Global phase portraits of Kukles differential systems with homogenous polynomial nonlinearities of degree 5 having a center

Abstract

We provide 22 different global phase portraits in the Poincaré disk of all centers of the so-called Kukles polynomial differential systems of the form $\dot{x} = -y$, $\dot{y} = x+Q_5(x,y)$, where $Q_5$ is a real homogeneous polynomial of degree 5 defined in $\mathbb{R}^2$.