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Applications of Hybrid Dynamical Systems to Dynamics of Equilibrium Problems

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Title: Applications of Hybrid Dynamical Systems to Dynamics of Equilibrium Problems
Author: Greenhalgh, Scott
Department: Department of Mathematics and Statistics
Program: Mathematics and Statistics
Advisor: Monica, Cojocaru
Abstract: Many mathematical models generally consist of either a continuous system like that of a system of differential equations, or a discrete system such as a discrete game theoretic model; however, there exist phenomena in which neither modeling approach alone is sufficient for capturing the behaviour of the intended real world system. This leads to the need to explore the use of combinations of such discrete and continuous processes, namely the use of mathematical modeling with what are known as hybrid dynamical systems. In what follows, we provide a blueprint for one approach to study several classes of equilibrium problems in non-equilibrium states through the direct use of hybrid dynamical systems. The motivation of our work stems from the fact that the real world is rarely, if ever, in a state of perfect equilibrium and that the behaviour of equilibrium problems in non-equilibrium states is just as complex and interesting (if not more so) than standard equilibrium solutions. Our approach consists of an association of classes of traffic equilibrium problems, noncooperative games, minimization problems, and complementarity problems to a class of hybrid dynamical system called projected dynamical systems. The purposed connection between equilibrium problems and projected dynamical system is made possible through mutual connections to the robust framework of variational inequalities. The results of our work include theoretical contributions such as showing how evolution solutions (non-equilibrium solutions) can be analyzed from a theoretical point of view and how they relate to equilibrium solutions; computational methods for tracking and visualizing evolution solutions and the development of numerical algorithms for simulation; and applications such as the effect of population vaccination decisions in the spread of infectious disease, dynamic traffic networks, dynamic vaccination games, and nonsmooth electrical circuits.
Date: 2012-08
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