2009, Vol. 19, Issue 1, an item 7

MALIČKÝ P., ZIMKA R., Can tori arise in a two-regional model with fixed exchange rates


In the paper a two-regional model describing the dynamic interaction of two regions connected through interregional trade and capital movement, which was introduced by T. Asada in [1], is investigated. The model describes the development of income, capital stock and money stock in the considered regions. In [1] T. Asada considered the existence of an equilibrium point for the model, found sufficient conditions for its local stability and also investigated the existence of business cycles around the equilibrium point. As the model under investigation is a five-dimensional dynamic system, the question of the existence of tori around the equilibrium point is legitimate. The paper gives the answer to this question. Tori can arise only in the case when the linear approximation matrix for the model has two pairs of purely imaginary eigenvalues at the equilibrium point. Theorem 1 gives sufficient conditions for the existence of two pairs of purely imaginary eigenvalues with the remaining one being negative. Theorem 2 comments on the existence of tori in a small neighbourhood of the equilibrium point. The model investigated in this paper can be applied to the analysis of the dynamic interaction of two countries with fixed exchange rates, for example any two countries within the Eurozone.

Keywords: dynamic model, equilibrium, linear approximatiomatrix, eigenvalues, normal form of differential equations on invariant surface, bifuraction equation, torus