Periodic orbits in singularly-perturbed systems

In this paper, we discuss a class of finite-dimensional singularly-perturbed ordinary differential equations. We present a technique that simultaneously yields the existence and local uniqueness of periodic orbits, as well as detailed information about their location in phase space and their stability type. In order to illustrate the method, and due to space constraints, we focus here on two simple example problems. The orbits include resonant multiple-pulse subharmonics and multiple-pulse periodic orbits, which have finitely many rapid transition layers in between successive passages (either long or short) near slow manifolds. The results for general systems, including systems with small or large dissipation, will be presented in [11]. One of the main technical tools is version of the Exchange Lemma with Exponentially Small Error, developed recently in [8], modified to treat the case of periodic orbits.