A New Normal Form for Monodromic Nilpotent Singularities of Planar Vector Fields

In this work, we present a new technique for solving the center problem for nilpotent singularities which consists of determining a new normal form conveniently adapted to study the center problem for this singularity. In fact, it is a pre-normal form with respect to classical Bogdanov–Takens normal formal and it allows to approach the center problem more efficiently. The new normal form is applied to several examples.The first and second authors are partially supported by Ministerio de Ciencia, Innovación y Universidades/ FEDER grant number PGC2018-096265- B-I00 and by the Consejería de Educación y Ciencia de la Junta de Andalucía (projects P12-FQM-1658, FQM-276). The third author is partially supported by a MINECO/ FEDER grant number PID2020-113758GB-I00 and an AGAUR (Generalitat de Catalunya) Grant number 2017SGR-1276