Quasi-periodic development of the economy

Abstract

The article considers the mathematical definition of the space of states of the macroeconomy, where each point represents a certain distribution of the growth of the main factors of production – labor, capital, and natural resources. We study the violation of the sustainable regime of economic development, described by the system of differential equations in partial derivatives. The phenomenon of stable development of the economy, as well as the presence of more than one stable regime of economic development, depending on political and international factors, is studied. The role of periodic development in the economy and the existence of multiple attractors that create various stable regimes are considered in detail. The various stable regimes that arise are solutions of non-linear differential equations. We study the process of the emergence of instability, when the economic number that defines a particular economy continues to increase after reaching its first critical value. A further increase in the economic number leads to the fact that periodic development becomes unsustainable. The trajectory analysis of production factors is performed by the method presented by L.D. Landau and E.M. Lifshitz. The emerging stabilization of the economy can be broken again as the economic number approaches the next critical value; instability reappears with the advent of a new frequency. After passing through several critical values of the economic number, a stable limit cycle can appear on the torus, which is a geometric interpretation of the trajectories of economic development. The appearance of a stable limit cycle means the manifestation of synchronization of oscillations the disappearance of the quasi-periodic and the establishment of a new periodic motion. The birth of a stable limit cycle prevents the emergence of a regime of superpositions of motions of trajectories with many incommensurable frequencies.