  A new diffusion mechanism from the neighborhood of elliptic equilibria for
Hamiltonian flows in three or more degrees of freedom is introduced. We thus
obtain explicit real entire Hamiltonians on $\R^{2d}$, $d\geq 4$, that have a
Lyapunov unstable elliptic equilibrium with an arbitrary chosen frequency
vector whose coordinates are not all of the same sign. For non-resonant
frequency vectors, our examples all have divergent Birkhoff normal form at the
equilibrium.
  On $\R^4$, we give explicit examples of real entire Hamiltonians having an
equilibrium with an arbitrary chosen non-resonant frequency vector and a
divergent Birkhoff normal form.
