ON THE LIMIT CYCLES FOR A CLASS OF GENERALIZED KUKLES DIFFERENTIAL SYSTEMS

In this paper, we consider the limit cycles of a class of polynomial differential systems of the form (x)over dot = -y, (y)over dot = x - f (x) - g(x)y - h(x)y(2) - l(x) y(3), where f (x) = epsilon f(1)(x) + epsilon(2) f(2) (x), g(x) = epsilon g(1)(x) + epsilon(2) g(2) (x), h(x) = epsilon h(1) (x) + epsilon(2)h(2)(x) and l(x) = epsilon l(1) (x) + epsilon(2)l(2) (x) where f(k)(x), g(k)(x), h(k )(x) and / l(k) (x) have degree n(1), n(2), n(3) and n(4), respectively for each k = 1,2, and epsilon is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center (x)over dot = -y, (y)over dot = x using the averaging theory of first and second order.