On Borel summation and Stokes phenomena of nonlinear differential systems

Abstract

In this paper we study analytic (linear or) nonlinear systems of ordinary differential equations, at an irregular singularity of rank one, under nonresonance conditions. It is shown that the formal asymptotic exponential series solutions (transseries solutions: countable linear combinations of formal power series multiplied by small exponentials) are Borel summable in a generalized sense along any direction in which the exponentials decay. Conversely, any solution that decreases along some direction is the Borel sum of a transseries. The summation procedure introduced is an extension of Borel summation which is linear, multiplicative, commutes with differentiation and complex conjugation. The summation algorithm uses the formal solutions alone (and not the differential equation that they solve). Along singular (Stokes) directions, the functions reconstructed by summation are shown to be given by Laplace integrals along special paths, a subset of Ecalle's median paths. The one-to-one correspondence established between actual solutions and generalized Borel sums of transseries is constant between Stokes lines and changes if a Stokes line is crossed (local Stokes phenomenon). We analyze the connection between local and classical Stokes phenomena.