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Quantifying Chaos by Various Computational Methods. Part 2: Vibrations of the Bernoulli–Euler Beam Subjected to Periodic and Colored Noise
Article

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

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Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowski St., 90-924 Lodz, Poland
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Cybernetic Institute, National Research Tomsk Polytechnic University, 30 Lenin Avenue, 634050 Tomsk, Russia
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Department of Applied Mathematics and Systems Analysis, Saratov State Technical University, 77 Politechnicheskaya, 410054 Saratov, Russia
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Department of Mathematics and Modeling, Saratov State Technical University, 77 Politechnicheskaya, 410054 Saratov, Russia
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Precision Mechanics and Control Institute, Russian Academy of Science, 24 Rabochaya Str., 410028 Saratov, Russia
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Author to whom correspondence should be addressed.
Entropy 2018, 20(3), 175; https://doi.org/10.3390/e20030175
Received: 17 January 2018 / Revised: 16 February 2018 / Accepted: 1 March 2018 / Published: 6 March 2018
(This article belongs to the Special Issue Entropy in Dynamic Systems)
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. View Full-Text
Keywords: Lyapunov exponents; Wolf method; Rosenstein method; Kantz method; neural network method; method of synchronization; Benettin method; Fourier spectrum; Gauss wavelets Lyapunov exponents; Wolf method; Rosenstein method; Kantz method; neural network method; method of synchronization; Benettin method; Fourier spectrum; Gauss wavelets
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MDPI and ACS Style

Awrejcewicz, J.; Krysko, A.V.; Erofeev, N.P.; Dobriyan, V.; Barulina, M.A.; Krysko, V.A. Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems. Entropy 2018, 20, 175. https://doi.org/10.3390/e20030175

AMA Style

Awrejcewicz J, Krysko AV, Erofeev NP, Dobriyan V, Barulina MA, Krysko VA. Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems. Entropy. 2018; 20(3):175. https://doi.org/10.3390/e20030175

Chicago/Turabian Style

Awrejcewicz, Jan; Krysko, Anton V.; Erofeev, Nikolay P.; Dobriyan, Vitalyj; Barulina, Marina A.; Krysko, Vadim A. 2018. "Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems" Entropy 20, no. 3: 175. https://doi.org/10.3390/e20030175

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