| We consider the discretization \begin{equation*} y(t+\varepsilon)=y(t-\varepsilon)+2\varepsilon\big(1-y(t)^{2}\big), \end{equation*} $\varepsilon>0$ 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 \begin{eqnarray*} \begin{cases} y'=2\big(1-v^{2}\big),\\ v'= 2\big(1-y^{2}\big). \end{cases} \end{eqnarray*} 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 distance in the sense of Hausdorff 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 Schäfke-Volkmer \cite{SV} using formal series and accurate estimates of the coefficients. |
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