The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential Systems

This paper studies the number of small limit cycles produced around an elementary center for Hamiltonian differential systems with the elliptic Hamiltonian function H = 1/2y(2) + 1/2x(2) - 2/3x(3) - 2/3x(3) + a/4x(4) (a not equal 0) under two types of polynomial perturbations of degree m, respectively. It is proved that the Hamiltonian system perturbed in Lienard systems can have at least [3m-1/4] small limit cycles near the center, where m <= 101, and that the related near-Hamiltonian system with general polynomial perturbations can have at least m + [m+1/2] - 2 small-amplitude limit cycles, where m <= 16. Furthermore, in any of the cases, the bounds for limit cycles can be reached by studying the isolated zeros of the corresponding first order Melnikov functions and with the help of Maple programs. Here, [.] represents the integer function.