Center cyclicity of a family of quartic polynomial differential system

Abstract

In this paper we study the cyclicity of the centers of the quartic polynomial family written in complex notation as z = i z + z z (A z^2 + B z z + C z^2 ), where A,B,C. We give an upper bound for the cyclicity of any nonlinear center at the origin when we perturb it inside this family. Moreover we prove that this upper bound is sharp.

The first and third authors are partially supported by a MINECO grant number MTM2014-53703-P and by a CIRIT grant number 2014 SGR 1204. The second author is partially supported by a MINECO grant MTM2013-40998-P, an AGAUR grant number 2014SGR 568, two FP7-PEOPLE-2012-IRSES numbers 316338 and 318999.