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Implicit Finite-Difference Scheme for a Duffing Oscillator with a Derivative of Variable Fractional Order of the Riemann-Liouville Type

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  • Valentine Aleksandrovich Kim

    (International Integrative Research Laboratory of Extreme Phenomena of Kamchatka, Vitus Bering Kamchatka State University, 4, Pogranichnaya St., Petropavlovsk-Kamchatskiy 683032, Russia
    Faculty of Applied Mathematics and Intelligent Technologies, National University of Uzbekistan Named after Mirzo Ulugbek, 4 Universitetskaya St., Tashkent 100174, Uzbekistan)

  • Roman Ivanovich Parovik

    (International Integrative Research Laboratory of Extreme Phenomena of Kamchatka, Vitus Bering Kamchatka State University, 4, Pogranichnaya St., Petropavlovsk-Kamchatskiy 683032, Russia
    Faculty of Applied Mathematics and Intelligent Technologies, National University of Uzbekistan Named after Mirzo Ulugbek, 4 Universitetskaya St., Tashkent 100174, Uzbekistan
    Laboratory for Simulation of Physical Processes, Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, 7, Mirnaya St., Kamchatka Krai, Yelizovsky District, Paratunka 684034, Russia)

  • Zafar Ravshanovich Rakhmonov

    (Faculty of Applied Mathematics and Intelligent Technologies, National University of Uzbekistan Named after Mirzo Ulugbek, 4 Universitetskaya St., Tashkent 100174, Uzbekistan)

Abstract

The article considers an implicit finite-difference scheme for the Duffing equation with a derivative of a fractional variable order of the Riemann–Liouville type. The issues of stability and convergence of an implicit finite-difference scheme are considered. Test examples are given to substantiate the theoretical results. Using the Runge rule, the results of the implicit scheme are compared with the results of the explicit scheme. Phase trajectories and oscillograms for a Duffing oscillator with a fractional derivative of variable order of the Riemann–Liouville type are constructed, chaotic modes are detected using the spectrum of maximum Lyapunov exponents and Poincare sections. Q-factor surfaces, amplitude-frequency and phase-frequency characteristics are constructed for the study of forced oscillations. The results of the study showed that the implicit finite-difference scheme shows more accurate results than the explicit one.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:558-:d:1042707
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    References listed on IDEAS

    as
    1. Valentine Kim & Roman Parovik, 2020. "Mathematical Model of Fractional Duffing Oscillator with Variable Memory," Mathematics, MDPI, vol. 8(11), pages 1-14, November.
    2. Gao, Xin & Yu, Juebang, 2005. "Chaos in the fractional order periodically forced complex Duffing’s oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 24(4), pages 1097-1104.
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    Cited by:

    1. Feiyun Pei & Guojiang Wu & Yong Guo, 2023. "Construction of Infinite Series Exact Solitary Wave Solution of the KPI Equation via an Auxiliary Equation Method," Mathematics, MDPI, vol. 11(6), pages 1-25, March.

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