Chapter 7.3, Problem 10P

The system

$\begin{array}{cc}x\text{'}=-y& y\text{'}=-\gamma y-x(x-0.15)(x-3)\end{array}$

Results from an approximation to the Hodgkin-Huxley equations, which model the transmission of neural impulses along an axon.

(a) Find the critical points and classify them by investigating the approximate linear system near each one.

(b) Draw phase portraits for $\gamma =0.8$ and for $\gamma =1.5$

(c) Consider the trajectories that leaves the critical points $(2,0).$

Find the value of $\gamma$ for which this trajectory ultimately approaches the origin as $t\to \infty$. Draw a phase portrait for this value of $\gamma$.