Exploiting the impact of ordering patterns in the Fisher-Shannon complexity plane

Abstract

The Fisher-Shannon complexity plane is a powerful tool to characterize complex dynamics. It locates, on a two-dimensional plane, a dynamical system based upon its entropy and its Fisher Information Measure (FIM). It has been recently shown that by using ordinal patterns to compute permutation entropy and FIM, this plane unveils inner details of the structure underlying the complex and chaotic dynamics of a system, not easily exposed by other methods. The entropy of the dynamics is invariant to the way the patterns are ordered, but the way FIM is defined is sensitive to the order in which we classify the ordinal patterns. In this paper we analyze in detail the impact that the sorting protocol used to calculate FIM has on the structure unveiled by the Fisher-Shannon plane. We show the importance of a suitable choice, which can lead to saving computational resources, but also to disclose details of the dynamics not accessible to other sorting protocols. Our results agree with previous research, and common characteristic fingerprints are found for the different chaotic maps studied. Our analysis also reveals the fractal behaviour of the chaotic maps studied. We extract some underlying symmetries that allow us to simulate the behaviour on the complexity plane for a wide range of the control parameters in the chaotic regimes.