The number of limit cycles of cubic Hamiltonian system with perturbation

In this paper, by using qualitative analysis, we investigate the number of limit cycles of perturbed cubic Hamiltonian system with perturbation in the form of (2n+2m)(2n+2m) or (2n+2m+1)(2n+2m+1)th degree polynomials . We show that the perturbed systems has at most (n+m)(n+m) limit cycles, and has at most n   limit cycles if m=1m=1. If m=1m=1, n=1n=1 and m=1m=1, n=2n=2, the general conditions for the number of existing limit cycles and the stability of the limit cycles will be established, respectively. Such conditions depend on the coefficients of the perturbed terms. In order to illustrate our results, two numerical examples on the location and stability of the limit cycles are given.