A Primer on the Exchange Lemma for Fast-Slow Systems

In this primer, we give a brief overview of the Exchange Lemma for fast-slow systems of ordinary differential equations. This Lemma has proven to be a useful tool for establishing the existence of homoclinic and heteroclinic orbits, especially those with many components or jumps, in a variety of traveling wave problems and perturbed near-integrable Hamiltonian systems. It has also been applied to models in which periodic orbits and solutions of boundary value problems are sought, including singularly perturbed two-point boundary value problems. The Exchange Lemma applies to fast-slow systems that have normally hyperbolic invariant manifolds (that are usually center manifolds). It enables one to track the dynamics of invariant manifolds and their tangent planes while orbits on them are in the neighborhood of a normally hyperbolic invariant manifold. The end result is a closeness estimate in the C1 topology of the tracked manifold to a certain submanifold of the normally hyperbolic’s local unstable manifold. We review the general version of the Exchange Lemma due to Tin [24] that treats problems in which there is both fast and slow evolution on the center manifolds. The main normal form used in the neighborhoods of the invariant manifolds is obtained from the persistence theory for normally hyperbolic invariant manifolds due to Fenichel [5, 6, 7]. The works of Jones and Tin [11, 15, 24] form the basis for this work, and [15, 24] contain full presentations of all of the results stated here.