On the distance between separatrices for the discretized pendulum equation
Résumé
We consider the discretization \begin{equation*} q(t+\varepsilon)+q(t-\varepsilon)-2q(t)=\varepsilon^{2}\sin\big(q(t)\big), \end{equation*} $\varepsilon>0$ a small parameter, of the pendulum equation $ q '' = \sin (q) $; in system form, we have the discretization \begin{equation*} q(t+\varepsilon)-q(t)=\varepsilon p(t+\varepsilon), \ p(t+\varepsilon)-p(t)=\varepsilon\sin\big(q(t)\big). \end{equation*} of the system \begin{equation*} q'=p,\ p'=\sin(q). \end{equation*} 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 Schäfke-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.
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