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

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 is 2 (taking into account their multiplicities) for h∈(0,1/6)h∈(0,1/6) and this upper bound is a sharp one. This implies that the number of limit cycles bifurcated from periodic orbits in the vicinity of the center of the unperturbed system for ε=0ε=0 inside an eye-figure loop is less than or equal to 2.