EFFECTIVE SCALING REGIME FOR COMPUTING THE CORRELATION DIMENSION FROMCHAOTIC TIME-SERIES

In the analysis of chaotic time series, a standard technique is to rec onstruct an image of the original dynamical system using delay coordin ates. If the original dynamical system has an attractor then the con e lation dimension D-2 of its image in the reconstruction can be estimat ed using the Grassberger-Procaccia algorithm. The quality of the recon struction can be probed by measuring the length of the linear scaling region used in this estimation, In this paper we show that the quality is constrained by both the embedding dimension m and, mon importantly , by the delay time tau. For a given embedding dimension and a finite time series, there exists a maximum allowed delay time beyond which th e size of the scaling region is no longer reliably discernible. We der ive an upper bound for this maximum delay time. Numerical experiments on several model chaotic time series support the theoretical argument. They also clearly indicate the different roles played by the embeddin g dimension and the delay time in the reconstruction. As the embedding dimension is increased, it is necessary to reduce the delay time subs tantially to guarantee a reliable estimate of D-2. Our results imply t hat it is the delay time itself, rather than the total observation tim e (m - 1)tau, which plays the most critical role in the determination of the correlation dimension. Copyright (C) 1998 Elsevier Science B.V.