ON DELAYED OSCILLATION IN NONSPATIALLY UNIFORM FITZHUGH-NAGUMO EQUATION

We study the problem of the slow passage through a Hopf bifurcation po int for the FitzHugh Nagumo equation (FHN) upsilon(t) = Dv(xx) - f(v)- w + phi(x)(I(i) + epsilont) (0.1a) w(t) = bv - bgammaw, (0.1b) where f has some properties so that the system has a Hopf bifurcation at I = I-when epsilon = 0 and I = I(i) + epsilont is regarded as a parameter independent of t. The experimental results of E. Jackobsson and R. Gu ttman (1981, in ''Biophysical Approach to Excitable Systems,'' Plenum, New York) showed that large amplitude oscillations occurred only afte r I reached a value well above I- when epsilon is positive and small. S. M. Baer, T. Erneux, and J. Rinzel (1989, SIAM Appl. 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 Equations 105, 180-215; 1990, ''Delayed Oscillation Phenomena in FitzH ugh Naguma Equation,'' Ph.D. thesis) proved the delayed oscillation ph enomena when phi(x) = 1. In this work, we show that delayed oscillatio ns occur when epsilon is small enough for any phi(x) > 0. (C) 1994 Aca demic Press, Inc.