The number of limit cycles of a quintic polynomial system

In this paper we consider the bifurcation of limit cycles of the system ẋ=y(x2−a2)(y2−b2)+εP(x,y),ẏ=−x(x2−a2)(y2−b2)+εQ(x,y) for εε sufficiently small, where a,b∈R−{0}a,b∈R−{0}, and P,QP,Q are polynomials of degree nn, we obtain that up to first order in εε the upper bounds for the number of limit cycles that bifurcate from the period annulus of the quintic center given by ε=0ε=0 are (3/2)(n+sin2(nπ/2))+1(3/2)(n+sin2(nπ/2))+1 if a≠ba≠b and n−1n−1 if a=ba=b. Moreover, there are systems with at least (3/2)(n+sin2(nπ/2))+1(3/2)(n+sin2(nπ/2))+1 if a≠ba≠b and, n−1n−1 limit cycles if a=ba=b.