The cyclicity of period annuli of a class of quintic Hamiltonian systems

In this paper, we prove that the cyclicity of period annuli for system d x d t = y , d y d t = − ( a x + b x 3 + c x 5 ) under perturbations of polynomials with degree n is not more than 54 n − 13 (taking into account the multiplicity), where a , b , c ∈ R with c ≠ 0 such that the unperturbed systems have at least one center.