Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank $k$

In this, paper, we give a complete modulus for germs of generic unfoldings of nonresonant linear differential systems with an irregular singularity of Poincaré rank $k$ at the origin, under analytic equivalence. The modulus comprises a formal part depending analytically on the parameters which, for generic values of the parameters, is equivalent to the set of eigenvalues of the residue matrices of the system at the Fuchsian singular points. The analytic part of the modulus is given by unfoldings of the Stokes matrices. For that purpose, we cover a fixed neighbourhood of the origin in the variable with sectors on which we have an almost unique linear transformation to a (diagonal) formal normal form. The comparison of the corresponding fundamental matrix solutions yields the unfolding of the Stokes matrices. The construction is carried on sectoral domains in the parameter space covering the generic values of the parameters corresponding to Fuchsian singular points.