Existence of Innitely Many Connecting Orbits in a Singularly Perturbed Ordinary Dierential Equation

We consider a one-parameter family of two-dimensional ordinary di erential equations with a slow parameter drift. Our equation assumes that when there is no parameter drift, there are two invariant curves consisting of equilibria, one of which is normally hyperbolic and whose stable and unstable manifolds intersect transversely. The slow parameter drift is introduced in a way that it creates two hyperbolic equilibria in the invariant normally hyperbolic curve that is persistent under perturbation. In this situation, we prove that the number of distinct orbits which connects these two equilibria changes from nite to in nite depending on the direction of the slow parameter drift. The proof uses the Conley index theory. The relation to a singular boundary value problem studied by W. Kath is, also, discussed. Research was supported in part by NSF Grant INT-9315117 and JSPS US-Japan Cooperative Research MPCR-294. y Research was supported in part by Grant-in-Aid for Scienti c Research (No. 06740150), Ministry of Education, Science and Culture, Japan. z Research was supported in part by NSF Grant DMS-9101412. x Research was supported in part by Science and Technology Fund for Research Grants of Ryukoku University.