Vestnik КRAUNC. Fiz.-Mat. nauki. 2024. vol. 49. no. 4. P. 24 – 35. ISSN 2079-6641

MATHEMATICAL MODELING
https://doi.org/10.26117/2079-6641-2024-49-4-24-35
Research Article
Full text in Russian
MSC 34A08, 34A34

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Study of Bifurcation Diagrams of Selkov’s Fractional Dynamic System to Describe Self-Oscillatory Modes of Microseisms

R. I. Parovik^{\ast}

Institute for Cosmophysical Research and Radio Propagation FEB RAS, 684034, v. Paratunka, Mirnaya st., 7, Russia

Abstract. The article studies the dynamic modes of the fractional Selkov system with variable heredity (memory). The effect of variable heredity means that heredity changes over time, i.e. the dependence of the current state of the system on the previous ones also depends on time. Variable heredity in the fractional Selkov system is described from the mathematical point of view using derivatives of fractional variables of the Gerasimov-Caputo type. The fractional dynamic Selkov system is studied using the Adams-Bashforth-Multon numerical method from the predictor-corrector family. Using the numerical algorithm, various bifurcation diagrams are constructed — dependences of the obtained numerical solution on various values of the parameters of the model equations. The Adams-Bashforth-Multon numerical algorithm and the construction of bifurcation diagrams were implemented in Python in the PyCharm 2024.1 environment. The study of bifurcation diagrams showed the presence of not only regular regimes: limit cycles and damped oscillations and chaotic oscillations, but also revealed a singularity — unlimited growth of the solution when changing the values of the orders of fractional derivatives in the model equation. Biffurcation diagrams may contain curve sections with and without spikes. Spikes may indicate relaxation oscillations or chaotic modes, the absence of spikes corresponds to damped oscillations or aperiodic modes.

Key words: mathematical modeling, fractional dynamic Selkov system, oscillogram, phase trajectory, bifurcation diagrams, statistical characteristics, fractional derivatives of variable order, hereditary, Python, PyCharm

Received: 25.10.2024; Revised: 18.11.2024; Accepted: 25.11.2024; First online: 27.11.2024

For citation. Parovik R. I. Study of bifurcation diagrams of Selkov’s fractional dynamic system to describe self-oscillatory modes of microseisms. Vestnik KRAUNC. Fiz.-mat. nauki. 2024, 49: 4, 24-35. EDN: ZNIAZE. https://doi.org/10.26117/2079-6641-2024-49-4-24-35.

Funding. The work was funded by Russian Science Foundation [grant number 22-11-00064 “Modeling dynamic processes in geospheres taking into account hereditarity” https://rscf.ru/project/22-11-00064].

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. The author participated in the writing of the article and is fully responsible for submitting the final version of the article to the press.

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

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

© Parovik R. I., 2024

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

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