Computing Center Manifolds: A Macroeconomic Example
When crossing the boundary of stability of a given dynamical system only indicates a bifurcation point and the type of the bifurcating solutions. But it doesn't tell us how and how many new solutions bifurcate or disappear in a bifurcation point. To answer this question one has to take into account the leading nonlinear terms. The Center manifold theorem helps to reduce the dimensionality of the phase space to the dimensionality of the so-called center manifold which in the bifurcation point is tangentially to the eigenspace of the marginal modes of the linear stability analysis. The center manifold theorem allows the dynamics to be projected onto the center manifold without loosing any significant aspect of the dynamics. Thus, the dynamics near a stationary co-dimension-one bifurcation can be decribed by an effective dynamics in a one-dimensional subspace. The dynamics projected onto the center manifold can by transformed to so-called normal forms by a nonlinear transformation of the phase space variables. In this paper we employ the techniques suggested by the center manifold theorem and explore the dynamics of an economic system as it moves away from the steady state. In particular we conisder the model by Azariadis, Bullard and Smith (2001) which is a dynamic general equilibrium model in which privately-issued liabilities may circulate, either by themselves, or alongside a stock of outside money which yileds a center manifold.
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