On Stability at the Hamiltonian Hopf Bifurcation

For a 2 d.o.f. Hamiltonian system we prove the Lyapunov stability of its equilibrium with two double pure imaginary eigenvalues and non-semisimple Jordan form for the linearization matrix, when some coefficient in the 4th order normal form is positive (the equilibrium is known to be unstable, if this coefficient is negative). Such the degenerate equilibrium is met generically in one-parameter unfoldings, the related bifurcation is called to be the Hamiltonian Hopf Bifurcation. Though the stability is known since 1977, proofs that were published are either incorrect or not complete. Our proof is based on the KAM theory and a work with the Weierstrass elliptic functions, estimates of power series and scaling.