Delayed bifurcation properties in the FitzHugh-Nagumo equation with periodic forcing

Abstract

We consider the initial value problems of the FitzHugh-Nagumo Equation (FHN) $$ \begin{align} \frac{ \partial v}{\partial t} & = Dv_{xx}-(v-a)v(v-1)-w+(I_i+\epsilon t)+g_1(I_i+ \epsilon t,t)\epsilon ^{\alpha +1} \quad \text{ for } 0<x <1,\ t>0 \\ \frac{ \partial w}{\partial t} & = bv-b\gamma w +g_2(I_i+\epsilon t,t)\epsilon ^{\alpha +1} \quad \text{ for } 0<x <1,\ t>0 \\ \frac{\partial v}{ \partial x} & = 0,\ \ \ \frac{\partial w}{ \partial x}=0 \quad \text{ for }x=0,1,\ t>0 \tag 0.1 \end{align} $$ where $\alpha \geq 0,$ $g_1$ and $g_2$ are analytic in both variables and periodic in $t$ with the period ${{2\pi }\over{\omega }}.$ We are interested in the delayed bifurcation phenomena of the system where the solutions have the property that while the linear stability of the solutions changes from stable to unstable after a critical point, the solutions remain dynamically stable (stay close to unstable solutions) until a point well above the critical point. In this work, we consider the delayed bifurcations in the FitzHugh-Nagumo Equation with the presence of some periodic forcing which models the membrane activities of the giant axon of a squid under certain external periodic influences. We show that if the frequency $\omega $ of $g_1,g_2$ and the Hopf bifurcation frequency of the FHN system are not in the ratio of 2:$n,$ then the delayed bifurcations occur.