Analytic nilpotent centers as limits of nondegenerate centers revisited

Abstract

We prove that all the nilpotent centers of planar analytic differential systems are limit of centers with purely imaginary eigenvalues, and consequently the Poincaré--Liapunov method to detect centers with purely imaginary eigenvalues can be used to detect nilpotent centers.

The first and third authors are partially supported by a MICINN grant number MTM2011-22877 and by a CIRIT grant number 2014 SGR 1204. The fourth author is partially supported by a MINECO/FEDER grant MTM2008-03437, a CIRIT grant number 2009SGR-410, an ICREA Academia, the grant FP7-PEOPLE-2012- IRSES 318999, and FEDER-UNAB10-4E-378.