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

1991; Cambridge University Press; Volume: 2; Issue: 2 Linguagem: Inglês

10.1017/s0956792500000449

1469-4425

J. G. B. Byatt‐Smith,

Advanced Differential Equations and Dynamical Systems

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.