KAM theory, Arnold diffusion and long-term stability

4 problems

• Optimal diffusion time compared with the Nekhoroshev stability bound

  Open

  Nekhoroshev proved that drift in nearly regular systems takes at least an exponentially long time. Whether typical systems actually drift that fast, and with which exponent, is open.

• Generic Lyapunov instability of elliptic equilibria in many degrees of freedom

  Partially resolved

  At an equilibrium where all nearby motions oscillate, orbits that start very close may still drift away very slowly. This is expected for typical systems with three or more degrees of freedom, but it is proven only for special constructions and for five or more degrees of freedom.

• Arnold diffusion for generic nearly integrable systems

  Partially resolved

  Arnold conjectured that in typical nearly regular systems with three or more degrees of freedom, some orbits drift by a finite amount no matter how small the perturbation. This is proven only in a weaker sense of typical, or for low numbers of degrees of freedom.

• KAM tori for the planetary N-body problem with actual masses

  Open

  KAM theory proves that planetary systems have many stable quasi-periodic orbits, but only for planets far lighter than real ones. Proving the same with the actual masses of Jupiter and Saturn is open.