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

In this paper we consider Lieńard equations of the form {ẋ=y,ẏ=−(x−2x3+x5)−ε(α+βx2+γx4)y where 0<|ε|≪10<|ε|≪1, (α,β,γ)∈Λ⊂R3(α,β,γ)∈Λ⊂R3 and ΛΛ is bounded. We prove that the least upper bound for the number of zeros of the related Abelian integrals I(h)=∮Γh(α+βx2+γx4)ydx for h∈(1/6,∞)h∈(1/6,∞) is three and for h∈(0,∞)h∈(0,∞) is four (counted with multiplicity) for all parameters α,βα,β and γγ. This implies that the number of limit cycles that bifurcated from periodic orbits of the unperturbed system for ε=0ε=0 outside an eye-figure loop is less than or equal to three.