Fenichel theory for multiple time scale singular perturbation problems

Abstract

This paper is concerned with a geometric study of singularly perturbed systems of ordinary differential equations expressed by (n-1)-parameter families of smooth vector fields on ℝl, where n ≥ 2. The inherent characteristic of such systems is the presence of an arbitrary number n of time scales. For n = 2, the proposed geometric approach in this paper reports to Fenichel theory of fast-slow systems [N. Fenichel, J. Differential Equations, 31 (1979), pp. 53-98]. We extend the three main theorems due to Fenichel [N. Fenichel, J. Differential Equations, 31 (1979), pp. 53-98] to systems involving any number of time scales.