Tracking Invariant Manifolds without Differential Forms

A frequently studied problem of geometric singular perturbation theory consists in establishing the presence of trajectories of certain types (homoclinic, heteroclinic, satisfying given boundary conditions, etc.) approximating singular ones for the unperturbed problem. A useful tool for this problem has been established in Jones et al. [2] and called “Exchange Lemma” by the authors. It resembles the well known λ-lemma (Palis et al. [4]) with critical elements of a dynamical system replaced by “slow manifolds” of a singularly perturbed differential equation. The degeneration of transversality in the unperturbed equation in important applications lead the authors of Jones et al. [3] establish a more precise version of the Exchange Lemma. The proof of the Exchange Lemma of Jones et al. [2], [3] involves differential equations for the evolution of differential forms of tangent vectors along trajectories. The purpose of this paper is to present an alternative proof which avoids differential forms. We believe that, except of being more elementary, it provides additional insight into the geometry of the problem. We refer the reader to Jones et al. [2], [3] for the motivation and the application of the Exchange Lemma. In order to facilitate the comparison of our result to Jones et al. [2], [3] we use freely their notation whenever possible. As in Jones et al. [2], [3] we consider a singularly perturbed system