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Limit cycles of Liénard polynomial systems type by averaging method

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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 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.