Limit cycles of polynomial differential systems with homogeneous nonlinearities of degree 4 via the averaging method

By using the averaging method, we study the limit cycles for a class of quartic polynomial differential systems as well as their global shape in the plane. More specifically, we analyze the global shape of limit cycles bifurcating from a Hopf bifurcation and also from periodic orbits with linear center x ˙ = - y , y ˙ = x . The perturbation of these systems is made inside the class of quartic polynomial differential systems without quadratic and cubic terms.