A Geometric Approach to Computing Center Manifolds

The area that has probably received most applications of qualitative dynamical systems is the study of business cycles and economic fluctuations. The majority of applications have employed the standard theory of bifurcations which deals with dynamical systems defined on open sets. In particular, this implies that all relevant trajectories, including steady states, must be contained in the interior of the state space. In the majority of cases the resulting planar economies explore the asymptotic behavior of an iterative process and provide a comprehensive description of the geometric structures arising from the evolution of the system. In general these applications just consider the evolution of the associated dynamical system near the steady state(s) characterizing the resulting invariant manifolds just locally. Typically the resulting dynamical systems are not able to generate eigenvalues in the unit circle for all the relevant parameter values. When center manifolds are present, some eigenvalues have moduli 1, the study of global dynamics is a bit more challenging and new numerical considerations are required. In particular, the center manifold theorem allows the dynamics to be projected onto the center manifold without loosing any significant aspect of the dynamics. Thus, dynamics near a stationary co-dimension one bifurcation can be described by an effective dynamics in a one-dimensional subspace. The dynamics projected onto the center manifold can be represented by the so called normal forms through a nonlinear transformation of the phase space variables. The basic goal of this paper is to apply computational techniques from the dynamical system literature in order to study the asymptotic behavior of an iterative process that has a center manifold, while providing a comprehensive description of the geometric structures arising from the evolution of an economy where private and public circulating liabilities coexist. In particular, we consider the model by Azariadis, Bullard and Smith (2001) which is a dynamic general equilibrium model where privately-issued liabilities may circulate, either by themselves, or alongside a stock of outside money which yields a center manifold.