# Vestnik КRAUNC. Fiz.-Mat. nauki. 2024. vol. 47. no. 2. P. 21 – 34. ISSN 2079-6641

**MATHEMATICAL MODELING**

https://doi.org/10.26117/2079-6641-2024-47-2-21-34

Research Article

Full text in Russian

MSC 26A33, 34A08

**Mathematical Model of Van der Pol-Airy Fractional Oscillator**

**A. I. Salimova¹, R. I. Parovik^\ast¹²**

¹National University of Uzbekistan named after Mirzo Ulugbek, 100174, Tashkent, Universitetskaya str., 4, Uzbekistan

²Institute of Cosmophysical Research and Radio Wave Propagation, Far Eastern Branch of the Russian Academy of Sciences, 684034, Paratunka, Mirnaya str., 7, Russia

Abstract. The paper proposes a mathematical model of the nonlinear Van der Pol-Airy oscillator taking into account heredity. The nonlinearity of the oscillator is due to the dependence of the friction coefficient on the square of the displacement function, which is typical for the Van der Pol oscillator. Also, the natural frequency of oscillations is a function of time, which increases linearly as it increases. The latter is typical for the Airy oscillator. Heredity effects are introduced into the model equation through fractional derivatives in the Gerasimov-Caputo sense. They indicate that the oscillatory system may have memory effects that manifest themselves depending on its current state from previous ones. For the proposed mathematical model, a numerical algorithm was developed based on an explicit first-order finite-difference scheme. The numerical algorithm was implemented in a computer program in the Maple language, with the help of which the simulation results were visualized. Oscillograms and phase trajectories were constructed for various values of the model parameters. It is shown that a fractional mathematical model can have various oscillatory modes: from self-oscillatory, damped and chaotic. An interpretation of the simulation results is given.

*Key words: mathematical model, Gerasimov-Caputo fractional derivative, oscillogram, phase trajectory, **limit cycle, numerical algorithm.*

Received: 29.03.2024; Revised: 15.05.2024; Accepted: 06.06.2024; First online: 25.08.2024

**For citation.** Salimova A. I., Parovik R. I. Mathematical model of Van der Pol-Airy fractional oscillator. Vestnik KRAUNC. Fiz.-mat. nauki. 2024, 47: 2, 21-34. EDN: QMOAXO. https://doi.org/10.26117/2079-6641-2024-47-2-21-34.

**Funding.** Institute of Cosmophysical Research and Radio Wave Propagation, Far Eastern Branch of the Russian Academy of Sciences (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.

^\ast**Correspondence**: E-mail: parovik@ikir.ru

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

© Salimova A. I., Parovik R. I., 2024

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

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**Information about the authors**

**Salimova Asal Iskandarovna** – 1st year master’s student “Applied Mathematics” , National University named after Mirzo Ulugbek, Tashkent, Uzbekistan, ORCID 0009-0003-9945-0991.

**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.