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In this, paper, we give a complete system of analytic invariants for the unfoldings of nonresonant linear differential systems with an irregular singularity of Poincaré rank 1 at the origin over a fixed neighborhood Dr. The unfolding parameter ǫ is taken in a sector S pointed at the origin of opening larger than 2π in the complex plane, thus covering a whole neighborhood of the origin. For each parameter value ǫ ∈ S, we cover Dr with two sectors and, over each sector, we construct a well chosen basis of solutions of the unfolded linear differential systems. This basis is used to find the analytic invariants linked to the monodromy of the chosen basis around the singular points. The analytic invariants give a complete geometric interpretation to the well-known Stokes matrices at ǫ = 0: this includes the link (existing at least for the generic cases) between the divergence of the solutions at ǫ = 0 and the presence of logarithmic terms in the solutions for resonance values of the unfolding parameter. Finally, we give a realization theorem for a given complete system of analytic invariants satisfying a necessary and sufficient condition, thus identifying the set of modules.

In the paper [8], we have identified the moduli space of generic unfoldings of linear differential systems with a nonresonant irregular singularity of Poincaré rank 1 for classification under analytic equivalence. The modulus of the unfolding of a linear differential system is the unfolding of the modulus of the system. It consists in formal invariants and an unfolding of the Stokes matrices. In the realization part, we have identified the realizable moduli. However, the necessary and sufficient condition for realizing unfoldings of Stokes matrices was quite obscure. In this paper we explore this condition and we determine the realizable moduli depending analytically on the parameter in dimensions 2 and 3. In dimension 2, all realizable unfoldings of Stokes matrices can be chosen depending analytically on the parameter. In dimension 3, not all pairs of Stokes matrices have realizable analytic unfoldings.

In this paper we study the confluence of two regular singular points of the hypergeometric equation into an irregular one. We study the consequence of the divergence of solutions at the irregular singular point for the unfolded system. Our study covers a full neighborhood of the origin in the confluence parameter space. In particular, we show how the divergence of solutions at the irregular singular point explains the presence of logarithmic terms in the solutions at a regular singular point of the unfolded system. For this study, we consider values of the confluence parameter taken in two sectors covering the complex plane. In each sector, we study the monodromy of a first integral of a Riccati system related to the hypergeometric equation. Then, on each sector, we include the presence of logarithmic terms into a continuous phenomenon and view a Stokes multiplier related to a 1-summable solution as the limit of an obstruction that prevents a pair of eigenvectors of the monodromy operators, one at each singular point, to coincide.

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