Overstable analytic solutions for non-linear systems of difference equations with small step size containing an additional parameter

Abir El Rabih , Reinhard Schäfke (1)

1 IRMA - Institut de Recherche Mathématique Avancée

We consider non linear systems of difference equations with smallstep size containing an additional parameter "a". Under the assumption that the Jacobian of our system is not invertible at some point x=x_0 and some transversality condition, we show the existence of a function "a" having a Gevrey-1 asymptotic expansion and for which the system admits an analytic solution bounded on a certain neighborhood of (0,x_0). This solution is indeed exponentially close to a quasi-solution obtained from a formal solution of the system. We also show the exponential closeness of any two quasi-solutions and hence of any two overstable solutions. Finally, as an application to this theory, we give two numerical examples with plots of canard solutions of some difference equations of second order.

Mots clés: difference equation; difference operator; inverse operator; fixed point theorem; Gevrey asymptotic; formal solution; quasi-solution; overstable solution; Nagumo norm; weighted norm; Borel-Laplace transform; Gronwall lemma

Domaines: Analyse classique [math.CA]

Abir El Rabih, Reinhard Schäfke. Overstable analytic solutions for non-linear systems of difference equations with small step size containing an additional parameter. Journal of Difference Equations and Applications, 2005, 11 (3), pp.183--213. ⟨hal-00144866⟩