Territory Development

Abstract

The article considers the application of an effective method for the analysis of nonlinear systems — the theory of bifurcations. The classical problem of fixed capital growth under the influence of investments is analyzed, where the volume of investments depends on the already existing capital and the economic number that assesses the level of economic development. The dependence of fixed capital on the economic number is considered in a state of equilibrium. When out of equilibrium, various types of bifurcations arise. The Taylor formula is used to find a solution for close values to the equilibrium point of the economic system. As a result, we obtain a new state of equilibrium that does not lead to a bifurcation of fixed assets. Analytical formulas are given for determining bifurcation points and boundary states. The state of the economies of three EU countries in the period 2007—2015 is considered, and an analysis is made of the occurrence of bifurcations of fixed assets. The conditions for the emergence of two possible solutions and the appearance of a supercritical bifurcation in the economies of these countries are determined. A system of two ordinary differential equations is considered that describes the growth of fixed assets and labor resources depending on the economic number and mutual influence on each other. Analytical formulas are derived to determine the moments of occurrence of bifurcations. This system of equations has Hopf type bifurcations. As a rule, bifurcations are associated with the appearance or disappearance of stationary solutions, and with the Hopf bifurcation, periodic regimes arise — a stable focus or a limit cycle. With an increase in the economic number, the focus loses stability and a limit cycle appears, the radius of which varies as the square root of the economic number.