Limit cycles of generalized Liénard polynomial differential systems via averaging theory

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.