Delayed Oscillation Phenomena in the FitzHugh Nagumo Equation

AbstractWe study the problem of the slow passage through a Hopf bifurcation point for the FitzHugh Nagumo equation (FHN) vt = Dvxx − ƒ(v) − w + I0 + ϵt, (0.1a)wt = bv − bγw, (0.1b) where ƒ has some properties so that the system has a Hopf bifurcation at I = I− when ϵ = 0 and I = I0 + ϵt is regarded as a parameter independent of t. The experimental results of Jakobsson and Guttman showed that large amplitude oscillations occurred only after I reaches a value well above I− when ϵ is positive and small. The paper of S. M. Baer, T. Erneux, and J. Rinzel (Siam J. Appl. Math.49, 1989, 55-71) studied these phenomena numerically, and produced a prediction of the ignition (jumping) time for the system. In this work, we provide a rigorous proof of the results conjectured by Baer, Erneux, and Rinzel (referenced above). We show that if we start the solution of (0.1) at any point near the frame solution, which is the zero of the right-hand side of (0.1), at any Ii < I−, then the solution stays near the frame solution until I reaches some Iq < I−. Furthermore, for those cases in which Ii is close to I−, we show that the solution moves from the frame solution to become a large amplitude solution after I moves above some Iq < I−