Center problem with characteristic directions and inverse integrating factors

Abstract

We consider the center-focus problem for analytic families of planar vector fields having a monodromic singularity with characteristic directions. We give a method to compute the center conditions of the family provided there is an inverse integrating factor whose expression in polar coordinates has the exceptional divisor of the first polar blow-up as isolated zero-set. Some non-trivial examples are analyzed.

The authors are partially supported by the Agencia Estatal de Investigación, Spain grant PID2020-113758GB-I00 and an AGAUR (Generalitat de Catalunya), Spain grant number 2017SGR 1276