Abstract:

The stability of an elliptic fixed point of a Hamiltonian system is a central question in mathematical physics and is one of the founding problems of dynamical systems. We explore this question from three points of view : topological stability (Lyapunov stability), statistical stability (KAM theory), and effective stability (finite time stability). We introduce in particular new diffusion mechanisms that give the first examples of real analytic Hamiltonians (in three or more degrees of freedom) with unstable elliptic fixed points, or unstable invariant quasi-periodic tori. We produce examples with arbitrary (including Diophantine) frequency vectors and with divergent or convergent (in the case of tori) Birkhoff Normal Forms. We also give examples of analytic Hamiltonians that are integrable on half of the phase space and diffusive on the other half.