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

Valentine Aleksandrovich Kim (), Roman Ivanovich Parovik () and Zafar Ravshanovich Rakhmonov ()
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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
Roman Ivanovich Parovik: International Integrative Research Laboratory of Extreme Phenomena of Kamchatka, Vitus Bering Kamchatka State University, 4, Pogranichnaya St., Petropavlovsk-Kamchatskiy 683032, 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

Mathematics, 2023, vol. 11, issue 3, 1-17

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.

Keywords: Duffing oscillator; Runge rule; Riemann-Liouville operator; Grunwald-Letnikov operator; amplitude-frequency response; phase-frequency response; Q-factor; Lyapunov exponents; Poincare sections; phase trajectory; oscillogram (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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