Stokes-Ramis matrices and connection constants for meromorphic linear differential systems with a single level: a perturbative approach

In the articleMatrices de Stokes-Ramis et constantes de connexion pour les systèmes différentiels linéaires de niveau unique (P. Remy), we considered linear differential systems with a unique but arbitrary level and we stated formulae to express all the Stokes multipliers in terms of connection constants in the Borel plane generalizing thus the calculations made in the article Resurgence, Stokes phenomenon and alien derivatives for level-one linear differential systems (M. LodayRichaud, P. Remy). In the present paper, we provide a new proof of these formulae. We perturb the given system in order that each Stokes value generate its own anti-Stokes direction. We state the connectionto-Stokes formulae for the perturbed system and we conclude by a limit process. We believe the method could provide an effi cient tool for the numerical calculation of the Stokes multipliers. As an illustration, we develop an example. No assumption of genericity is made.