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Singular perturbations on the infinite time interval

Abstract : We consider the slow and fast systems that belong to a small neighborhood of an unperturbed problem. We study the general case where the slow equation has a compact positively invariant subset which is asymptotically stable, and meanwhile the fast equation has asymptotically stable equilibria (Tykhonov’s theory) or asymptotically stable periodic orbits (Pontryagin–Rodygin’s theory). The description of the solutions is by this way given on infinite time interval. We investigate the stability problems derived from this results by introducing the notion of practical asymptotic stability. We show that some particular subsets of the phase space of the singularly perturbed systems behave like asymptotically stable sets. Our results are formulated in classical mathematics. They are proved within Internal Set Theory which is an axiomatic approach to Nonstandard Analysis.
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Karim Yadi. Singular perturbations on the infinite time interval. Revue Africaine de la Recherche en Informatique et Mathématiques Appliquées, INRIA, 2008, 9, pp.537-560. ⟨hal-01277906⟩

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