THREE TIME SCALE SINGULAR PERTURBATION PROBLEMS AND NONSMOOTH DYNAMICAL SYSTEMS

Abstract

In this paper we study three time scale singular perturbation problemsepsilon x ' = f(x, epsilon, delta), y ' = g(x, epsilon, delta), z ' = delta h(x, delta, delta),where x = (x, y, z) is an element of R-n x R-m x R-p, epsilon and delta are two independent small parameters (0 < epsilon, delta << 1), and f, g, h are C-r functions, where r is big enough for our purposes. We establish conditions for the existence of compact invariant sets (singular points, periodic and homoclinic orbits) when epsilon, delta > 0. Our main strategy is to consider three time scales which generate three different limit problems. In addition, we prove that double regularization of nonsmooth dynamical systems with self-intersecting switching variety provides a class of three time scale singular perturbation problems.