Detection function method and its application to a perturbed quintic Hamiltonian system

We employ the methods of both qualitative analysis and numerical exploration to investigate the bifurcation of limit cycles of a perturbed quintic Hamiltonian system. With the help of the detection functions for the perturbed Hamiltonian system, we study the perturbation of a quintic Hamiltonian system with 25 finite singular points and 4 infinite singular points. We first classify the phase portraits of the unperturbed system and categorize the closed orbits, then obtain the detection functions for the perturbed system, from which the detection curves and the number and the distribution of limit cycles are obtained. As application examples, we numerically compute the detection curves and draw the distribution of limit cycles. It seems that our methods are very effective on the study of the bifurcation of limit cycles of quintic Hamiltonian systems.