The Influence of Spatial Effects on the Dynamics of Solutions in Keynes' Mathematical Model of the Business Cycle.

We consider the special case of the mathematical model of Keynes' business cycle with the spatial interaction. In this model we assume that macroeconomic factors affect a certain geographical region and economic indicators. The dependence occurs on the spatial (geographical) coordinates in addition to the dependence on the temporal evolution, even if the economic subject is externally homogeneous in space. Spatial interaction led us to analyze a system of two differential equations of the 'reaction-diffusion' type, which replaces a system of two ordinary differential equations. This method is often used to analyze the dynamics of complex nonlinear systems and macroeconomic entities. An analysis of such a mathematical model is based on the use of modern methods of the theory of dynamical systems indicate the presence of new nonlinear effects in addition to those used in the traditional version of the Keynes model. We encountered the loss of stability of the homogeneous economic equilibrium state and the occurrence of economic cycles for some values of the parameters while investigating the characteristics of such a system. Meanwhile, another version of instability of a homogeneous economic equilibrium state with a different choice of parameters occurs, which in many cases leads to the appearance of a spatially non-homogeneous equilibrium state, which is characterized by the dependence of the corresponding economic indicators on the spatial (geographical) coordinates of the area in which the assigned macroeconomic entity is located.

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.