Quantifying Chaos by Various Computational Methods. Part 1: Simple 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 b...
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| Format: | Article |
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MDPI AG
2018-03-01
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| Series: | Entropy |
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| Online Access: | http://www.mdpi.com/1099-4300/20/3/175 |
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| author | Jan Awrejcewicz Anton V. Krysko Nikolay P. Erofeev Vitalyj Dobriyan Marina A. Barulina Vadim A. Krysko |
| author_facet | Jan Awrejcewicz Anton V. Krysko Nikolay P. Erofeev Vitalyj Dobriyan Marina A. Barulina Vadim A. Krysko |
| author_sort | Jan Awrejcewicz |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-10T08:05:52Z |
| format | Article |
| id | doaj.art-c54f64bb63fb4568a53e2110ba5346c2 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-12-10T08:05:52Z |
| publishDate | 2018-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-c54f64bb63fb4568a53e2110ba5346c22022-12-22T01:56:41ZengMDPI AGEntropy1099-43002018-03-0120317510.3390/e20030175e20030175Quantifying Chaos by Various Computational Methods. Part 1: Simple SystemsJan Awrejcewicz0Anton V. Krysko1Nikolay P. Erofeev2Vitalyj Dobriyan3Marina A. Barulina4Vadim A. Krysko5Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowski St., 90-924 Lodz, PolandCybernetic Institute, National Research Tomsk Polytechnic University, 30 Lenin Avenue, 634050 Tomsk, RussiaDepartment of Mathematics and Modeling, Saratov State Technical University, 77 Politechnicheskaya, 410054 Saratov, RussiaDepartment of Mathematics and Modeling, Saratov State Technical University, 77 Politechnicheskaya, 410054 Saratov, RussiaPrecision Mechanics and Control Institute, Russian Academy of Science, 24 Rabochaya Str., 410028 Saratov, RussiaDepartment of Mathematics and Modeling, Saratov State Technical University, 77 Politechnicheskaya, 410054 Saratov, RussiaThe 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.http://www.mdpi.com/1099-4300/20/3/175Lyapunov exponentsWolf methodRosenstein methodKantz methodneural network methodmethod of synchronizationBenettin methodFourier spectrumGauss wavelets |
| spellingShingle | Jan Awrejcewicz Anton V. Krysko Nikolay P. Erofeev Vitalyj Dobriyan Marina A. Barulina Vadim A. Krysko Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems Entropy Lyapunov exponents Wolf method Rosenstein method Kantz method neural network method method of synchronization Benettin method Fourier spectrum Gauss wavelets |
| title | Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems |
| title_full | Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems |
| title_fullStr | Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems |
| title_full_unstemmed | Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems |
| title_short | Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems |
| title_sort | quantifying chaos by various computational methods part 1 simple systems |
| topic | Lyapunov exponents Wolf method Rosenstein method Kantz method neural network method method of synchronization Benettin method Fourier spectrum Gauss wavelets |
| url | http://www.mdpi.com/1099-4300/20/3/175 |
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