Bifurcation of limit cycles for a family of perturbed Kukles differential systems
In this paper, the authors consider an integrable non-Hamiltonian system, which belongs to the quadratic Kukles differential systems having a center surrounded by a bounded period annulus. They study polynomial perturbations of such a Kukles system inside the Kukles family and apply averaging theory to study the limit cycles that bifurcate from the period annulus and from the center of the unperturbed system. First, it is shown that the periodic orbits of the period annulus can be parametrized explicitly through the Lambert function. Then, it is proved that at most one limit cycle bifurcates from the period annulus, under quadratic perturbations. Moreover, conditions are given for the non-existence, existence, and stability of the bifurcated limit cycle. Finally, by using averaging theory of seventh order, it is proved that there are cubic systems, close to the unperturbed system, with one and two small limit cycles.
scientific article; zbMATH DE number 6919414