On the distance between separatrices for the discretized logistic differential equation

TL;DR

This paper analyzes the discretized logistic differential equation, showing that the stable and unstable manifolds of saddle points do not coincide, with their distance being exponentially small but asymptotically estimated.

Contribution

It provides an asymptotic estimate of the exponentially small distance between manifolds in a discretized logistic system, using formal series and coefficient estimates.

Findings

The stable and unstable manifolds do not coincide in the discretized system.

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

An asymptotic estimate of this distance is derived.

Abstract

We consider the discretization y(t+\epsilon)=y(t-\epsilon)+2\epsilon\big(1-y(t)^{2}\big), $\epsilon>0$ 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 Sch\"afke-Volkmer \cite{SV} using formal series and accurate estimates of the coefficients.