Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems

Abstract

The first part of the paper was aimed at analyzing the given nonlinear problem by different methods of computation of the Lyapunov exponents (Wolf method [1], Rosenstein method [2], Kantz method [3], method based on the modification of a neural network [4, 5], and the synchronization method [6, 7]) for the classical problems governed by difference and differential equations (Hénon map [8], hyper-chaotic Hénon map [9], logistic map [10], Rössler attractor [11], Lorenz attractor [12]) and with the use of both Fourier spectra and Gauss wavelets [13]. It was shown that a modification of the neural network method [4, 5] makes it possible to compute a spectrum of Lyapunov exponents, and then to detect a transition of the system regular dynamics into chaos, hyper-chaos, hyper hyper-chaos and deep chaos [14-16]. Different algorithms for computation of Lyapunov exponents were validated by comparison with the known dynamical systems spectra of the Lyapunov exponents. The carried out analysis helps comparatively estimate the employed methods in order to choose the most suitable/optimal one to study different kinds of dynamical systems and different classes of problems in both this and the next paper parts.

Synchronization Measure Based on a Geometric Approach to Attractor Embedding Using Finite Observation Windows

“…Note that the maximal Lyapunov exponents computed directly from time series X 1 and X 2 are underestimated if compared to the maximal exponents for the coupled systems in (26). This effect is well known for synchronized differential equations [38]. Nevertheless, both methods demonstrate that the coupled systems become less chaotic when the coupling parameter is increased.…”

EEG Analysis in Structural Focal Epilepsy Using the Methods of Nonlinear Dynamics (Lyapunov Exponents, Lempel–Ziv Complexity, and Multiscale Entropy)

“…e values of the highest Lyapunov exponent calculated by the Rosenstein method, the Wolf method, and the Sano-Sawada method for classical problems are in good agreement with each other. Along with the indicated methods for calculating Lyapunov exponents, there is the Kantz method [9] and the modified neural network method [36]. It is worth noting that the calculation of the Lyapunov exponent spectrum using the modification of neural networks takes a longer time compared to the Sano-Sawada method.…”

Surrogate model approach for investigating the stability of a friction-induced oscillator of Duffing’s type

“…In recent years several numerical strategies have been suggested for estimating LEs in case of non-smooth dynamic systems. A thorough review can be found in [20]. Recently an approximate numerical method based on the estimation of the Jacobian matrix by perturbation method of the initial orthogonal vectors involving an Euler forward scheme has been proposed [14].…”