  We consider the discretization
  y(t+\epsilon)=y(t-\epsilon)+2\epsilon\big(1-y(t)^{2}\big),
  $\epsilon>0$ a small parameter, of the logistic differential equation
$y'=2\big(1-y^{2}\big)$, which can also be seen as a discretization of the
system
  {y'=2\big(1-v^{2}\big),
  v'= 2\big(1-y^{2}\big).
  This system has two saddle points at $A=(1,1)$, $B=(-1, -1)$ and there exist
stable and unstable manifolds. We will show that the stable manifold
$W_{s}^{+}$ of the point $A=(1,1)$ and the unstable manifold $W_{i}^{-}$ of the
point $B=(-1, -1)$ for the discretization do not coincide. The vertical
distance between these two manifolds is exponentially small but not zero, in
particular we give an asymptotic estimate of this distance. For this purpose we
will use a method adapted from the paper of Sch\"afke-Volkmer \cite{SV} using
formal series and accurate estimates of the coefficients.

Dynamical Systems