  Properties of an infinite system of nonlinearly coupled ordinary differential
equations are discussed. This system models some properties present in the
equations of motion for an inviscid fluid such as the skew symmetry and the
3-dimensional scaling of the quadratic nonlinearity. In a companion paper [6]
it is proved that every solution for the system with forcing blows up in finite
time in the Sobolev $H^{5/6}$ norm. In this present paper, it is proved that
after the blow-up time all solutions stay in $H^s$, $s<5/6$ for almost all time
and the energy dissipates.
  Moreover, it is proved that the unique equilibrium is an exponential global
attractor.
