The Hamiltonian-dissipative decomposition of normal forms of vector fields

We consider dynamical systems in two variables with nilpotent linearization at the origin. We show that the behavior of the equilibria of such systems is determined by a modified Hamiltonian function which is constructed from an appropriate normal form for the vector field. In particular, the equilibria of the dynamical system correspond to critical points of the modified Hamiltonian and the local behavior of the vector field near an equilibrium is determined by the second variation of the modified Hamiltonian and its time derivative.