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 consisting of two pieces of slow manifolds and connections between them, whereas slow waves are perturbations of homoclinic orbits of the unperturbed system. We unfold a degenerate point where the two types of singular orbits coalesce forming a heteroclinic orbit of the unperturbed system. Letcdenote the wave speed and the singular perturbation parameter. We show that there exists aC2smooth curve of homoclinic orbits of the form (c, (c)) connecting the fast wave branch to the slow wave branch. Additionally we show that this curve has a unique non-degenerate maximum. Our analysis is based on a Shilnikov coordinates result, extending the Exchange Lemma of Jones and Kopell. We also prove the existence of inclination-flip points for the travelling wave equation thus providing the evidence of the existence ofn-homoclinic orbits (n-pulses for the FitzHugh–Nagumo equation) for arbitraryn.