Analytic Classification of Families of Linear Differential Systems Unfolding a Resonant Irregular Singularity

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 Poincaré 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 the 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.