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In this article, we consider the Liénard system of the form x ˙ = y , y ˙ = x ( x − 1 ) ( x + 1 ) ( x 2 − 3 ) + ε ( α + β x 2 + γ x 4 ) y with 0 < ε ≪ 1 , a, b and c are real bounded parameters. We prove that the least upper bound of the number of isolated zeros of the corresponding Abelian integral I ( h ) = ∮ Γ h ( α + β x 2 + γ x 4 ) y d x is four (counting the multiplicity). This implies that the number of limit cycles that bifurcated from periodic orbits of the unperturbed system for ε = 0 is less than or equal to four. MSC:34C05, 34C07, 34C08.

Bifurcation of limit cycles from a hyper-elliptic Hamiltonian system with a double heteroclinic loops