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

Slow and fast systems gain their special structure from the presence of two time scales. Their analysis is achieved with the help of Singular PerturbationTheory. ThefundamentaltoolisTykhonov’stheoremwhich describesthelimitingbehaviour,forcompactintervaloftime, ofsolutions of the perturbedsystem which is a one-parameter deformations of the socalled 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 innite time interval. Our results are formulated in classical mathematics. TheyareprovedwithinInternalSetTheorywhichisanaxiomatic approach to Nonstandard Analysis. "_

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