Characterization of chaotic maps using the permutation Bandt-Pompe probability distribution

Abstract

By appealing to a long list of different nonlinear maps we review the characterization of time series arising from chaotic maps. The main tool for this characterization is the permutation Bandt-Pompe probability distribution function. We focus attention on both local and global characteristics of the components of this probability distribution function. We show that forbidden ordinal patterns (local quantifiers) exhibit an exponential growth for pattern-length range 3 ≤ D ≤ 8, in the case of finite time series data. Indeed, there is a minimum D; min-value such that forbidden patterns cannot appear for D < Dmin. The system’s localization in an entropy-complexity plane (global quantifier) displays typical specific features associated with its dynamics’ nature. We conclude that a more “robust” distinction between deterministic and stochastic dynamics is achieved via the present time series’ treatment based on the global characteristics of the permutation Bandt-Pompe probability distribution function.