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Singularly perturbed ordinary differential equations with dynamic limits

Published online by Cambridge University Press:  14 November 2011

Zvi Artstein  and
Alexander Vigodner
Zvi Artstein
Affiliation:
Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel
Alexander Vigodner
Affiliation:
Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel
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Extract

Coupled slow and fast motions generated by ordinary differential equations are examined. The qualitative limit behaviour of the trajectories as the small parameter tends to zero is sought after. Invariant measures of the parametrised fast flow are employed to describe the limit behaviour, rather than algebraic equations which are used in the standard reduced order approach. In the case of a unique invariant measure for each parameter, the limit of the slow motion is governed by a chattering type equation. Without the uniqueness, the limit of the slow motion solves a differential inclusion. The fast flow, in turn, converges in a statistical sense to the direct integral, respectively the set-valued direct integral, of the invariant measures.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 3 , 1996 , pp. 541 - 569
Copyright
Copyright © Royal Society of Edinburgh 1996

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