INVARIANT-MANIFOLDS AND SINGULARLY PERTURBED BOUNDARY-VALUE-PROBLEMS

Boundary value problems for singularly perturbed equations often have singular solutions with internal and boundary layers. This paper addre sses the question of when singular solutions to such problems correspo nd to nearby actual ones. We give a geometric method based on one vers ion of the ''exchange lemma,'' which is used to track an invariant man ifold (such as the manifold of solutions satisfying one set of boundar y conditions) as it passes close to a slow manifold. The estimates pro vided by the lemma are used to establish transversality between the tw o manifolds of solutions each satisfying one of the two sets of bounda ry conditions. The authors also discuss other uses of the exchange lem ma.