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.
Highlights
Economic systems may not always satisfy the usual smoothness condition everywhere
This paper considers a dynamical system defined by a set of ordinary autonomous differential equations with discontinuous right-hand side
The novelty of the stability theory discussed in this paper is that it is independent of the explicit solution of the system
Summary
Economic systems may not always satisfy the usual smoothness condition everywhere. In particular, a discontinuity in an economic system may occur due to a change in economic regime or policy implementation. Little is known about stability results of non-smooth periodic orbits Such results depend on the explicit calculation of the periodic orbit and employ a global stability theory based on Poincaré’s map. Since such explicit calculations may not always be possible, we want to establish existence and exponentially asymptotical stability of a nonsmooth periodic orbit without its calculation. The advantage of such a local stability theory would allow economists to derive analytic results for the purpose of economic policy analysis.
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