Bifurcation set and distribution of limit cycles for a class of seven-order Hamiltonian system with fifteen-order perturbed terms

A class of seven-order planar Z7-equivariant Hamiltonian system with rotation symmetry and the fifteen-order perturbed term are investigated. By using the bifurcation theory, qualitative method, detection function and numerical simulations, we obtain: (1) explicit formulas for the global and local bifurcations of the flow; (2) 15 limit cycles with distribution 7C12 C271. These results in this paper are useful for the study of the weakened Hilbert 16th problem.