The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor

Abstract We consider analytic families of planar vector fields depending analytically on the parameters in Λ \Lambda that guarantee the existence of a (may be degenerate and with characteristic directions) monodromic singularity. We characterize the structure of the asymptotic Dulac series of the Poincaré map associated to the singularity when the family possesses a Puiseux inverse integrating factor in terms of its multiplicity and index. This characterization is only valid in a restricted monodromic parameter space Λ \ Λ ∗ \Lambda \backslash {\Lambda }^{\ast } associated to the nonexistence of local curves with zero angular speed. As a byproduct, we are able to study the center-focus problem (under the assumption of the existence of some Cauchy principal values) in very degenerated cases where no other tools are available. We illustrate the theory with several nontrivial examples.