Método de Rosenstein para a estimativa do LLE em sistemas dinâmicos

Abstract

Lyapunov exponents play a central role in the characterization of dynamical regimes, and are widely used to identify and quantify chaos in complex systems. Among these, the largest Lyapunov exponent is particularly relevant, since it indicates the average rate of change of nearby trajectories in phase space. In this context, the search for efficient and accurate methods to estimate this parameter has become an area of intense study in the theory of dynamical systems. A milestone in this field was the development of the Wolf algorithm, which demonstrated high accuracy in estimating the largest Lyapunov exponent. However, its practical application is challenging due to the high computational cost and complexity of its implementation. To overcome these difficulties, Sato and, later, Rosenstein proposed an alternative approach, based on the supervision of the phase space by means of delayed coordinates. This method, which preserves the properties of the original system by canonical transforms, allows the estimation of the largest Lyapunov exponent directly from time series. This work aims to present a detailed description of the delayed coordinates method, highlighting its main features and advantages. Furthermore, a comprehensive evaluation of the accuracy and applicability of this technique in different contexts is carried out. This is expected not only to contribute to the methodological advancement in the estimation of the largest Lyapunov exponent, but also to provide more accessible tools for the analysis of complex dynamical systems in several areas, such as physics, biology and engineering.