Hierarchical aggregation of linear systems with multiple time scales

Abstract

In this paper we carry out a detailed analysis of the multiple time scale behavior of singularly perturbed linear systems of the form \dot{x}^{\epsilon}(t) = A(\epsilon)x^{\epsilon}(t) where A(\epsilon) is analytic in the small parameter e. Our basic result is a uniform asymptotic approximation to \exp A(\epsilon)t that we obtain under a certain multiple semistability condition. This asymptotic approximation gives a complete multiple time scale decomposition of the above system and specifies a set of reduced order models valid at each time scale. Our contribution is threefold. 1) We do not require that the state variables be chosen so as to display the time scale structure of the system. 2) Our formulation can handle systems with multiple ( > 2) time scales and we obtain uniform asymptotic expansions for their behavior on [ 0, \infty ]. 3) We give an aggregation method to produce increasingly simplified models valid at progressively slower time scales.