Moscow Mathematical Journal

Volume 14, Issue 2, April–June 2014 pp. 309–338.

Complete System of Analytic Invariants for Unfolded Differential Linear Systems with an Rank k Irregular Singularity of Poincaré

Authors: Jacques Hurtubise (1), Caroline Lambert, and Christiane Rousseau (2)
Author institution:(1) Department of Mathematics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal (Qc), H3A 2K6, Canada
(2) Département de mathématiques et de statistique, Université de Montréal, C.P. 6128, Succursale Centre-ville, Montréal (Qc), H3C 3J7, Canada

Summary:

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.

2010 Mathematics Subject Classification. Primary: 34M35, 34M40, 34M03; Secondary: 37G10, 34E10, 37G05.

Keywords: Stokes phenomenon, irregular singularity, unfolding, confluence, divergent series, monodromy, analytic classification, summability, flags.

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