Instability of pole singularities for the Chazy equation
Résumé
We prove that the negative resonances of the Chazy equation (in the
sense of Painlev\'e analysis) can be related directly to its
group-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.
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