= Solution
At an <equilibrium> $(v_*,kv_*)$, set $p=f'(v_*)$. The <Jacobian matrix>, its <trace> and its <determinant> are
$$
J_* =\begin{pmatrix}p&-1\\\beta&-\gamma\end{pmatrix},\qquad
T=p-\gamma,\qquad \Delta=\beta-\gamma p=\gamma(k-p).
$$
The <eigenvalues> and their <discriminant> are
$$
s_\pm=\frac{p-\gamma\pm\sqrt{(p+\gamma)^2-4\beta}}2,\qquad
\mathcal D=(p+\gamma)^2-4\beta.
$$
Thus the complete local <linear stability> test is
$$
\boxed{\begin{array}{ll}
p>k:&\text{saddle},\\
p<k,\ p<\gamma:&\text{asymptotically stable},\\
\gamma<p<k:&\text{unstable}.
\end{array}}
$$
In the last two cases, $\mathcal D>0$ gives a <node>, while $\mathcal D<0$ gives a <focus>; the sign of $T$ distinguishes inward and outward trajectories. At $\mathcal D=0$, a nonzero repeated <eigenvalue> gives a degenerate <node>. At $p=\gamma<k$ the <eigenvalues> are purely imaginary, so <linear stability analysis> alone does not decide <asymptotic stability>; with the required crossing and nondegeneracy this is a <Hopf bifurcation>. At $p=k$ there is a zero <eigenvalue>, requiring a nonlinear test and generically marking the <saddle-node bifurcation> in the preceding answer.
For a unique simple <equilibrium>, $p<k$, so it can be a <sink equilibrium> or a <source equilibrium>. The number of <equilibria> alone does not decide <stability>. If $k\le\gamma$, the unique simple <equilibrium> is always a <sink equilibrium>. If $\gamma<k$, it can become a <source equilibrium> where $\gamma<p<k$. The same conclusion follows particularly simply if $\gamma>M$: the <trace> is negative at every nonsaddle <equilibrium>.
For three simple <equilibria> ordered by their $v$ coordinates, the middle one has $p>k$ and is a <saddle equilibrium>; the two outer ones have $p<k$. When both outer <equilibria> are <sink equilibria>, each has an attracting neighborhood, giving <bistability>. In the usual two-basin portrait, the <stable manifold> of the <saddle equilibrium> separates their basins. Each branch of its <unstable manifold> initially leaves towards a different side; its eventual destination requires a global argument. If $k\le\gamma$, both outer <equilibria> must be <sink equilibria>. If $\gamma<k$, either or both outer <equilibria> can instead be <source equilibria>; the same test $p\lessgtr\gamma$ applies separately to each. The three lower panels illustrate all three possibilities up to exchanging left and right.
The arrows in the <phase plane> follow directly from the <nullclines>: below $w=f(v)+I$ the horizontal component points right, and above it left; below $w=kv$ the vertical component points up, and above it down. Near a <stable focus> trajectories spiral inward; near an unstable <focus> they spiral outward. Near a <saddle equilibrium> they approach along its <stable manifold> and leave along its <unstable manifold>. These directions, together with the local tests, determine the sketches.
\Image[/past-exam-of-the-further-mathematical-biology-course-of-the-university-of-oxford/oxford-2023-further-mathematical-biology-phase-portraits.png]
{title=<Phase portraits> of the <FitzHugh-Nagumo model> with one or three <equilibria>, including changes of <linear stability>}
There is also a useful global conclusion when the unique <equilibrium> is a <source equilibrium>. With
$$
E(v,w)=\frac{\beta v^2+w^2}{2},\qquad
\dot E=\beta v f(v)+\beta Iv-\gamma w^2,
$$
the leading term in $v$ is $-\beta v^4$, so sufficiently large ellipses are <trapping regions>. A small repelling neighborhood of the hyperbolic <source equilibrium> can be removed to form an invariant annulus. The <Poincaré-Bendixson theorem> then guarantees at least one <periodic orbit> in that annulus. The numerical sketch illustrates the resulting sustained oscillation; this argument does not assert uniqueness of a <limit cycle>. A stable <equilibrium> can instead represent a resting <membrane potential>, with perturbations producing damped relaxation; near a threshold, a sufficiently large perturbation can produce a large excursion before recovery. Global basin boundaries and coexistence of <periodic orbits> need more information than the number and local <stability> of <equilibria>.
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