Bifurcations of Relaxation oscillations Near Folded saddles

Relaxation oscillations are periodic orbits of multiple time scale dynamical systems that contain both slow and fast segments. The slow-fast decomposition of these orbits is defined in the singular limit. Geometric methods in singular perturbation theory classify degeneracies of these decompositions that occur in generic one parameter families of relaxation oscillations. This paper investigates the bifurcations that are associated with one type of degeneracy that occurs in systems with two slow variables, namely orbits that become homoclinic to a folded saddle.