A Geometric Method for Periodic Orbits in Singularly-Perturbed Systems

In this work, we establish a modular geometric method to demonstrate the existence of periodic orbits in singularly perturbed systems of differential equations. These orbits have alternating fast and slow segments, reflecting the two time scales in the problems. The method involves converting the periodic orbit problem into a boundary value problem in an appropriately augmented system, and it employs several versions of the exchange lemmas due to Jones, Kopell, Kaper and Tin. It is applicable to models that arise in a wide variety of scientific disciplines, and applications are given to the FitzHugh-Nagumo, Hodgkin-Huxley, and Gray-Scott systems.