Chapter 7.5, Problem 21P

The system

$\begin{array}{l}x\text{'}=3\left(x+y-\frac{1}{3}{x}^3-k\right),\\ y\text{'}=-\frac{1}{3}\left(x+0.8y-0.7\right)\end{array}$

Is a special case of the Fitzhugh-Nagumo equations, which model the transmission of neural impulses along an axon. The parameter $k$ is the external stimulus.

(a) Show that the system has one critical point regardless of the value of $k$.

(b) Find the critical point for $k=0$ and show that it is an asymptotically stable spiral point. Repeat the analysis for $k=0.5$ and show that the critical point is now an unstable spiral point. Draw a phase portrait for the system in each case.

(c) Find the value $k_0$ where the critical point changes from asymptotically stable to unstable. Find the critical point and draw a phase portrait for the system for $k=k_0$.

(d) For $k>k_0$, the system exhibits an asymptotically stable limit cycle; the system has a Hopf bifurcation point at $k_0$. Draw a phase portrait for $k=0.4,0.5,$ and $0.6$. Observe that the limit cycle is not small when $k$ is near $k_0$. Also plot $x$ versus $t$ and estimate the period $T$ in each case.

(e) As $k$ increases further, there is a value $k_1$ at which the critical point again becomes asymptotically stable and the limit cycle vanishes. Find $k_1$