Limit cycles and invariant parabola in a kukles system of degree three

In this article, the authors consider a class of Kukles planar polynomial differential system of degree three having an invariant parabola. For this class of second-order differential systems, it is shown that for certain values of the parameters the invariant parabola coexists with a center. For other values it can coexist with one, two or three small amplitude limit cycles which are constructed by Hopf bifurcation. This result gives an answer for the question given in [4], about the existence of limit cycles for such class of system.