Coexistence of limit cycles and invariant curves for Extended Kukles systems

Abstract

We work with a certain class of extended Kukles system of arbitrary degree n with at least three invariant straight lines. We show that for a certain values of the parameters, the system has an lower bound of limit cycles. By writing the system as a perturbation of a Hamiltonian system, we show that the first Poincar´e-Melnikov integral of the system is a polynomial whose coefficients are the Lyapunov quantities. The maximum number of simples zero of this polynomial gives the maximum number of the global limit cycles; the multiplicity of the origin as a root the polynomial gives the maximum weakness that the weak focus at the origin. On the other hand, we also work with a certain extended Kukles system of order four with a invariant circumference. We show that for certain values of the parameters the system has an lower bound of limit cycles at the origin.