Fenichel Theory for Multiple Time Scale Singular Perturbation Problems

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.