edges of $S$.

We now compute the Jacobian matrix $J\left(  x,y\right)  $ to establish the dynamical nature of the fixed points in the usual way.

We have $$J\left(  x,y\right)  =\left[
\begin{array}
[c]{cc}%
1 & 0\\
0 & 1
\end{array}
\right]  + \varepsilon\left[
\begin{array}
[c]{cc}%
2\cos x+\cos\left(  x-y\right)  & -\cos\left(  x-y\right)  +\ \cos y\\
\cos x-\cos\left(  x-y\right)  & \cos\left(  x-y\right)  +2\cos y
\end{array}
\right]  . \label{Jacobmatrix}%$$

We first consider the fixed points of $F$ in the interior of $S$. We start with $\left(  \frac{2}{3} \pi,\frac{4}{3} \pi\right)  $ and $\left(  \frac
{4}{3} \pi,\frac{2}{3} \pi\right)  $. The Jacobian is exactly the same $$\left[
\begin{array}
[c]{cc}%
1-3\frac{\sqrt{3}}{2}\varepsilon & 0\\
0 & 1-3\frac{\sqrt{3}}{2}\varepsilon
\end{array}
\right]  ,$$ meaning that those two points are locally asymptotically stable for $\varepsilon$ sufficiently small.

The Jacobian matrix of $F$ at $\left(  \pi\text{,}\pi\right)  $ is $$\left[
\begin{array}
[c]{cc}%
1-\varepsilon & -2\varepsilon\\
-2\varepsilon & 1-\varepsilon
\end{array}
\right]  ,$$ with eigenvalues $1-3\varepsilon$ and $1+\varepsilon$, which qualifies $\left(  \pi\text{,}\pi\right)  $ as a saddle point. The stable manifold has direction $\left(  1,1\right)  $, and the unstable manifold is tangent at $\left(  \pi\text{,}\pi\right)  $ to the vector $\left(  -1,1\right)  $.

We now consider now the points placed at the vertexes of $S$. The Jacobian matrix of 