Entropia de permutação espacial multivariada e sua aplicação para análise da complexidade de sistemas caóticos

Abstract

The analysis of time series that model nonlinear dynamic systems is a topic that has been widely explored in the scientific world, due to the wide range of applications of complexity measures that are used to characterize and study these systems. In particular, chaos is one of the behaviors that can be observed in nonlinear systems depending on the values that are determined for its parameters and the initial conditions established. Due to its great sensitivity to these conditions, its trajectory can diverge exponentially with small parameter changes. Thus, the identification of chaotic regimes and the quantification of the degree of chaoticity are essential to determine the limits within which it is possible to make predictions. To carry out this characterization, the concept of the Causal Complexity-Entropy Plan (CECP) was introduced, capable of detecting the underlying complexity of dynamic systems and extracting the structure of the time series according to the probability distribution function constructed from its ordinal patterns. The algorithm of this technique is simple and has the ability to distinguish between periodic, chaotic and stochastic signals. However, in this approach, the evaluation is performed on univariate time series, not guaranteeing an adequate characterization for multivariate systems, since the technique adopted disregards the existing correlation between the component series of the system. Furthermore, as it only considers the ordinal patterns intrinsic to the time series, this approach implies the loss of some details of amplitude information from the original series. In this work, a multivariate CECP approach is proposed for the characterization of dynamic multidimensional systems, through the Complexity-Entropy Multivariate Causal Plan (MvCECP). In addition, the probability distribution function is added to the identification of spatial subpatterns, making the approach sensitive to scale changes in the evaluated datasets. To ensure the effectiveness of the method, its ability to distinguish the chaotic, stochastic and periodic behavior of multivariate systems is demonstrated, being even more accurate than the univariate approach in the characterization of chaotic dynamics.