Stability and instability of elliptic fixed points in Hamiltonian systems

It is well-known that an elliptic fixed point of a system
of ODE can be turned into a stable or unstable one by a small
Would the same be true if we assume that both the initial
system and the perturbation are Hamiltonian?
The answer is much more involved and depends on what we mean by

I will speak about different types of stability: Lyapunov stability,
stability in a probabilistic sense (KAM-stability) and effective
stability, and present some of the related results in this field.