Electronic Journal of Differential Equations,
Vol. 1998(1998), No. 19, pp. 1-22.

Title: On Tykhonov's theorem for convergence of solutions 
       of slow and fast systems

Authors: Claude Lobry (Centre Int. de Math. Pures et Appl., Nice, France)
         Tewfik Sari (Univ. de Haute Alsace, Mulhouse, France)
         Sefiane Touhami (Univ. de Haute Alsace, Mulhouse, France)

Abstract: 
 Slow and fast systems gain their special structure from the presence of two
 time scales. Their analysis is achieved with the help of Singular
 Perturbation Theory. The fundamental tool is Tykhonov's theorem which
 describes the limiting behaviour, for compact interval of time, of solutions
 of the perturbed system which is a one-parameter deformations of the
 so-called unperturbed system. Our aim here is to extend this description to
 the solutions of all systems that belong to a small neighbourhood of the
 unperturbed system. We investigate also the behaviour of solutions on the
 infinite time interval. Our results are formulated in classical mathematics.
 They are proved within Internal Set Theory which is an axiomatic approach to
 Nonstandard Analysis.
 
Submitted September 30, 1997. Published July 9, 1998.
Math Subject Classification: 34D15, 34E15, 03H05.
Key Words: singular perturbations; deformations;
           asymptotic stability; nonstandard analysis.