Numerical and experimental study of the effects of noise on the permutation entropy

We analyze the effects of noise on the permutation entropy of dynamical systems. We take as numerical examples the logistic map and the Rössler system. Upon varying the noise strength we find a transition from an almost-deterministic regime, where the permutation entropy grows slower than linearly with the pattern dimension, to a noise-dominated regime, where the permutation entropy grows faster than linearly with the pattern dimension. We perform the same analysis on experimental time-series by considering the stochastic spiking output of a semiconductor laser with optical feedback, and find that the permutation entropy always increases faster than linearly. Nevertheless, the analysis allows to detect regularities of the underlying dynamics and model simulations are in a good agreement with the empirical data. By comparing the results of these three different examples, we discuss the possibility of determining from a time series whether the underlying dynamics is dominated by noise or not.