The number of limit cycles of a class of polynomial differential systems

In this paper, we consider the bifurcation of limit cycles of a class of polynomial differential systems of the form ẋ=−y(1+x4)+εPn(x,y),ẏ=x(1+x4)+εQn(x,y), where PnPn, QnQn are arbitrary polynomials of degree nn. We prove that there is a system of the above form having at least 3[n+12]−2 limit cycles in Hopf bifurcation. Then applying the Argument Principle, we obtain that up to first order in εε the number of limit cycles that bifurcate from period annulus surrounding the origin of this system is at most 5[n+14]+[n+12] for n≥5n≥5.