Stability of Equilibrium and Bifurcation Behavior of the Production and Economic System

Abstract

This study considers some key problems of analysis of nonlinear dynamic systems on the example of production and economic objects. The construction of a mathematical model of systems, functioning in the market environment, is aimed at qualitative forecasting (along the development trajectory) of the behavioral properties of such a system. The conceptual orientation of the study involves the analysis of structural instability of equilibrium states within the framework of the proposed model by the presence of characteristic combinations of the most important economic parameters that have a significant impact on the static and dynamic characteristics of the production and economic system. Critical modes of functioning of the object have been identified and its stability area has been built in three-dimensional space depending on significant parameters. An example of such dynamic modes, which were revealed in the process of analysis using the proposed model, is an unstable boundary cycle that provokes the so-called «hard» mechanism of excitation of self-oscillations around the equilibrium state of the «focus» type. Also in this system, the global bifurcation of the saddle joint in the presence of a loop of the «saddle» separatrix around the state of equilibrium was found and studied. Such modes are rather dangerous, since there are cyclic processes with very long periods, which significantly affects the accuracy of predicting the behavior of the object under study.