  We consider examples of discrete nonlinear Schroedinger equations in Z
admitting ground states which are orbitally but not asymptotically stable in l
^2(Z). The ground states contain internal modes which decouple from the
continuous modes. The absence of leaking of energy from discrete to continues
modes leads to an almost conservation and perpetual oscillation of the discrete
modes. This is quite different from what is known for nonlinear Schroedinger
equations in eucliedean spaces. We do not investigate connections with work on
quasi periodic solutions as in M.Johansson & S.Aubry and of Bambusi & Vella
