Global bifurcations in a perturbed cubic system withZ2-symmetry

In this paper we consider global and local bifurcations in disturbed planar Hamiltonian vector fields which are invariant under a rotation over π. All calculation formulas of bifurcation curves have been obtained. Various possible distributions and the existence of limit cycles and singular cycles in different parameter regions have been determined. It is shown that for a planar cubic differential system there are infinitely many parameters in the three-parameter space such that Hilbert numberH(3)≥11.