Synchronization of Fluctuations in the Interaction of Economies within the Framework of the Keynes's Business Cycle Model.

In this paper, we will study two independent economies in a country (national, regional and urban), where the dynamics of fluctuations in each economy is described by Keynes's mathematical business cycle model. This is an interaction of two economies which include trade and competition. In the resulting system that consists of two independent economic entities, we show that fluctuations can emerge as two possible types of economic indicators (synchronous and antiphase) when the peaks and downturns of business activities in each of the economies are completely synchronized or on the contrary when the rise of one economy is accompanied by a recession (antiphase cycles). Our aim is to examine the stability question of solutions of the cognate mathematical model. Our analysis of the mathematical model will render methods of the theory of dynamical systems, such as the method of integral manifolds and the Poincare normal forms. This approach will provide a sufficient analysis of the dynamics of solutions of a system of differential equations, which is used as a mathematical model. Asymptotic formulas will be obtained for solutions that depict economic cycles.