Vestnik KRAUNC. Fiz.-Mat. Nauki. 2022. vol. 41. no. 4. pp. 146–166. ISSN 2079-6641

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MSC 86-10

Research Article

Investigation of the Selkov fractional dynamical system

R. I. Parovik

Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, 684034, Paratunka, Mirnaya str., 7, Russia


A fractional nonlinear Selkov dynamic system is proposed to describe microseismic phenomena. This system is known for the presence of self-oscillatory regimes and is used in biology to describe glycolytic oscillations of the  substrate and product.The Selkov dynamic system can also, by analogy, describe the interaction of two types of cracks in an elastic-brittle medium. The first type is seed cracks with less energy, which are not recorded by seismic equipment, and the second type is large cracks that generate microseisms. The first type of cracks are triggers for cracks of the second type. However, the reverse transition is also possible. For example, when large cracks lose their energy and partially become seed cracks. Further, after increasing their concentration, the process is repeated, providing the self-oscillating nature of microseismic sources. The Selkov fractional dynamical system takes into account the effect of hereditarity and is described using derivative fractional orders. The heredity of oscillatory systems is studied within the framework of hereditary mechanics and indicates that a dynamic system can «remember» some time, the impact on it, which is typical for viscoelastic and plastic media. The orders of fractional derivatives are related to the hereditarity of the system and are responsible for the intensity of energy dissipation emitted by cracks of the first and second types. In this paper, the Selkov fractional dynamic model is investigated using the Adams-Bashforth-Moulton numerical method, oscillograms and phase trajectories are constructed, and rest points are investigated. It is shown that a fractional dynamic model can have relaxation and damped oscillations, as well as chaotic modes.

Key words: Selkov dynamic system, self-oscillating mode, oscillograms, phase trajectories, bifurcation diagrams, Adams-Bashforth-Multon method

DOI: 10.26117/2079-6641-2022-41-4-146-166

Original article submitted: 07.12.2022

Revision submitted: 14.12.2022

For citation. Parovik R. I. Investigation of the Selkov fractional dynamical systems. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 41: 4, 146-166. DOI: 10.26117/2079-6641-2022-41-4-146-166

Competing interests. The author declare that there is no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. The author is solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by the author.

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

© Parovik R. I., 2022

Funding. The work was carried out with the financial support of the Russian Science Foundation (project No. 22-11-00064).


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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.