Vestnik KRAUNC. Fiz.-Mat. Nauki. 2022. vol. 40. no. 3. pp. 179–198. ISSN 2079-6641

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MSC 26A33, 34C15

Research Article

Implicit finite-difference scheme for a Duffing oscillator with a derivative of variable fractional order of the RiemannLiouville type

V. A. Kim¹, R. I. Parovik¹²

¹Vitus Bering Kamchatka State University, 683032, Petropavlovsk-Kamchatskiy, Pogranichnaya str., 4, Russia
²Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, 7, Mirnaya st., Kamchatka Krai, Yelizovsky district, 684034, c. Paratunka, Russia
E-mail: valentinekim@mail.ru, romanparovik@gmail.com

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.

Key words: Duffing oscillator, Runge rule, Riemann-Liouville operator, Grunwald-Letnikov operator, amplitude-frequency response, phase-frequency response, Q-factor, Lyapunov exponents, Poincare sections, oscillogram

DOI: 10.26117/2079-6641-2022-40-3-179-198

Original article submitted: 24.11.2022

Revision submitted: 05.12.2022

For citation. Kim V. A., Parovik R. I. Implicit finite-difference scheme for a Duffing oscillator with a derivative of variable fractional order of the Riemann-Liouville type. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 40: 3, 179-198. DOI: 10.26117/2079-6641-2022-40-3-179-198

Competing interests. The authors declare that there are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

Funding. Financial support was provided within the framework of the grant of the President of the Russian Federation, No. MD-758.2022.1.1

The content is published under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/deed.ru)

© Kim V. A., Parovik R. I., 2022

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Kim Valentine Aleksandrovich – Junior researcher of the integrative Scientific Research Laboratory of Natural Disasters of Kamchatka — earthquakes and volcanic eruptions, Petropavlovsk-Kamchatskiy, Russia, ORCID 0000-0001-8895-6821.

Parovik Roman Ivanovich – D. Sci. (Phys. & Math.), Associate Professor, Prof., Dep. of Informatics and Mathematics, Vitus Bering Kamchatka State University, Petropavlovsk-Kamchatsky; Leading Researcher, Laboratory for Simulation of Physical Processes, Institute for Cosmophysical Research and Radio Wave Propagation FEB RAS, Paratunka, Russia, ORCID 0000-0002-1576-1860.

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