Bifurcation of Limit Cycles in a Cubic Hamiltonian System with Some Special Perturbed Terms

This paper presents an analysis on the bifurcation of limit cycles for a cubic Hamiltonian system with quintic perturbed terms using both qualitative analysis and numerical exploration. The perturbed terms considered here is in the form of R(x, y, λ) = S(x, y, λ) = mx+ny+kxy−λ, where m, n, k, and λ are all variable. The investigation is based on detection functions which are particularly effective for the perturbed cubic Hamiltonian system. The study reveals that, for the Hamiltonian system [equation (1.2) in the introduction] with the perturbed terms mentioned above, there are 15 limit cycles if 0.630393 < λ < 0.630867; and 7 limit cycles if 0.630247 < λ < 0.630393 (or 0.630867 < λ < 0.632824 ). The results presented here are helpful for further investigating the bifurcation behavior of the Hilbert’s 16th problem.