<jupyter_start><jupyter_text><NAME>\begin{align}  \varepsilon \dot{x} &= x(-a - x)(x - a) - y \\  \dot{y} &= x - x_{c}\end{align}At the fixed point, the Jacobian Matrix is\begin{equation}J = \frac{1}{\varepsilon}  \begin{bmatrix}    a^2 - 3 x_c^2 & -1 \\    \varepsilon & 0   \end{bmatrix}\end{equation}We have\begin{equation}{\rm Tr}(J) = \frac{a^2 - 3 x_c^2}{\varepsilon}, \quad\vert J \vert = \frac{1}{\varepsilon}\end{equation}The eigenvalues are\begin{equation}  \lambda_{\pm} = \frac{1}{2\varepsilon}\left[a^2 - 3x_c^2 \pm \sqrt{\left(a^2 - 3x_c^2\right)^2 - 4\varepsilon}\right]\end{equation}The value of $x_c$ that causes a change in stability is $$ x_c = \pm \frac{a}{\sqrt{3}}$$<jupyter_code>params = {
    'I': -0.25,
    'xc': -0.8/sqrt(3.) + 0.9, 
    'epsilon': 0.02,
    'a': 0.8}

model = FHNmodel(**params)
y0 = (-0.25, -0.45)
tpoints = linspace(0, 6, 300)
sol = model.gslint(tpoints, y0)

fig = figure(1, [15, 5])

fig.add_subplot(121)
Xnc = linspace(-1, 1, 100)
Ync = model.nullclines(Xnc)
plot(Xnc, Ync[0], 'orange', lw=2, label='$\dot{x}=0$')
plot(Xnc, Ync[1], 'c', lw=2, label='$\dot{y}=0$')
plot(model.xc, model.yc, 'ko')

plot(sol['x'][0], sol['y'][0], 'kx')
plot(sol['x'], sol['y'], '--k')
ylim(-0.5, 0.3)
xlabel('$x$', fontsize=24)
ylabel('$y$', fontsize=24)
legend(fontsize=24);

fig.add_subplot(122)
plot(sol['t'], sol['x'], label='$x(t)$')
plot(sol['t'], sol['y'], label='$y(t)$')
xlabel('$t$', fontsize=24)
legend(fontsize=24);<jupyter_output><empty_output>
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