The classical mathematical model of S.V. Dubovsky and some of its modifications for describing K-waves

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.