关于离散化摆方程的分界线距离
On the distance between separatrices for the discretized pendulum equation
摘要：
考虑摆方程$ q '' = \sin (q) $的离散化q（t+ε）+q（t-ε）-2q（t）=ε^{2} \sin \big（q（t）\big），$ε>0$为小参数;在系统形式下，离散化q（t+ε）-q（t）=εp（t+ε），p（t+ε）-p（t）=ε \sin \big（q（t）\big）.系统q '= p，p'= \sin（q）.后一类常微分方程组在$A=(0,0)$和$B=(2π, 0)$处有两个鞍点，在这两个鞍点附近存在稳定流形和不稳定流形。它也允许一个异宿轨道连接由$q_0(t)=4\arctan\big(e^{-t}\big)$参数化的驻点$B$和$A$，它包含该系统在$A$的稳定流形以及在$B$的不稳定流形。我们证明了点$A$的稳定流形和点$B$的不稳定流形对于离散化是不重合的。更确切地说，我们表明，这两个流形之间的垂直距离是指数小，但不为零，特别是我们给这个距离的渐近估计。为此，我们使用的方法改编自文章的Schäfke-Schummer \cite{SV}使用正式系列和准确的估计系数。我们的结果与Lazutkin et. \cite{LS} ;然而，我们的证明方法却完全不同。
英文摘要：
We consider the discretization q(t+ε)+q(t-ε)-2q(t)=ε^{2}\sin\big(q(t)\big), $ε>0$ a small parameter, of the pendulum equation $ q '' = \sin (q) $; in system form, we have the discretization q(t+ε)-q(t)=εp(t+ε), p(t+ε)-p(t)=ε\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 $A=(0,0)$, $B=(2π, 0)$ 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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