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Switching Regimes in Economics: The Contraction Mapping and the &omega;-Limit Set
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作者 Pascal Stiefenhofer Peter Giesl 《Applied Mathematics》 2019年第7期513-520,共8页
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 mor... 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. 展开更多
关键词 NON-SMOOTH Periodic ORBIT Differential Equation CONTRACTION Mapping Economic Regimes NON-SMOOTH DYNAMICAL System
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