Bifurcations of limit cycles from quintic Hamiltonian systems with an eye-figure loop

In this paper we consider Lie´nard equations of the form

˙ x = y,
˙y = −(x − 2x3 + x5) − "( + x2 +
x4)y,
where 0 < |"|  1, ( , ,) 2   R3 and  is bounded. We prove that the least upper bound for the number of zeros of the
related Abelian integrals
I (h) =
I