Limit cycles of Liénard polynomial systems type by averaging method

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) + ∊²f₂(x), g(x) = ∊g₁(x) + ∊²g₂(x) and h(x) = ∊h₁(x) + ∊²h₂(x) where l_k(x) has degree m and f_k(x), g_k(x) and h_k(x) have degree n for each k = 1, 2, and ∊ is a small parameter.