Abstract

The use of power series is very common in applied mathematics: periodic orbits of vector fields, stable manifolds of hyperbolic points, solutions near fixed points, ... In case that they are divergent, we still can use them to obtain some information. The Borel transform allows us to associate an analytic germ to a divergent series and, depending on its analytic continuation, the Borel-Laplace resummation yields di®erent functions that are asymptotic to the initial series. Jean Ecalle introduced the so-called Resurgence Theory to use in these situations. It consists of an algebra of functions and a differential calculus (the alien derivations), that allows to follow the different resummations of a series and, for example, compute the difference between them. This theory has been applied recently to several problems and it has been very useful to understand exponentially small phenomena. In this talk, we are going to show in a very briefly way two different problems where this theory is applied.