Bifurcations of limit cycles for a cubic Hamiltonian system under quartic perturbations

This paper is concerned with the number of limit cycles of a cubic system with quartic perturbations. Fifteen limit cycles are found and their distributions are studied by using the methods of bifurcation theory and qualitative analysis. It gives rise to the conclusion: H(4) 15, where H(n) is the Hilbert number for the second part of Hilbertʹs 16th problem.