Abstract
Using the averaging theory of first and second order we study the maximum number of limit cycles of the polynomial differential systemsẋ=y,ẏ=-x-ε(h1(x)+p1(x)y+q1(x)y2)-ε2(h2(x)+p2(x)y+q2(x)y2),which bifurcate from the periodic orbits of the linear center ẋ=y,ẏ=-x, where ε is a small parameter. If the degrees of the polynomials h1,h2,p1,p2,q1 and q2 are equal to n, then we prove that this maximum number is [n/2] using the averaging theory of first order, where [·] denotes the integer part function; and this maximum number is at most n using the averaging theory of second order.
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