Intrinsic character of Stokes matrices

TL;DR

This paper investigates how the collection of Stokes matrices, which classify certain linear differential systems, depends on the choice of rotation and ordering of eigenvalues, revealing intrinsic properties of these matrices.

Contribution

It establishes the dependence of Stokes matrices on the rotation in the complex plane and the ordering of eigenvalues, clarifying their intrinsic character.

Findings

Stokes matrices depend on the rotation in the complex plane.

The ordering of eigenvalues affects the structure of Stokes matrices.

The paper clarifies the intrinsic properties of Stokes matrices beyond arbitrary choices.

Abstract

Two germs of linear analytic differential systems $x^{k+1}Y^\prime=A(x)Y$ with a non resonant irregular singularity are analytically equivalent if and only if they have the same eigenvalues and equivalent collections of Stokes matrices. The Stokes matrices are the transition matrices between sectors on which the system is analytically equivalent to its formal normal form. Each sector contains exactly one separating ray for each pair of eigenvalues. A rotation in $S$ allows supposing that $\mathbb R^+$ lies in the intersection of two sectors. Reordering of the coordinates of $Y$ allows ordering the real parts of the eigenvalues, thus yielding triangular Stokes matrices. However, the choice of the rotation in $x$ is not canonical. In this paper we establish how the collection of Stokes matrices depends on this rotation, and hence on a chosen order of the projection of the eigenvalues on a line through the origin.