Centers and limit cycles for a generalized kukles differential systems of degree 8

Abstract In this paper we provide all the global phase portraits of the generalized kukles differential systems x= y; y = x + ax8 + bx6y2 + cx4y4 + dx2y6 + ey8; symmetric with respect to the x{axis, with a2 + b2 + c2 + d2 + e2 6= 0, and by using the averaging theory up to seven order, we give the upper bounds of limit cycles which can bifurcate from its center when we perturb it inside the class of all polynomial differential systems of degree 8. The main tool used for proving these results is based in the first integrals of the systems which form the discontinuous piecewise differential systems.