{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T22:16:29Z","timestamp":1778278589157,"version":"3.51.4"},"reference-count":35,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2018,3,6]],"date-time":"2018-03-06T00:00:00Z","timestamp":1520294400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002674","name":"Russian Academy of Sciences","doi-asserted-by":"publisher","award":["RSF 16-19-10290"],"award-info":[{"award-number":["RSF 16-19-10290"]}],"id":[{"id":"10.13039\/501100002674","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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 (H\u00e9non map, hyperchaotic H\u00e9non map, logistic map, R\u00f6ssler 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.<\/jats:p>","DOI":"10.3390\/e20030175","type":"journal-article","created":{"date-parts":[[2018,3,6]],"date-time":"2018-03-06T12:16:27Z","timestamp":1520338587000},"page":"175","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":42,"title":["Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0387-921X","authenticated-orcid":false,"given":"Jan","family":"Awrejcewicz","sequence":"first","affiliation":[{"name":"Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1\/15 Stefanowski St., 90-924 Lodz, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9389-5602","authenticated-orcid":false,"given":"Anton V.","family":"Krysko","sequence":"additional","affiliation":[{"name":"Cybernetic Institute, National Research Tomsk Polytechnic University, 30 Lenin Avenue, 634050 Tomsk, Russia"},{"name":"Department of Applied Mathematics and Systems Analysis, Saratov State Technical University, 77 Politechnicheskaya, 410054 Saratov, Russia"}]},{"given":"Nikolay P.","family":"Erofeev","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Modeling, Saratov State Technical University, 77 Politechnicheskaya, 410054 Saratov, Russia"}]},{"given":"Vitalyj","family":"Dobriyan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Modeling, Saratov State Technical University, 77 Politechnicheskaya, 410054 Saratov, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3867-648X","authenticated-orcid":false,"given":"Marina A.","family":"Barulina","sequence":"additional","affiliation":[{"name":"Precision Mechanics and Control Institute, Russian Academy of Science, 24 Rabochaya Str., 410028 Saratov, Russia"}]},{"given":"Vadim A.","family":"Krysko","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Modeling, Saratov State Technical University, 77 Politechnicheskaya, 410054 Saratov, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2018,3,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/0167-2789(85)90011-9","article-title":"Determining Lyapunov exponents from a time series","volume":"16","author":"Wolf","year":"1985","journal-title":"Phys. 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