Christophe LETELLIER
20/04/2023

With Inmaculada Leyva and Irene Sendina-Nadal, we proposed a metric to characterize the complex behavior of a dynamical system and to distinguish between organized and disorganized complexity. [1] The approach combines two quantities that separately assess the degree of unpredictability of the dynamics and the lack of describability of the structure in the Poincaré plane constructed from a given time series. For the former, we use the permutation entropy *S*_{p}, while for the later, we introduce an indicator, the structurality ∆, which accounts for the fraction of visited points in the Poincaré plane. The complexity measure thus defined as the sum of those two components is validated by classifying in the (*S*_{p},∆) space the complexity of several benchmark dissipative and conservative dynamical systems. As an application, we show how the metric can be used as a powerful biomarker for different cardiac pathologies and to distinguish the dynamical complexity of two electrochemical dissolutions.

Here is a code computing the dynamical complexity from a data file (two columns) for a first-return map made of N = 50,000 pts (this parameter can be easily adjusted). It returns the permutation entropy, the structurality and the dynamical complexity for that map.

Find below a code for computing the structurality from a first-return map to a Poincaré section of the Rössler system. This code call XmGrace (available under common Linux distribution). There is a style file (rosmap.par) which is also provided below.

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[1] **C. Letellier, I. Leyva & I. Sendiña-Nada**l
Dynamical complexity measure to distinguish organized from disorganized dynamics
*Physical Review E*, **101**, 022204, 2020. Online