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

HTML  XML Download Download as PDF (Size: 367KB)  PP. 513-520  
DOI: 10.4236/am.2019.107035    572 Downloads   1,152 Views  Citations

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.

Share and Cite:

Stiefenhofer, P. and Giesl, P. (2019) Switching Regimes in Economics: The Contraction Mapping and the ω-Limit Set. Applied Mathematics, 10, 513-520. doi: 10.4236/am.2019.107035.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.