Goodwin's business cycle model and synchronization of oscillations of two interacting economies

Three well-known versions of the mathematical model of the business cycle proposed by R. Goodwin are considered. To analyze them, such methods of the theory of dynamical systems as the method of integral manifolds and normal forms of A. Poincare were used. It is shown that only one of the versions of this model can have a stable cycle in the neighborhood of the state of economic equilibrium. In the other two models, cycles exist, but they are unstable. For all three considered variants of R. Goodwin's model, asymptotic formulas are obtained.
The question of the competitive interaction of the two economies is also considered. It is shown that the problem can be interpreted as the problem of synchronization of oscillations for two self-oscillating systems in the presence of weak coupling. Two types of such a coupling are considered. The analysis of such a problem was reduced to the analysis of the Poincare normal form. As a result, for one of the studied problems, oscillations of two types were identified: synchronous and antiphase oscillations. The issue of their stability has been investigated. For such periodic solutions, asymptotic formulas are obtained.
When constructing normal forms, the correspondingly modified Krylov — Bogolyubov algorithm was used in all cases.