Dubovsky’s Class of Mathematical Models for Describing Economic Cycles with Heredity Effects

The article is devoted to the study of economic cycles within the framework of the theory of Kondratieff's long waves or K-waves. The object of the study is Dubovsky's fractional mathematical models, which consist of two nonlinear ordinary differential equations of fractional order and describe the dynamics of the efficiency of new technologies and capital productivity, taking into account constant and variable heredity. Fractional mathematical models also take into account the dependence of the accumulation rate on capital productivity, the influx of external investment and new technological solutions. The effects of heredity lead to a delayed effect of the reaction of the system in question to the impact. The property of heredity in mathematical models is taken into account using fractional derivatives of constant and variable orders, which are understood in the sense of Gerasimov-Caputo. Dubovsky's fractional mathematical models are studied numerically using the Adams-Bashforth-Moulton algorithm. Using a numerical algorithm, oscillograms and phase trajectories were constructed for various values of the model parameters. It is shown that Dubovsky's fractional mathematical models can have limit cycles, and there are no self-oscillatory modes.