Finite Cyclicity of Graphics with a Nilpotent Singularity of Saddle or Elliptic Type

In this paper we prove finite cyclicity of several of the most generic graphics through a nilpotent point of saddle or elliptic type of codimension 3 inside C∞ families of planar vector fields. In some cases our results are independent of the exact codimension of the point and depend only on the fact that the nilpotent point has multiplicity 3. By blowing up the family of vector fields, we obtain all the limit periodic sets. We calculate two different types of Dulac maps in the blown-up family and develop a general method to prove that some regular transition maps have a nonzero higher derivative at a point. The finite cyclicity theorems are derived by a generalized derivation–division method introduced by Roussarie.