On Borel summation and Stokes phenomena for rank one nonlinear systems of ODE’s

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 Écalle’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. We study the analytic properties of the Borel (formal inverse Laplace) transform of the series contained in the transseries of the transseries and give a systematic description of their singularities. These Borel transforms satisfy a hierarchy of convolution equations, for which we give the general solution in a space of hyperfunctions. In addition, we show that they are resurgent functions in the sense of Écalle. The summation procedure is not unique; we classify all proper extensions of Borel summation to transseries solutions of nonresonant systems. We find formulas connecting the different series contained in the transseries among themselves (resurgence equations). Resurgence turns out to be closely linked to the local Stokes phenomenon.