Lobry, Claude; Sari, Tewfik; Touhami, Sefiane

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

Electron. J. Differ. Equ. 1998, Paper No. 19, 22 p. (1998)


Summary: 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.

Mathematics Subject Classification

34D15, 34E15, 03H05


singular perturbations, deformations, asymptotic stability, nonstandard analysis