On the limit cycles of a class of Kukles type differential systems

In this paper we study the limit cycles of two families of differential systems in the plane. These systems are obtained by polynomial perturbations with arbitrary degree on the second component of the standard linear center. The classes under consideration are polynomial generalizations of certain canonical form of a Kukles system with an invariant ellipse, previously studied in the literature. We provide, in both cases, an accurate upper bound of the maximum number of limit cycles that the perturbed system can have bifurcating from the periodic orbits of the linear center, using the averaging theory of first, second and third order. These upper bounds are presented in terms of the degree of the respective systems. Moreover, the existence of a weak focus with the highest order is also studied.