Resurgence, Stokes phenomenon and alien derivatives for level-one linear differential systems

TL;DR

This paper provides a detailed analysis of the singularities of Borel transforms for solutions of level-one linear differential systems, connecting resurgence theory, Stokes phenomena, and alien derivatives without assuming generic conditions.

Contribution

It offers a precise description of singularities, compares Stokes and alien derivative viewpoints, and explicitly computes Stokes-Ramis matrices for specific examples.

Findings

Explicit description of Borel transform singularities

Comparison of Stokes cocycle and alien derivatives

Explicit Stokes-Ramis matrices for examples

Abstract

A precise description of the singularities of the Borel transform of solutions of a level-one linear differential system is deduced from a proof of the summable-resurgence of the solutions by the perturbative method of J. \'Ecalle. Then we compare the meromorphic classification (Stokes phenomenon) from the viewpoint of the Stokes cocycle and the viewpoint of alien derivatives. We make explicit the Stokes-Ramis matrices as functions of the connection constants in the Borel plane and we develop two examples. No assumption of genericity is made.