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On the existence of homoclinic and heteroclinic orbits for differential equations with a small parameter

Published online by Cambridge University Press:  16 July 2009

John G. Byatt-Smith
John G. Byatt-Smith
Affiliation:
Department of Mathematics, James Clerk Maxwell Building, King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK
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Abstract

Low order differential equations typically have solutions which represent homoclinic or heteroclinic orbits between singular points in the phase plane. These orbits occur when the stable manifold of one singular point intersects or coincides with its unstable manifold, or the unstable manifold of another singular point. This paper investigates the persistence of these orbits when small dispersion is added to the system. In the perturbed system the stable manifold of a singular point passes through an exponentially small neighbourhood of a singular point and careful analysis is required to determine whether a homoclinic or heteroclinic connection is achieved.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 2 , Issue 2 , June 1991 , pp. 133 - 158
Copyright
Copyright © Cambridge University Press 1991

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