### 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 |
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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.