Optimizing Time Delays for Reconstructing Attractors of Dynamical Systems from Time Series Signals

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Identifying deterministic chaos and its quantitative characterization are very important from the viewpoint of time series analysis based on nonlinear dynamical systems theory. For quantitative characterization of deterministic chaos, there are many statistics, for example, the fractal dimensions, the Lyapunov exponents, the metric entropies and so on.In order to estimate these nonlinear statistics, the first step is the reconstructior of attractors from observed time series'. In order to accomplish the above issue, the embedding theorems by Takens (1981) and Embedology theory by Sauer et al.(1991) are lmportant from theoretical justification.For practical reconstruction of dynamical systems from an observed single variable time series, the theory of embedding proved that the transformation by the time delay coordinates is an embedding if the dimension of reconstructing attractors is at least larger than twice of the box counting dimension of an underlying dynamical system. Although the time delay value in this reconstrution is an arbitrary value under the theoretical situation, in the case of analyzing real time series, which is corrupted by noise and whose length and measurement precision are finite, it is important to set an appropriate value, particularly for continuous data.We propose a novel criterion of deciding time delays for reconstructing attractors from a single variable time series. Our criterion considers higher order correlation functions, and finds the convergence values of the extrema. In order to see how our method works well, we analyze two numerical examples by observing the shapes of reconstructed attractors and calculating phase space continuities.Our method can be a good criterion to set appropriate time delays on reconstructing attractors as a result.