In this paper, we consider the number of limit cycles of the Lienard system of the form x = y; y = -x(x~2 + bx - 1) + εf_m(x)y , where b > 0, f_m(x) = ∑_(i=0)~m a_ix~i is a polynomial of x with degree not greater than m and 0 < ε ? 1. By studying the number of isolated zeros of the corresponding Abelian integral I(h) = ∮_mf_m(x)ydx; we obtain the upper bound of the number of limit cycles that bifurcated from periodic orbits of the unperturbed system for ε = 0 .
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