An inviscid dyadic model of turbulence: the global attractor

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.