Stokes phenomenon for single-level linear differential systems: a perturbative approach

Given a meromorphic linear differential system with an arbitrary single level $r \geq 1$, we build a regular holomorphic perturbation which preserves the single level and we show that the Stokes-Ramis matrices of the initial system are limits of convenient products of the perturbed one. As an application, we provide an alternative method for the effective calculation of the Stokes multipliers of the initial system illustrated on two examples. No assumption of genericity is made on the initial system