Relaxation oscillations in R3

The existence of periodic relaxation oscillations in singularly perturbed systems with two slow and one fast variable is analyzed geometrically. It is shown that near a singular periodic orbit a return map can be defined which has a one-dimensional slow manifold with a stable invariant foliation. Under a natural hyperbolicity assumption on the singular periodic orbit this allows to prove the existence of a periodic relaxation orbit for small values of the perturbation parameter. Additionally the existence of an invariant torus is proved for the periodically forced van der Pol oscillator. The analysis is based on methods from geometric singular perturbation theory. The blow-up method is used to analyze the dynamics near the foldcurves.