Exchange Lemmas for Singular Perturbation Problems with Certain Turning Points

In this work, singular perturbation problems with a certain type of turning point are studied from a geometric point of view. We first describe the delay phenomenon, initially studied by Pontryagin, of dynamics near the slow manifold possessing the turning points. Based on the local properties of such turning points and the center manifold theory for general invariant manifolds developed by Chow, Liu, and Yi, we extend the well-known exchange lemma, first formulated by Jones and Kopell for problems with normally hyperbolic slow manifolds, to problems with this type of turning point. Applications to singular boundary value problems with turning points are discussed