Quantifying chaos by various computational methods. Part 1: Simple systems

Jan Awrejcewicz, Anton V. Krysko, Nikolay P. Erofeev, Vitalyj Dobriyan, Marina A. Barulina, Vadim A. Krysko

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10 Citations (Scopus)

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.

Original languageEnglish
Article number175
JournalEntropy
Volume20
Issue number3
DOIs
Publication statusPublished - 1 Mar 2018

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Keywords

  • Benettin method
  • Fourier spectrum
  • Gauss wavelets
  • Kantz method
  • Lyapunov exponents
  • Method of synchronization
  • Neural network method
  • Rosenstein method
  • Wolf method

ASJC Scopus subject areas

  • Physics and Astronomy(all)

Cite this

Awrejcewicz, J., Krysko, A. V., Erofeev, N. P., Dobriyan, V., Barulina, M. A., & Krysko, V. A. (2018). Quantifying chaos by various computational methods. Part 1: Simple systems. Entropy, 20(3), [175]. https://doi.org/10.3390/e20030175