On the distance between separatrices for the discretized logistic differential equation

Hocine Sellama (1)

1 IRMA - Institut de Recherche Mathématique Avancée

We consider the discretization \begin{equation*} y(t+\varepsilon)=y(t-\varepsilon)+2\varepsilon\big(1-y(t)^{2}\big), \end{equation*} $\varepsilon>0$ 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 \begin{eqnarray*} \begin{cases} y'=2\big(1-v^{2}\big),\\ v'= 2\big(1-y^{2}\big). \end{cases} \end{eqnarray*} 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.

Mots clés: Difference equation; Manifolds; Linear operator; Formal solution; Gevrey asymptotic; Quasi-solution

Domaines: Systèmes dynamiques [math.DS]

Hocine Sellama. On the distance between separatrices for the discretized logistic differential equation. 2007. ⟨hal-00143589v2⟩