Claude Lobry, Tewfik Sari, & Sefiane Touhami
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.
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Tewfik Sari
Laboratoire de Mathematiques, Universite de Haute Alsace
4, rue des Fr\`eres Lumiere
68093 Mulhouse, France.
e-mail: T.Sari@univ-mulhouse.fr
Sefiane Touhami
Laboratoire de Mathematiques, Universite de Haute Alsace
4, rue des Fr\`eres Lumiere
68093 Mulhouse, France.
e-mail: S.Touhami@univ-mulhouse.fr