  We consider the discretization
  q(t+\epsilon)+q(t-\epsilon)-2q(t)=\epsilon^{2}\sin\big(q(t)\big),
  $\epsilon>0$ a small parameter, of the pendulum equation $ q '' = \sin (q) $;
in system form, we have the discretization
  q(t+\epsilon)-q(t)=\epsilon p(t+\epsilon),
p(t+\epsilon)-p(t)=\epsilon\sin\big(q(t)\big).
  of the system
  q'=p, p'=\sin(q).
  The latter system of ordinary differential equations has two saddle points at
$A=(0,0)$, $B=(2\pi, 0)$ and near both, there exist stable and unstable
manifolds. It also admits a heteroclinic orbit connecting the stationary points
$B$ and $A$ parametrised by $q_0(t)=4\arctan\big(e^{-t}\big)$ and which
contains the stable manifold of this system at $A$ as well as its unstable
manifold at $B$. We prove that the stable manifold of the point $A$ and the
unstable manifold of the point $B$ do not coincide for the discretization. More
precisely, we show that the vertical distance between these two manifolds is
exponentially small but not zero and in particular we give an asymptotic
estimate of this distance. For this purpose we use a method adapted from the
article of Sch\"afke-Volkmer \cite{SV} using formal series and accurate
estimates of the coefficients. Our result is similar to that of Lazutkin et.
al. \cite{LS}; our method of proof, however, is quite different.
