On the distance between separatrices for the discretized pendulum equation

TL;DR

This paper analyzes the effect of discretization on the pendulum equation, showing that the stable and unstable manifolds of the discretized system do not coincide, with their distance being exponentially small but non-zero.

Contribution

It provides an asymptotic estimate of the non-zero distance between manifolds in the discretized pendulum, using a novel method based on formal series and coefficient estimates.

Findings

The stable and unstable manifolds do not coincide after discretization.

The distance between manifolds is exponentially small but non-zero.

The method differs from previous approaches by employing formal series and precise coefficient bounds.

Abstract

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.