Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 139, pp. 1-17.

Title: On Pontryagin-Rodygin's theorem for convergence of 
       solutions of slow and fast systems

Authors: Tewfik Sari (Univ. de Haute Alsace, Mulhouse, France)
         Karim Yadi  (Univ. de Haute Alsace, Mulhouse, France)

Abstract: 
 In this paper we study fast and slow systems for which the fast
 dynamics has limit cycles, for all fixed values of the slow variables.
 The fundamental tool is the Pontryagin and Rodygin theorem which
 describes the limiting behavior of the solutions in the continuously
 differentiable case, when the cycles are  exponentially stable.
 We extend this result to the continuous case, and  exponential
 stability is replaced by asymptotic stability. We give two  examples
 with numerical simulations to illustrate the problem. Our results are
 formulated in classical mathematics. They are proved using Nonstandard
 Analysis.

Submitted May 12, 2004. Published November 26, 2004.
Math Subject Classifications: 34D15, 34E15, 34E18.
Key Words: Singular perturbations; asymptotic stability; 
           nonstandard analysis.