Gevrey series in compensators linearizing a planar resonant vector field and its unfolding

Abstract

We consider a planar vector field X near a saddle type p : -q resonant singular point. Assuming that it has a normal form with a Gevrey-d expansion (like d = p + q which is in particular the case when starting from an analytic vector field) we show that X can be linearized working with a change of coordinates that is of Gevrey order d in certain log-like variables, called compensators or also tags, multiplied by the first integral u = x^qy^p of the linear part. Next we consider the unfolding of such a resonance, and provide (weaker) Gevrey-type linearization using compensators.