Limit cycles of Liénard polynomial systems type by averaging method

Abstract We apply the averaging theory of first and second order for studying the limit cycles of generalized polynomial Linard systems of the form x˙=y-1(x)y,  y˙=-x-f(x)-g(x)y-h(x)y2, \dot x = y - 1\left( x \right)y,\,\,\dot y = - x - f\left( x \right) - g\left( x \right)y - h\left( x \right){y^2}, where l(x) = ∊l1(x) + ∊2l2(x), f (x) = ∊ f1(x) + ∊2 f2(x), g(x) = ∊g1(x) + ∊2g2(x) and h(x) = ∊h1(x) + ∊2h2(x) where lk(x) has degree m and fk(x), gk(x) and hk(x) have degree n for each k = 1, 2, and ∊ is a small parameter.