离散Logistic微分方程的分离点距离
On the distance between separatrices for the discretized logistic differential equation
摘要：
考虑Logistic微分方程$y'=2\big(1-y^{2}\big)$的离散化y（t+ε）=y（t-ε）+2ε \big（1-y（t）^{2} \big），$ε>0$是小参数，它也可以看作是系统{y '= 2 \big（1-v^{2} \big），v'= 2 \big（1-y^{2} \big）的离散化.该系统在$A=(1,1)$，$B=(-1, -1)$有两个鞍点，存在稳定流形和非稳定流形.我们将证明点$A=(1,1)$的稳定流形$W_{s}^{+}$和点$B=(-1, -1)$的不稳定流形$W_{i}^{-}$对于离散化并不重合。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。为此目的，我们将使用一种方法改编自文件的Schäfke-Schummer \cite{SV}使用正式的系列和准确的估计系数。
英文摘要：
We consider the discretization y(t+ε)=y(t-ε)+2ε\big(1-y(t)^{2}\big), $ε>0$ 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 Schäfke-Volkmer \cite{SV} using formal series and accurate estimates of the coefficients.
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