Vestnik КRAUNC. Fiz.-Mat. nauki. 2024. vol. 46. no. 1. P. 52-69. ISSN 2079-6641

Research Article
Full text in Russian
MSC 34A34

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The Classical Mathematical Model of S.V. Dubovsky and Some of its Modifications for Describing K-Waves

D. V. Makarov^\ast¹²

¹Vitus Bering Kamchatka State University, Petropavlovsk-Kamchatsky, st. Pogranichnaya, 4, Russia
²National University of Uzbekistan named after Mirzo Ulugbek, 100174, Tashkent, st. Universitetskaya, 4, Uzbekistan

Abstract. In this work, the classical mathematical model of S.V. was investigated. Dubovsky to describe long waves N.D. Kondratiev (K-waves). This model describes the dynamics of free fluctuations in the efficiency of new technologies and the efficiency of capital productivity. From the point of view of mathematics, it is a system of nonlinear ordinary differential equations of the first order. The purpose of the research is to visualize the results of the solution using numerical modeling of a modification of the mathematical model of S.V. Dubovsky, which consists in taking into account the dependence of the accumulation rate on capital productivity and external inflow of investments and new technological models. It was also shown using the Bendixson test that the classical model of S.V. Dubovsky can generate closed phase trajectories, which indicates its use in describing economic crises and cycles. Similarly, it was shown that within the framework of the modified mathematical model S.V. Dubovsky can also have closed phase trajectories. It is shown using computer modeling that the dependence of the accumulation rate on capital productivity can influence the period of cyclical fluctuations, which is important when modeling real economic cycles and crises. Taking into account the external influx of investment and new technologies (managerial decisions) using harmonic functions significantly complicates the appearance of phase trajectories, however, closed phase trajectories are also possible here. These harmonic functions determine forced fluctuations in the efficiency of new technologies and the efficiency of capital productivity, and here resonance effects may occur, which were shown using computer modeling in this article. Computer simulation was carried out in the computer algebra environment Matlab.

Key words: K-waves, oscillogram, phase trajectory, economic cycles and crises, accumulation rate, Bendixson criterion, model, Adams-Bashforth-Moulton method

Received: 19.02.2024; Revised: 04.03.2024; Accepted: 06.03.2024; First online: 07.03.2024

For citation. Makarov D. V. The classical mathematical model of S.V. Dubovsky and some of its modifications for describing K-waves. Vestnik KRAUNC. Fiz.-mat. nauki. 2024, 46: 1, 52-69. EDN:

Funding. The research was carried out within the framework of the “Priority 2030. Far East program”.

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:

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

© Makarov D. V., 2024

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


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Information about author

Makarov Danil Vasilyevich – Junior Researcher at the International Integrative Research Laboratory for Extreme Phenomena in Kamchatka, Petropavlovsk-Kamchatsky, Russia, ORCID 0000-0003-1033-7611.