$F$ at $\left(  0\text{,}0\right)  $, $\left(  0,2\pi\right)  $, $\left(  2\pi,0\right)  $ and $\left(  2\pi,2\pi\right)  $ is, for all of them, the following $$\left[
\begin{array}
[c]{cc}%
1+3\varepsilon & 0\\
0 & 1+3\varepsilon
\end{array}
\right]  ,$$ which qualifies all the vertexes of $S$ as a repellers.

On the vertical edges of $S$ we have the fixed points $\left(  0,\pi\right)
$, and $\left(  2\pi,\pi\right)  $, at which the Jacobian matrix of $F$ is $$\left[
\begin{array}
[c]{cc}%
1+\varepsilon & 0\\
2 \varepsilon & 1-3 \varepsilon
\end{array}
\right]  ,$$ which qualifies $\left(  0,\pi\right)  $, and $\left(  2\pi,\pi\right)  $ as saddle points. The stable manifold has the direction of the $y$ axis and the unstable manifold is tangent at $\left(  0\text{,}\pi\right)  $ and $\left(
2\pi,\pi\right)  $ to the vector $\left(  2,1\right)  $.

Finally, at the horizontal edges of $S$ we have the Jacobian matrix of $F$ at $\left(  \pi,0\right)  $, and $\left(  \pi,2\pi\right)  $ $$\left[
\begin{array}
[c]{cc}%
1-3 \varepsilon & 2 \varepsilon\\
0 & 1+ \varepsilon
\end{array}
\right]  ,$$ which qualifies $\left(  \pi,0\right)  $ and $\left(  \pi,2\pi\right)  $ again as saddle points. The stable manifold is the direction of the $x$ axis and the unstable manifold is tangent at $\left(  \pi,0\right)  $ and $\left(  \pi
,2\pi\right)  