Instability of pole singularities for the Chazy equation

TL;DR

This paper investigates the stability of pole singularities in the Chazy equation, linking negative resonances to instability and showing how parameter variations can lead to boundary formation or singularity splitting.

Contribution

It establishes a direct connection between negative resonances and pole instability in the Chazy equation, providing new insights into their behavior and stability conditions.

Findings

Negative resonances indicate pole instability.

Unstable poles can form natural boundaries or split into singularities.

Pole stability correlates with the number of positive resonances.

Abstract

We prove that the negative resonances of the Chazy equation (in thesense of Painlev\'e analysis) can be related directly to it sgroup-invariance properties. These resonances indicate in this case the instability of pole singularities. Depending on the value of a parameter in the equation, an unstable isolated pole may turn into the familiar natural boundary, or split into several isolated singularities. In the first case, a convergent series representation involving exponentially small corrections can be given. This reconciles several earlier approaches to the interpretation of negative resonances. On the other hand, we also prove that pole singularities with the maximum number of positive resonances are stable. The proofs rely on general properties of nonlinear Fuchsian equations.