Switching Regimes in Economics: The Contraction Mapping and the ω-Limit Set

Abstract

This paper considers a dynamical system defined by a set of ordinary autonomous differential equations with discontinuous right-hand side. Such systems typically appear in economic modelling where there are two or more regimes with a switching between them. Switching between regimes may be a consequence of market forces or deliberately forced in form of policy implementation. Stiefenhofer and Giesl [1] introduce such a model. The purpose of this paper is to show that a metric function defined between two adjacent trajectories contracts in forward time leading to exponentially asymptotically stability of (non)smooth periodic orbits. Hence, we define a local contraction function and distribute it over the smooth and nonsmooth parts of the periodic orbits. The paper shows exponentially asymptotical stability of a periodic orbit using a contraction property of the distance function between two adjacent nonsmooth trajectories over the entire periodic orbit. Moreover it is shown that the ω-limit set of the (non)smooth periodic orbit for two adjacent initial conditions is the same.

Review on contraction analysis and computation of contraction metrics

“…In Giesl 2007 [55], a converse theorem is proved, showing the existence of such a metric. Stiefenhofer & Giesl 2019 [174,175] generalize the sufficient condition to two spatial dimensions.…”

Review on contraction analysis and computation of contraction metrics

“…In Giesl 2007 [55], a converse theorem is proved, showing the existence of such a metric. Stiefenhofer & Giesl 2019 [174,175] generalize the sufficient condition to two spatial dimensions. Dahlquist 1958 [35] applies contractivity in numerics for solving ODEs.…”