On Delayed Oscillation in Nonspatially Uniform Fitzhugh Nagumo Equation

We study the problem of the slow passage through a Hopf bifurcation point for the FitzHugh Nagumo equation (FHN) vt = Dvxx − f(v) − w + φ(x)(Ii + εt) (0.1a)wt = bv − bγw, (0.1b) where f has some properties so that the system has a Hopf bifurcation at I = I− when ε = 0 and I = Ii + εt is regarded as a parameter independent of t. The experimental results of E. Jackobsson and R. Guttman (1981, in "Biophysical Approach to Excitable Systems," Plenum, New York) showed that large amplitude oscillations occurred only after I reached a value well above I− when ε is positive and small. S. M. Baer, T. Erneux, and J. Rinzel (1989, SIAMAppl. Math.49, 55-71) studied these phenomena numerically and produced a prediction of the ignition (jumping) time for the system. J. Su (1993, J. Differential Equations105, 180-215; 1990, "Delayed Oscillation Phenomena in FitzHugh Naguma Equation," Ph.D. thesis) proved the delayed oscillation phenomena when φ(x) = 1. In this work, we show that delayed oscillations occur when ε is small enough for any φ(x) > 0.