Toward an Optimal Strategy for Sampling Chaotic Time Series Using the Smale-Williams Dynamical System.

摘要

Chaotic dynamical systems such as the atmosphere can be characterized by their associated geometrical structures in phase space that are known as strange attractors. Time series data obtained from these dynamical systems must be sampled properly in order to estimate the fractal dimensions of these strange attractors with the greatest possible accuracy. The Smale-Williams attractor is used here because the value of the correlation dimension for this sttractor can be determined analytically. Two types of sampling strategies are considered to determine how a time series might be best used to estimate the characteristics of the attractor. This attractor can be characterized by either a histogram of distances from the origin to each point on the attractor or by an estimate of its correlation dimension. Differences between histograms of a sufficiently long control series and subsets of this series are used to quantify the accuracy of representations of the attractor. Differences between analytic and estimated values of the correlation dimension are also used. Finally, the independence of the time series data is quantified in a new way and the relationship between the independence of sampled data and the accuracy of characterizations of the attractor by histograms of sampled data or by estimates of the correlation dimension is examined.