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 dxdt=y,dydt=−(ax+bx3+cx5) under perturbations of polynomials with degree nn is not more than 54n−1354n−13 (taking into account the multiplicity), where a,b,c∈Ra,b,c∈R with c≠0c≠0 such that the unperturbed systems have at least one center.