Consider a Hamiltonian dynamical system having more degrees of freedom than invariant quantities (Suppose that the number of degrees of freedom is $D>1$ and that the only invariant quantity is the Hamiltonian). Let $z^*$ be a fixed point of the motion equations.
Assume that the eigenvalues of the Jacobian matrix (evaluated at $z^*$) are all such that their real part is null. The fixed point $z^*$ is thus called “elliptic” in the context of linear stability analysis.
What can be said concerning the behaviour of trajectories starting from a point of the phase space very close to $z^*$? Will they remain in the neighbourhood of the fixed point (elliptic periodic orbits)?
If yes, it means that we are observing a regular motion in a system which is non-integrable (recall the hypothesis: more degrees of freedom than conserved quantities). Is this possible? Is there something similar to an invariant torus, despite the system is non-integrable?
If no, it means that chaos eventually shows up due to non-linearities, thus completely destroying the memory of the initial condition. So, what’s the purpose of studying the linear stability analysis of a fixed point?
On top of that, It comes to my mind the ghost of Arnold diffusion, which says that for $D>2$ the invariant tori (if any) cannot topologically separate the phase space into two disconnected regions and thus cannot prevent chaos from percolating in all the phase space. Is this phenomenon related to the issue?