Abstract
The aim of the paper was to analyze the given nonlinear problem by different methods of computation of the Lyapunov exponents (Wolf method, Rosenstein method, Kantz method, the method based on the modification of a neural network, and the synchronization method) for the classical problems governed by difference and differential equations (Hénon map, hyperchaotic Hénon map, logistic map, Rössler attractor, Lorenz attractor) and with the use of both Fourier spectra and Gauss wavelets. It has been shown that a modification of the neural network method makes it possible to compute a spectrum of Lyapunov exponents, and then to detect a transition of the system regular dynamics into chaos, hyperchaos, and others. The aim of the comparison was to evaluate the considered algorithms, study their convergence, and also identify the most suitable algorithms for specific system types and objectives. Moreover, an algorithm of calculation of the spectrum of Lyapunov exponents based on a trained neural network has been proposed. It has been proven that the developed method yields good results for different types of systems and does not require a priori knowledge of the system equations.
Highlights
The first part of the present work is focused on the numerical investigation of classical dynamical systems to estimate velocity of divergence of the neighborhood trajectories with the help of a measure coupled with the Kolmogorov entropy [1]
We began with the numerical investigation of the Kolmogorov entropy of the Hénon-Heiles model
We have proposed a novel counterpart method to compute Lyapunov exponents (LEs) based on a modification of the neural network method
Summary
The first part of the present work is focused on the numerical investigation of classical dynamical systems to estimate velocity of divergence of the neighborhood trajectories with the help of a measure coupled with the Kolmogorov entropy [1] (or metrics). The method proposed by Wolf [1] is most widely used to verify and study chaotic dynamics. The Rosenstein [4] and Kantz [5] methods are often employed to estimate the largest Lyapunov exponents. The state-of-the-art of papers devoted to the theoretical background of the Lyapunov exponents and methods of their computations has been carried out by Awrejcewicz et al [6]. The method of the choice of an embedding dimension has been described. The method of the correlating dimension, the false nearest neighbor method and the Entropy 2018, 20, 175; doi:10.3390/e20030175 www.mdpi.com/journal/entropy
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