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 $$\dot{z} = i z + z \bar{z}\big(A z^2 + B z \bar{z} + C \bar{z}^2 \big),$$ where \({A,B,C \in \mathbb{C}}\). 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.