Kolmogorov-Smirnov Test Distinguishes Attractors with Similar Dimensions

Recent advances in nonlinear dynamics have led to more informative characterizations of complex signals making it possible to probe correlations in data to which traditional linear statistical and spectral analyses were not sensitive. Many of these new tools require detailed knowledge of small scale structures of the attractor; knowledge that can be acquired only from relatively large amounts of precise data that are not contaminated by noise-not the kind of data one usually obtains from experiments. There is a need for tools that can take advantage of \u27\u27coarse-grained\u27\u27 information, but which nevertheless remain sensitive to higher-order correlations in the data. We propose that the correlation integral, now much used as an intermediate step in the calculation of dimensions and entropies, can be used as such a tool and that the Kolmogorov-Smirnov test is a convenient and reliable way of comparing correlation integrals quantitatively. This procedure makes it possible to distinguish between attractors with similar dimensions. For example, it can unambiguously distinguish (p \u3c 10(-8)) the Lorenz, Rossler, and Mackey-Glass (delay = 17) attractors whose correlation dimensions are within 1% of each other. We also show that the Kolmogorov-Smirnov test is a convenient way of comparing a data set with its surrogates