Complete system of analytic invariants for unfolded differential linear systems with an rank \(k\) irregular singularity of Poincaré (Q2876644)

The paper contains a complete modulus for germs of generic unfoldings NEWLINE\[NEWLINEp_{\varepsilon}(x)y'=A(\varepsilon ,x)yNEWLINE\]NEWLINE of nonresonant linear differential systems with an irregular singularity of Poincaré rank \(k\) at the origin NEWLINE\[NEWLINEx^{k+1}y'=A(x)YNEWLINE\]NEWLINE under analytic equivalence. Here \(A(0,x)=A(x)\) and \(p_{\varepsilon}(x)=x^{k+1}+\sum _{j=0}^{k-1}\varepsilon _jx^j\), \(\varepsilon, x\in ({\mathbb C},0)\), \(A\) are \(n\times n\)-matrices analytically depending on \(x\) or \((\varepsilon ,x)\), the eigenvalues of \(A(0)\) are distinct. 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, a fixed neighbourhood of the origin in the variable is covered with sectors on which there is 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.