Effective scaling regime for computing the correlation dimension from chaotic time series

In the analysis of chaotic time series, a standard technique is to reconstruct an image of the original dynamical system using delay coordinates. If the original dynamical system has an attractor, then the correlation dimension D₂ of its image in the reconstruction can be estimated using the Grassberger-Procaccia algorithm. The quality of the reconstruction 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, more importantly, by the delay time τ. For a given embedding dimension and a finite time series, there exists a maximum allowed delay time beyond which the size of the scaling region is no longer reliably discernible. We derive 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 embedding dimension and the delay time in the reconstruction. As the embedding dimension is increased, it is necessary to reduce the delay time substantially to guarantee a reliable estimate of D₂. Our results imply that it is the delay time itself, rather than the total observation time (m - 1)τ, which plays the most critical role in the determination of the correlation dimension.