The existence of chaotic regimes of the fractional analogue of the Duffing-type oscillator
- Authors: Parovik R.I.1,2
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Affiliations:
- Institute of Cosmophysical Researches and Radio Wave Propagation, Far East Division, Russian Academy of Sciences
- Kamchatka State University named after Vitus Bering
- Issue: Vol 23, No 2 (2019)
- Pages: 378-393
- Section: Articles
- URL: https://ogarev-online.ru/1991-8615/article/view/20634
- DOI: https://doi.org/10.14498/vsgtu1678
- ID: 20634
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Abstract
In this paper, we study the chaotic regimes of the fractional Duffing oscillator. To do this, using the Wolf algorithm with Gram-Schmidt orthogonalization, we calculated the spectra of maximum Lyapunov exponents depending on the values of the control parameters, on the basis of which bifurcation diagrams were constructed. Bifurcation diagrams made it possible to determine areas in which a chaotic oscillatory regime exists. Phase trajectories were also constructed, which confirmed the research results.
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##article.viewOnOriginalSite##About the authors
Roman Ivanovich Parovik
Institute of Cosmophysical Researches and Radio Wave Propagation, Far East Division, Russian Academy of Sciences; Kamchatka State University named after Vitus Bering
Email: romano84@mail.ru
Candidate of physico-mathematical sciences, Associate professor 7, Mirnaya st., Paratunka, Kamchatkiy kray, 684034, Russian Federation; 4, Pogranichnaya st., Petropavlovsk-Kamchatskiy, 683032, Russian Federation
References
- Ахромеева Т С., Курдюмов С. П., Малинецкий Г. Г., Самарский А. А., Структуры и хаос в нелинейных средах, Физматлит, M., 2007, 488 с.
- Федоров В. К., Федянин В. В., "Особенности режимов детерминированного хаоса преобразователей постоянного напряжения для ветро- и гелиоэлектростанций", Известия Томского политехнического университета. Инжиниринг георесурсов, 327:3 (2016), 47-56
- Аливер В. Ю., "Хаотические режимы в непрерывных динамических системах", Вестник МГТУ им. Н. Э. Баумана. Сер. Приборостроение, 2006, № 1, 65-84
- Beninca E, Ballantine B., Ellner S. P., Huisman J., "Species fluctuations sustained by a cyclic succession at the edge of chaos", Proc. Natl. Acad. Sci., 112:20 (2015), 6389-6394
- Solé R. V., Valls J., "On structural stability and chaos in biological systems", J. Theor. Biol., 155:1 (1992), 87-102
- Bodalea I., Oancea V. A., "Chaos control for Willamowski-Rössler model of chemical reactions", Chaos, Solitons and Fractals, 78 (2015), 1-9
- Palanivel J., Suresh K., Sabarathinam S., Thamilmaran K., "Chaos in a low dimensional fractional order nonautonomous nonlinear oscillator", Chaos, Solitons and Fractals, 95 (2017), 33-41
- Паровик Р. И., "Математическое моделирование нелокальной колебательной системы Дуффинга с фрактальным трением", Вестник КРАУНЦ. Физ.-мат. науки, 2015, № 1(10), 18-24
- Syta A., Litak G., Lenci S., Scheffler M., "Chaotic vibrations of the Duffing system with fractional damping", Chaos, 24:1 (2014), 013107
- Liu Q. X., Liu J. K., Chen Y. M., "An analytical criterion for jump phenomena in fractional Duffing oscillators", Chaos, Solitons & Fractals, 98 (2017), 216-219
- Паровик Р. И., "Хаотические режимы фрактального нелинейного осциллятора", Вестн. Сам. гос. техн. ун-та. Сер. Физ.-мат. науки, 22:2 (2018), 364-379
- Герасимов А. Н., "Обобщение линейных законов деформирования и его применение к задачам внутреннего трения", ПММ, 12:3 (1948), 251-260
- Caputo M., Elasticità e dissipazione, Zani-Chelli, Bologna, 1969, 150 pp.
- Diethelm K., The analysis of fractional differential equations. An application-oriented exposition using differential operators of Caputo type, Lecture Notes in Mathematics, 2004, Springer, Berlin, 2010, viii+247 pp.
- Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, Elsevier Science B. V., Amsterdam, 2006, xvi+523 pp.
- Паровик Р. И., "Существование и единственность задачи Коши для фрактального нелинейного уравнения осциллятора", Узб. мат. ж., 2017, № 4, 110-118
- Parovik R. I., "Mathematical model of a wide class memory oscillators", Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming & Computer Software, 11:2 (2018), 108-122
- Wolf A., Swift J. B., Swinney H. L., Vastano J. A., "Determining Lyapunov exponents from a time series", Physica D: Nonlinear Phenomena, 16:3 (1985), 285-317
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