Discriminating chaotic and stochastic dynamics through the permutation spectrum test

Abstract

In this paper, we propose a new heuristic symbolic tool for unveiling chaotic and stochastic dynamics: the permutation spectrum test. Several numerical examples allow us to confirm the usefulness of the introduced methodology. Indeed, we show that it is robust in situations in which other techniques fail (intermittent chaos, hyperchaotic dynamics, stochastic linear and nonlinear correlated dynamics, and deterministic non-chaotic noise-driven dynamics). We illustrate the applicability and reliability of this pragmatic method by examining real complex time series from diverse scientific fields. Taking into account that the proposed test has the advantages of being conceptually simple and computationally fast, we think that it can be of practical utility as an alternative test for determinism. The importance of distinguishing between periodic, chaotic, and stochastic dynamics from time series analysis is well-recognized for understanding the mechanisms that govern the regarded complex systems. In this work, we have introduced a conceptually simple and computationally fast symbolic visual test for discriminating chaotic and stochastic dynamics, called the permutation spectrum test. Because the symbolization is made by implementing the Bandt and Pompe methodology, all the advantages associated with this natural encoding (simplicity, extremely fast calculation, robustness, and invariance with respect to monotonous transformations) are inherited by the permutation spectrum test. We have shown that this pragmatic approach is robust in situations in which other tests fail. We have also confirmed its practical utility by examining several experimental and natural time series.