Frameworks Application for Estimation of Lyapunov Exponents for Systems with Periodic Coeffi cients
https://doi.org/10.17587/mau.21.3-13
Abstract
The identification problem of Lyapunov exponents is considered for dynamic systems with periodic coefficients under uncertainty. Indexes identification is based on the analysis of a special class of frameworks describing dynamics of indexes change. The method of frameworks obtaining is described. The adequacy concept of obtained estimations Lyapunov exponents is introduced. The adequacy criterion is based on the analysis of the structure definition domain. The domain which belongs to the set of Lyapunov exponents estimates is determined. The method proposed for the order estimation of the system. The method is based on the properties analysis of almost periodic to Bohr functions and proposed frameworks. The case when lineals for Lyapunov exponents are crossed is considered. WE obtain to an infinite spectrum of Lyapunov exponents. Upper bound for the smallest index and mobility limit for the large index are obtained and the index set of the system is determined. The graphics criteria based on the analysis of framework properties are proposed for the adequacy estimation of obtained indexes. The histogram method is applied to check of estimations set. It is shown that a dynamic system with periodic coefficients can have a set of Lyapunov exponents. The extension of almost periodic functions on Bohr is proposed to the problem solve of Lyapunov exponents evaluation. The system order estimation is obtained on the basis of the framework property analysis.
References
1. Thamilmaran K., Senthilkumar D. V., Venkatesan A., Lakshmanan M. Experimental realization of strange nonchaotic attractors in a quasiperiodically forced electronic circuit, Physical Review E., 2006, vol. 74, no. 9, pp. 036205.
2. Porcher R., Thomas G. Estimating Lyapunov exponents in biomedical time series. Physical Review E., 2001, vol. 64, no. 1, pp. 010902(R).
3. Hołyst J. A., Urbanowicz K. Chaos control in economical model by time-delayed feedback method. Physica A: Statistical Mechanics and its Applications, 2000, vol. 287, no. 3—4, pp. 587—598.
4. Macek W. M., Redaelli S. Estimation of the entropy of the solar wind flow, Physical Review E., 2000, vol. 62, no. 5, pp. 6496—6504.
5. Skokos Ch. The Lyapunov Characteristic Exponents and Their Computation, Lect. Notes Phys., 2010, vol. 790, pp. 63—135.
6. Gencay R., Dechert W. D. An algorithm for the n Lyapunov exponents of an n-dimensional unknown dynamical system, Physica D., 1992, vol. 59, pp. 142—157.
7. Takens F. Detecting strange attractors in turbulence. Dynamical Systems and Turbulence. Lecture Notes in Mathematics, Eds D. A. Rand, L.-S. Young. Berlin, Springer-Verlag, 1980, vol. 898, pp. 366—381.
8. Wolf A., Swift J. B., Swinney H. L., Vastano J. A. Determining Lyapunov exponents from a time series, Physica 16D, 1985, no. 16, pp. 285—301.
9. Rosenstein M. T., Collins J. J., De Luca C. J. A practical method for calculating largest Lyapunov exponents from small data sets Source, Physica D., 1993, vol. 65, iss. 1—2, pp. 117—134.
10. Bespalov A. V., Polyakhov N. D. Comparative analysis of methods for estimating the first Lyapunov exponent, Modern problems of science and education, 2016, no. 6, 8 p. (in Russian).
11. Perederiy Y. A. Method for calculation of lyapunov exponents spectrum from data series, Izvestiya VUZ. Applied nonlinear dynamics, 2012, iss. 20, no. 1, pp. 99—104 (in Russian).
12. Moskalenko O., Koronovskii A. A., Hramov A. E. Lyapunov exponent corresponding to enslaved phase dynamics: Estimation from time series, Physical review E 92, 2015, pp. 012913.
13. Cvitanovi´c P., Artuso R., Mainieri R., Tanner G., Vattay G. Chaos: Classical and Quantum. ChaosBook.org version 16.0. 2017.
14. Filatov V. V. Structural characteristics of geophysical fields anomalies and their use in forecasting, Geophysics, 2013, no. 4(16), pp. 34—41.
15. Bylov B. F., Vinograd R. E., Grobman D. M., Nemytsky V. V. Theory of Lyapunov indexes and its application to stability problems, Moscow, Nauka, 1966 (in Russian).
16. Moskalenko O., Pavlov A. S. A method of evaluating zero conditional Lyapunov exponent time serie, Pis’ma v Zhurnal Tekhnicheskoi Fiziki, 2014, vol. 40, no. 12, pp. 66—72.
17. Karabutov N. Structural methods of estimation Lyapunov exponents linear dynamic system, International journal of intelligent systems and applications, 2015, vol. 7, no. 10, pp. 1—11.
18. Karabutov N. Frameworks in identification problems: Design and analysis, Moscow, URSS/Lenand, 2018, 306 p. (in Russian).
19. Karabutov N. About structural identifiability of nonlinear dynamic systems under uncertainty, Global journal of science frontier research: (A) Physics and Space Science, 2018, vol. 18, iss. 11, pp. 51—61.
20. Karabutov N. N. Structural identification of static objects: Fields, frameworks, methods, Moscow, URSS/Lenand, 2011, 152 p. (in Russian).
21. Demidovich B. P. Lektion on mathematical stability theory, Moscow, Nauka, 1967.
22. Karabutov N. About Lyapunov exponents identification for systems with periodic coefficients, International journal of intelligent systems and applications, 2018, vol. 10, no. 11, pp. 1—10.
Review
For citations:
Karabutov N.N. Frameworks Application for Estimation of Lyapunov Exponents for Systems with Periodic Coeffi cients. Mekhatronika, Avtomatizatsiya, Upravlenie. 2020;21(1):3-13. (In Russ.) https://doi.org/10.17587/mau.21.3-13
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