The number of limit cycles of a quintic polynomial system with center

In this paper we study the global bifurcation of limit cycles for the system ẋ=y(1+x4),ẏ=−x(1+x4)+εy2m−1Σj=0lajxj for εε sufficiently small, where l=2n+2or2n+3, m,nm,n are arbitrary positive integers and a0,a1,…,ala0,a1,…,al are real. We have used Argument principle to give estimate of an upper bound for the number of limit cycles that can bifurcate from period annulus of this system for ε=0ε=0. Furthermore we have shown that there exist a set of constant a0,a1,…,ala0,a1,…,al where the related abelian integral of this system has at least 3m+n−23m+n−2 isolated zeros. We have, in order to prove our result applied the Argument Principle to a complex extension of the Abelian integral.