Bifurcations of limit cycles in Kukles systems of arbitrary degree with invariant ellipse

In this paper, for a certain class of Kukles polynomial systems of arbitrary degree n with an invariant ellipse, we show that for certain values of the parameters, the system has an upper bound of limit cycles, where one of the limit cycle is given by an invariant ellipse as an algebraic limit cycle. Writing the system as a perturbation of a Hamiltonian system, we show that the first Poincaré–Melnikov integral of the system is a polynomial whose coefficients are the Lyapunov quantities. The maximum number of simple zeros of this polynomial, gives the maximum number of the global limit cycles and the multiplicity of the origin as a root of the polynomial, minus one, gives the maximum weakness that may have the weak focus at the origin.