Centers and limit cycles of generalized Kukles polynomial differential systems: phase portraits and limit cycles

In this work, we give the seven global phase portraits in the Poincaré disc of the Kukles differential system given by
\begin{equation*} \begin{array}{l} \dot{x} = -y,\\ \dot{y}= x + a x^8 + b x^4 y^4 + cy^8, \end{array} \end{equation*}
where $x, y \in \mathbb{R}$ and $a, b, c \in \mathbb{R}$ with $a^2 + b^2 + c^2 \neq 0$.
Moreover, we perturb these system inside all classes of polynomials of eight degrees, then we use the averaging theory up sixth order to study the number of limit cycles which can bifurcate from the origin of coordinates of the Kukles differential system.