### 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 language | English |
---|---|

Pages (from-to) | 404-424 |

Number of pages | 21 |

Journal | Physica. D, Nonlinear Phenomena |

Volume | 69 |

Issue number | 3-4 |

Publication status | Published - 15 Dec 1993 |

### Keywords

- SMALL DATA SETS
- STRANGE ATTRACTORS
- SYSTEMS

### Cite this

*Physica. D, Nonlinear Phenomena*,

*69*(3-4), 404-424.

**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 .

Research output: Contribution to journal › Article

*Physica. D, Nonlinear Phenomena*, vol. 69, no. 3-4, pp. 404-424.

}

TY - JOUR

T1 - ESTIMATING CORRELATION DIMENSION FROM A CHAOTIC TIME-SERIES - WHEN DOES PLATEAU ONSET OCCUR

AU - DING, M Z

AU - GREBOGI, C

AU - OTT, E

AU - SAUER, T

AU - YORKE, J A

PY - 1993/12/15

Y1 - 1993/12/15

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).

KW - SMALL DATA SETS

KW - STRANGE ATTRACTORS

KW - SYSTEMS

M3 - Article

VL - 69

SP - 404

EP - 424

JO - Physica. D, Nonlinear Phenomena

JF - Physica. D, Nonlinear Phenomena

SN - 0167-2789

IS - 3-4

ER -