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Mathematical Model of Fractional Duffing Oscillator with Variable Memory

Author

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  • Valentine Kim

    (Department of Mathematics and Physics, Vitus Bering Kamchatka State University, Pogranichnaya, 4, 683032 Petropavlovsk-Kamchatskiy City, Russia
    Department of Control Systems, Kamchatka State Technical University, Kluchevskaya, 35, 683003 Petropavlovsk-Kamchatskiy City, Russia
    These authors contributed equally to this work.)

  • Roman Parovik

    (Department of Mathematics and Physics, Vitus Bering Kamchatka State University, Pogranichnaya, 4, 683032 Petropavlovsk-Kamchatskiy City, Russia
    Department of Control Systems, Kamchatka State Technical University, Kluchevskaya, 35, 683003 Petropavlovsk-Kamchatskiy City, Russia
    Institute of Cosmophysical Research and Radio Wave Propagation, Far East Branch, Russian Academy of Sciences, Mirnaya, 7, 684034 Paratunka, Russia
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Valentine Kim & Roman Parovik, 2020. "Mathematical Model of Fractional Duffing Oscillator with Variable Memory," Mathematics, MDPI, vol. 8(11), pages 1-14, November.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:11:p:2063-:d:447597
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    References listed on IDEAS

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    1. Liu, Q.X. & Liu, J.K. & Chen, Y.M., 2017. "An analytical criterion for jump phenomena in fractional Duffing oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 98(C), pages 216-219.
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    Cited by:

    1. Valentine Aleksandrovich Kim & Roman Ivanovich Parovik & Zafar Ravshanovich Rakhmonov, 2023. "Implicit Finite-Difference Scheme for a Duffing Oscillator with a Derivative of Variable Fractional Order of the Riemann-Liouville Type," Mathematics, MDPI, vol. 11(3), pages 1-17, January.

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