Vestnik КRAUNC. Fiz.-Mat. nauki. 2024. vol. 46. no. 1. P. 70-88. ISSN 2079-6641

Research Article
Full text in Russian
MSC 34A34

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Mathematical Model of a Fractional Nonlinear Mathieu Oscillator

A.Zh. Otenova¹, R. I. Parovik^\ast¹²

¹National University of Uzbekistan named after Mirzo Ulugbek, 100174, Tashkent, st. Universitetskaya, 4, Uzbekistan
²Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, 684034, Paratunka, st. Mirnaya, 4, Russia

Abstract. The work studies the fractional nonlinear Mathieu oscillator using numerical analysis methods in order to establish its various oscillatory modes. Mathieu’s fractional nonlinear oscillator is an ordinary nonlinear differential equation with fractional derivatives in the Gerasimov-Caputo sense and local initial conditions (Cauchy problem). Gerasimov-Caputo fractional derivatives characterize the presence of the heredity effect in an oscillatory system. In such a system, its current state depends on the previous history. To study the Cauchy problem, a numerical method from the predictor-corrector family was used – the Adams-Bashforth-Moulton method, the algorithm of which was implemented in the Matlab computer mathematics system. Using a numerical algorithm, oscillograms and phase trajectories were constructed for various values of the parameters of the Mathieu fractional nonlinear oscillator. It is shown that in the absence of an external periodic influence, self-oscillations can arise in the oscillatory system under consideration, which are characterized by limit cycles on the phase trajectory. A study of limit cycles was carried out using computer simulation. It has been shown that aperiodic regimes can also arise, i.e. modes that are not oscillatory. Therefore, the orders of fractional derivatives can be influenced by the oscillatory mode of a nonlinear fractional Mathieu oscillator: from oscillations with a constant amplitude to damped ones and disappearing completely.

Key words: model, nonlinear Mathieu oscillator, fractional order derivative, numerical modeling, oscillograms, phase trajectories

Received: 15.02.2024; Revised: 29.02.2024; Accepted: 01.03.2024; First online: 07.03.2024

For citation. Otenova A. Zh., Parovik R. I. Mathematical model of a fractional nonlinear Mathieu oscillator. Vestnik KRAUNC. Fiz.-mat. nauki. 2024, 46: 1, 70-88. EDN: MQEHDX.

Funding. The work was supported by IKIR FEB RAS State Task (subject registration No. 124012300245- 2).

Competing interests. 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.

^\astCorrespondence: E-mail:

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Otenova A. Zh., Parovik R. I., 2024

© Institute of Cosmophysical Research and Radio Wave Propagation, 2024 (original layout, design, compilation)


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Information about authors

Otenova Aysanem Zhebegen qizi – 2nd year master’s student “Applied Mathematics” , National University named after Mirzo Ulugbek, Tashkent, Uzbekistan, ORCID 0009-0004-1225-1832.

Parovik Roman Ivanovich – D. Sci. (Phys. & Math.), Associate Professor, Leading researcher, laboratory of modeling physical processes Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Paratunka, Russia, ORCID 0000-0002-1576-1860.