STABILITY ANALYSIS FOR THE SLOW TRAVELING PULSE OF THE FITZHUGH-NAGUMO SYSTEM

This paper is concerned with the existence and stability of the slow traveling pulse for the FitzHugh-Nagumo system u(t) = u(xx) + u(1-u)(u-a)-w, w(t) = epsilon(u-gamma-w). This traveling wave is obtained as a perturbation of the standing wave of the Nagumo equation u(t) = u(xx) + u(1-u)(u-a). Its existence is established by analyzing how the unstable manifold of the origin exits a suitable block. This geometric proof is an alternative approach to the singular perturbation expansion proposed by Casten, Cohen, and Lagerstrom [Quart. Appl. Math., 32 (1975), pp. 335-367], as well as to the existence proof of Hastings [SIAM J. Appl. Math., 42 (1982), pp. 247-260]. The method also allows use of the techniques developed by Evans [Indiana Univ. Math. J., 24 (1985), pp. 193-226] to analyze the spectrum of the variational equation around the traveling wave. It is shown that there is exactly one unstable mode.