addition, we discuss new findings showing how practical factors such as a lack of data and observational noise can produce results that may seem to be inconsistent with the theoretically predicted plateau onset at m = Ceil(D-2).

AB - Suppose that a dynamical system has a chaotic attractor A with a correlation dimension D-2. A common technique to probe the system is by measuring a single scalar function of the system state and reconstructing the dynamics in an m-dimensional space using the delay-coordinate technique. The estimated correlation dimension of the reconstructed attractor typically increases with m and reaches a plateau (on which the dimension estimate is relatively constant) for a range of large enough m values. The plateaued dimension value is then assumed to be an estimate of D-2 for the attractor in the original full phase space. In this paper we first present rigorous results which state that, for a long enough data string with low enough noise, the plateau onset occurs at m = Ceil(D-2), where Ceil(D-2), standing for ceiling of D-2, is the smallest integer greater than or equal to D-2. We then show numerical examples illustrating the theoretical prediction. In addition, we discuss new findings showing how practical factors such as a lack of data and observational noise can produce results that may seem to be inconsistent with the theoretically predicted plateau onset at m = Ceil(D-2).

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KW - STRANGE ATTRACTORS

KW - SYSTEMS

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JO - Physica. D, Nonlinear Phenomena

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