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Mathematical Model of Fractional Duffing Oscillator with Variable Memory

by 1,2,† and 1,2,3,*,†
1
Department of Mathematics and Physics, Vitus Bering Kamchatka State University, Pogranichnaya, 4, 683032 Petropavlovsk-Kamchatskiy City, Russia
2
Department of Control Systems, Kamchatka State Technical University, Kluchevskaya, 35, 683003 Petropavlovsk-Kamchatskiy City, Russia
3
Institute of Cosmophysical Research and Radio Wave Propagation, Far East Branch, Russian Academy of Sciences, Mirnaya, 7, 684034 Paratunka, Russia
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2020, 8(11), 2063; https://doi.org/10.3390/math8112063
Received: 30 October 2020 / Revised: 15 November 2020 / Accepted: 16 November 2020 / Published: 19 November 2020
(This article belongs to the Special Issue Mathematical Modeling of Hereditarity Oscillatory Systems)
The article investigates a mathematical model of the Duffing oscillator with a variable fractional order derivative of the Riemann–Liouville type. The study of the model is carried out using a numerical scheme based on the approximation of the fractional derivative of the Riemann–Liouville type by a discrete analog—the fractional derivative of Grunwald–Letnikov. The adequacy of the numerical scheme is verified using specific examples. Using a numerical algorithm, oscillograms and phase trajectories are constructed depending on the values of the model parameters. Chaotic regimes of the Duffing fractional oscillator are investigated using the Wolf–Bennetin algorithm. The forced oscillations of the Duffing fractional oscillator are investigated using the harmonic balance method. Analytical formulas for the amplitude-frequency, phase-frequency characteristics, and also the quality factor are obtained. It is shown that the fractional Duffing oscillator possesses different modes: regular, chaotic, multi-periodic. The relationship between the order of the fractional derivative and the quality factor of the oscillatory system is established. View Full-Text
Keywords: Riemann–Liouville derivative; Grunwald–Letnikov derivative; Lyapunov exponents; Runge rule; phase trajectories; amplitude-frequency characteristic; phase-frequency characteristic; Q-factor Riemann–Liouville derivative; Grunwald–Letnikov derivative; Lyapunov exponents; Runge rule; phase trajectories; amplitude-frequency characteristic; phase-frequency characteristic; Q-factor
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MDPI and ACS Style

Kim, V.; Parovik, R. Mathematical Model of Fractional Duffing Oscillator with Variable Memory. Mathematics 2020, 8, 2063. https://doi.org/10.3390/math8112063

AMA Style

Kim V, Parovik R. Mathematical Model of Fractional Duffing Oscillator with Variable Memory. Mathematics. 2020; 8(11):2063. https://doi.org/10.3390/math8112063

Chicago/Turabian Style

Kim, Valentine; Parovik, Roman. 2020. "Mathematical Model of Fractional Duffing Oscillator with Variable Memory" Mathematics 8, no. 11: 2063. https://doi.org/10.3390/math8112063

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