Dulac functions and monodromic singularities

We are interested in bound the maximum number of small amplitude limit cycles that an analytic planar vector field can have bifurcating from any monodromic singularity as well as its stability and hyperbolic nature. We do not use the Poincaré map approach to this problem. Instead, we propose an algorithmic procedure to construct, under some assumptions, a Dulac function in a neighborhood (may be punctured) of the singularity. This approach is based on the existence of a real analytic invariant curve passing through the singularity which allows us to overcome the usual difficulty seeking for the candidates to be a Dulac function. We finally apply our results to a degenerate polynomial monodromic family.