FAST AND SLOW WAVES IN THE FITZHUGH-NAGUMO EQUATION

It is known that the FitzHugh-Nagumo equation possesses fast and slow travelling waves. Fast waves are perturbations of singular orbits cons isting of two pieces of slow manifolds and connections between them, w hereas slow waves are perturbations of homoclinic orbits of the unpert urbed system. We unfold a degenerate point where the two types of sing ular orbits coalesce forming a heteroclinic orbit of the unperturbed s ystem. Let c denote the wave speed and e the singular perturbation par ameter. We show that there exists a C-2 smooth curve of homoclinic orb its of the form (c, e(c)) connecting the fast wave branch to the slow wave branch. Additionally we show that this curve has a unique non-deg enerate maximum. Our analysis is based on a Shilnikov coordinates resu lt, extending the Exchange Lemma of Jones and Kopell. We also prove th e existence of inclination-flip points for the travelling wave equatio n thus providing the evidence of the existence of n-homoclinic orbits (n-pulses for the FitzHugh-Nagumo equation) for arbitrary n. (C) 1997 Academic Press