Statistical Complexity of Sampled Chaotic Attractors

TL;DR

This paper investigates how the statistical complexity of sampled chaotic attractors varies with sampling period, revealing a maximum at a specific period when using Bandt and Pompe's method, and relates this to reconstruction times.

Contribution

It demonstrates the existence of a specific sampling period maximizing statistical complexity using Bandt and Pompe's method, linking it to known reconstruction times in chaos analysis.

Findings

Maximum statistical complexity occurs at a specific sampling period tM.

tM aligns with recommended delay times in Takens' reconstruction.

Results are validated on three chaotic systems and supported by experimental data.

Abstract

We analyze the statistical complexity vs. entropy plane-representation of sampled chaotic attractors as a function of the sampling period {\tau}. It is shown 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 period tM. If the usual histogram approach is used instead in order to assign the PDF to the time series, the SCM remains almost constant at any sampling period {\tau}. The significance of tM is further investigated by comparing it with typical times given in the literature for the two main reconstruction processes: the Takens' one in a delay-time embedding, and the exact Nyquist-Shannon reconstruction. 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.