Versal unfolding of planar Hamiltonian systems at fully degenerate equilibrium

In this paper we study bifurcations of a planar Hamiltonian system at a fully degenerate equilibrium, which has a zero linearization. Since the Poincaré normal form theory is not applicable to such a degenerate system, we investigate its restrictive normal forms in the class of Hamiltonian fields and prove that such a degenerate system is of codimension 3 degeneracy in the class, so that we introduce three parameters to versally unfold the degenerate system in the class. In order to discuss further the qualitative properties of the versal unfolding, we use the Poincaré index to determine the number and distribution of hyperbolic sectors near the degenerate equilibrium. We display its all bifurcations such as pitchfork bifurcation, saddle-center bifurcation and the Bogdanov–Takens bifurcation within Hamiltonian systems.