Centers and limit cycles of polynomial differential systems of degree 4 via averaging theory

In this paper we classify the phase portraits in the Poincaré disc of the centers of the generalized class of Kukles systems ẋ=−y,ẏ=x+ax3y+bxy3, symmetric with respect to the y-axis, and we study, using the averaging theory up to sixth order, the limit cycles which bifurcate from the periodic solutions of these centers when we perturb them inside the class of all polynomial differential systems of degree 4.