Analytic classification of families of linear differential systems unfolding a resonant irregular singularity

TL;DR

This paper provides a comprehensive classification of parametric families of 2D linear differential systems with resonant irregular singularities, detailing their invariants, normal forms, and the geometric phenomena involved.

Contribution

It introduces a complete analytic classification framework for these systems, including explicit normal forms and a description of the moduli space based on formal and analytic invariants.

Findings

Classification of systems using formal and analytic invariants

Description of moduli space of equivalence classes

Analysis of confluence phenomena and Stokes geometry

Abstract

We give a complete classification of analytic equivalence of germs of parametric families of systems of complex linear differential equations unfolding a generic resonant singularity of Poincare rank 1 in dimension $n = 2$ whose leading matrix is a Jordan bloc. The moduli space of analytic equivalence classes is described in terms of a tuple of formal invariants and a single analytic invariant obtained from the trace of monodromy, and analytic normal forms are given. We also explain the underlying phenomena of confluence of two simple singularities and of a turning point, the associated Stokes geometry, and the change of order of Borel summability of formal solutions in dependence on a complex parameter.