Computing the dynamical complexity

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.

C++ code for computing the dynamical complexity of a first-return 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.

C++ code for computing the structuratility

Style file to format the first-return map

Any question ? Please write me