On the limit cycles of polynomial differential systems with homogeneous nonlinearities of degree 3 via the averaging method

We study the limit cycles of a class of cubic polynomialdifferential systems in the plane and their global shape using the averaging theory. More specifically, we analyze the global shape of the limit cycles which bifurcate: first, from a Hopf bifurcation; second, from periodic orbits of the linear center ẋ = -y, ẏ = x; and finally from periodic orbits of the cubic centers ẋ = -yh(x, y), ẏ = xh(x, y) where h(x, y) = 0 is a conic. The perturbation of these systems is made inside the class of cubic polynomial differential systems havingnon quadratic terms. © Watnam Press.