Sampling period, statistical complexity, and chaotic attractors

We analyze the statistical complexity measure vs. entropy plane-representation of sampled chaotic attractors as a function of the sampling period τ and show that, if the Bandt and Pompe procedure is used to assign a probability distribution function (PDF) to the pertinent time series, the statistical complexity measure (SCM) attains a definite maximum for a specific sampling periodtM. On the contrary, the usual histogram approach for assigning PDFs to a time series leads to essentially constant SCM values for any sampling period τ. The significance of tM is further investigated by comparing it with typical times found in the literature for the two main reconstruction processes: the Takens’ one in a delay-time embedding, on one hand, and the exact Nyquist–Shannon reconstruction, on the other one. It is shown that tM is compatible with those times recommended as adequate delay ones in Takens’ reconstruction. The reported results correspond to three representative chaotic systems having correlation dimension 2<D2<3. One recent experiment confirms the analysis presented here.