Associated to analytic Hamiltonian vector fields in ${ℂ}^4$ having an equilibrium point satisfying a non semisimple $1:-1$ resonance, we construct two constants that are invariant with respect to local analytic symplectic changes of coordinates. These invariants vanish when the Hamiltonian is integrable. We also prove that one of these invariants does not vanish on an open and dense set.

This is an expository article, based on the talk with the same title, given at the 2011 FASDE II Conference in Będlewo, Poland. In the introduction we define Multiple Zeta Values and certain historical remarks are given. Then we present several results on Multiple Zeta Values and, in particular, we introduce certain meromorphic differential equations associated to their generating function. Finally, we make some conclusive remarks on generalisations of Multiple Zeta Values as well as the meromorphic...

In this expository paper we consider various approaches to multisummability. We apply it to nonlinear ODE's and give a somewhat modified proof of multisummability of formal solutions of ODE's with levels 1 and 2 via Écalle's method involving convolution equations.

This paper is divided in two parts. In the first part we study a convergent $q$-analog of the divergent Euler series, with $q\in (0,1)$, and we show how the Borel sum of a generic Gevrey formal solution to a differential equation can be uniformly approximated on a convenient sector by a meromorphic solution of a corresponding $q$-difference equation. In the second part, we work under the assumption $q\in (1,+\infty )$. In this case, at least four different $q$-Borel sums of a divergent power series solution of an irregular singular...

This work is devoted to the study of a Cauchy problem for a certain family of q-difference-differential equations having Fuchsian and irregular singularities. For given formal initial conditions, we first prove the existence of a unique formal power series X̂(t,z) solving the problem. Under appropriate conditions, q-Borel and q-Laplace techniques (firstly developed by J.-P. Ramis and C. Zhang) help us in order to construct actual holomorphic solutions of the Cauchy problem whose q-asymptotic expansion...

On étudie les systèmes différentiels singulièrement perturbés de dimension 3 du type$$\{\begin{array}{ccc}\dot{x}\hfill & =& f(x,y,z,\epsilon ),\hfill \\ \dot{y}\hfill & =& g(x,y,z,\epsilon ),\hfill \\ \epsilon \dot{z}\hfill & =& h(x,y,z,\epsilon ),\hfill \end{array}$$où $f$, $g$, $h$ sont analytiques quelconques. Les travaux antérieurs étudiaient les points réguliers où la surface lente $h=0$ est transverse au champ rapide vertical. C’est le domaine d’application du théorème de Tikhonov. Dans d’autres travaux antérieurs, on étudiait les singularités de certains types : plis et fronces de la surface lente, ainsi que certaines singularités plus compliquées, analogues aux points tournants...

We study the resurgent structure associated with a Hamilton-Jacobi equation. This equation is obtained as the inner equation when studying the separatrix splitting problem for a perturbed pendulum via complex matching. We derive the Bridge equation, which encompasses infinitely many resurgent relations satisfied by the formal solution and the other components of the formal integral.

The Euler-MacLaurin summation formula compares the sum of a function over the lattice points of an interval with its corresponding integral, plus a remainder term. The remainder term has an asymptotic expansion, and for a typical analytic function, it is a divergent (Gevrey-1) series. Under some decay assumptions of the function in a half-plane (resp. in the vertical strip containing the summation interval), Hardy (resp. Abel-Plana) prove that the asymptotic expansion is a Borel summable series,...

Dans cet article, nous établissons le caractère résurgent-sommable de séries formelles ramifiées solutions d’une classe d’équations différentielles linéaires. Nous analysons d’une part le problème de la dépendance analytique des sommes de Borel de telles séries par rapport aux paramètres de cette classe d’équations différentielles linéaires d’ordre deux, et d’autre part, nous analysons la structure résurgente complète associée à ces séries formelles via l’outil des singularités générales (ou microfonctions)....

We study germs of singular holomorphic vector fields at the origin of ${ℂ}^n$ of which the linear part is $1$-resonant and which have a polynomial normal form. The formal normalizing diffeomorphism is usually divergent at the origin but there exists holomorphic diffeomorphisms in some “sectorial domains” which transform these vector fields into their normal form. In this article, we study the interplay between the small divisors phenomenon and the Gevrey character of the sectorial normalizing diffeomorphisms....

Nous abordons dans cet article la question de la sommation effective d’une somme de Borel d’une série par la série de factorielles associée. Notre approche fournit un contrôle de l’erreur entre la somme de Borel recherchée et les sommes partielles de la série de factorielles. Nous généralisons ensuite cette méthode au cadre des séries de puissances fractionnaires, après avoir démontré un analogue d’un théorème de Nevanlinna de sommation de Borel fine pour ce cadre.

This article studies the summability of first integrals of a $C^{\omega }$-non-integrable resonant Hamiltonian system. The first integrals are expressed in terms of formal exponential transseries and their Borel sums. Smooth Liouville integrability and a relation to the Birkhoff transformation are discussed from the point of view of the summability.