| In this dissertation, we consider solutions of hybrid systems in which both continuous dynamics and discrete events coexists. One of the main ingredients of our models is the two-time-scale formulation. Under broad conditions, asymptotic expansions are developed for the solutions of the systems of backward equations for switching diffusion in two classes of models, namely, fast switching systems and fast diffusion systems. To prove the validity of the asymptotic expansions, uniform error bounds are obtained.;In the second part of the dissertation, a singular linear system is considered. Again a two-time-scale formulation is used. Under suitable conditions, the system has a limit. Using the limit system as a guide, our effort is devoted to deriving a sufficient condition for the stability of the original system. These results present a perspective on reduction of complexity from stability and asymptotic analysis points of view. |
