Unstable equilibria of Hamiltonian systems

It is shown that a Hamiltonian system in the neighbourhood of an equilibrium may be given a special normal form in case four of the eigenvalues of the linearized system are of the form λ1,−λ1, λ2,−λ2, with λ1 and λ2 independent over the reals, i.e., λ1/λ2 /∈ R. That is, for a real Hamiltonian system and concerning the variables x1, y1, x2, y2 the equilibrium is of either type center–saddle or complex–saddle. The normal form exhibits the existence of a four–parameter family of solutions which has been previously investigated by Moser. This paper completes Moser’s result in that the convergence of the transformation of the Hamiltonian to a normal form is proven.