summary:We consider limit cycles of a class of polynomial differential systems of the form {x˙=y,y˙​=−x−ε(g21​(x)y2α+1+f21​(x)y2β)−ε2(g22​(x)y2α+1+f22​(x)y2β),​ where β and α are positive integers, g2j​ and f2j​ have degree m and n, respectively, for each j=1,2, and ε is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center x˙=y, y˙​=−x using the averaging theory of first and second order