Bifurcations of limit cycles for a quintic Hamiltonian system with a double cuspidal loop

In this work we consider the number of limit cycles that can bifurcate from periodic orbits located inside a double cuspidal loop of the quintic Hamiltonian vector field XH=y∂∂x−x3(x2−1)∂∂y under small perturbations of the form ε(α+βx2+γx4)y∂∂y, where 0<∣ε∣≪10<∣ε∣≪1 and α,β,γα,β,γ are real constants. Using Picard–Fuchs equations for related abelian integrals, asymptotic expansion of these integrals about critical level curves of HH, and some geometric properties of the curves defined by ratios of two especial integrals, we show that the least upper bound for the number of limit cycles appeared in this bifurcation is two.