Stokes phenomenon and confluence in non-autonomous Hamiltonian systems

TL;DR

This paper investigates how regular singular points merge into an irregular singularity in non-autonomous Hamiltonian systems, elucidating the relation between monodromy and Stokes phenomena during degeneration, with implications for Painleve equations.

Contribution

It provides a sectoral normalization theorem for time-dependent Hamiltonian systems, linking monodromy operators and Stokes phenomena in the confluence process.

Findings

Established a formal normal form for the family of systems.

Connected local monodromy with non-linear Stokes phenomena.

Addressed the analytic classification of the systems.

Abstract

This article studies a confluence of a pair of regular singular points to an irregular one in a generic family of time-dependent Hamiltonian systems in dimension 2. This is a general setting for the understanding of the degeneration of the sixth Painleve equation to the fifth one. The main result is a theorem of sectoral normalization of the family to an integrable formal normal form, through which is explained the relation between the local monodromy operators at the two regular singularities and the non-linear Stokes phenomenon at the irregular singularity of the limit system. The problem of analytic classification is also addressed. Key words: Non-autonomous Hamiltonian systems; irregular singularity; non-linear Stokes phenomenon; wild monodromy; confluence; local analytic classification; Painleve equations.