Limit cycles of polynomial differential systems bifurcating from the periodic orbits of a linear differential system in RdRd

Let Pk(x1,…,xd)Pk(x1,…,xd) and Qk(x1,…,xd)Qk(x1,…,xd) be polynomials of degree nknk for k=1,2,…,dk=1,2,…,d. Consider the polynomial differential system in RdRd defined byx˙1=−x2+εP1(x1,…,xd)+ε2Q1(x1,…,xd),x˙2=x1+εP2(x1,…,xd)+ε2Q2(x1,…,xd),x˙k=εPk(x1,…,xd)+ε2Qk(x1,…,xd), for k=3,…,dk=3,…,d.Suppose that nk=n⩾2nk=n⩾2 for k=1,2,…,dk=1,2,…,d. Then, by applying the first order averaging method this system has at most (n−1)nd−2/2(n−1)nd−2/2 limit cycles bifurcating from the periodic orbits of the same system with ε=0ε=0; and by applying the second order averaging method it has at most (n−1)(2n−1)d−2(n−1)(2n−1)d−2 limit cycles bifurcating from the periodic orbits of the same system with ε=0ε=0. We provide polynomial differential systems reaching these upper bounds.In fact our results are more general, they provide the number of limit cycles for arbitrary nknk.