Mathematical Model of Fractional Duffing Oscillator with Variable Memory

Valentine Kim and Roman Parovik
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Valentine Kim: Department of Mathematics and Physics, Vitus Bering Kamchatka State University, Pogranichnaya, 4, 683032 Petropavlovsk-Kamchatskiy City, Russia
Roman Parovik: Department of Mathematics and Physics, Vitus Bering Kamchatka State University, Pogranichnaya, 4, 683032 Petropavlovsk-Kamchatskiy City, Russia

Mathematics, 2020, vol. 8, issue 11, 1-14

Abstract: 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.

Keywords: Riemann–Liouville derivative; Grunwald–Letnikov derivative; Lyapunov exponents; Runge rule; phase trajectories; amplitude-frequency characteristic; phase-frequency characteristic; Q-factor (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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