Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 006, 46 pages      arXiv:1301.5228      https://doi.org/10.3842/SIGMA.2020.006

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

Martin Klimeš
Independent Researcher, Prague, Czech Republic

Received March 13, 2019, in final form December 21, 2019; Published online January 23, 2020

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 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.

Key words: linear differential equations; confluence of singularities; Stokes phenomenon; monodromy; analytic classification; moduli space; biconfluent hypergeometric equation.

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