  We show that for any fixed accuracy and time length $T$, a {\it finite}
number of $T$-time length pieces of the complete trajectories on the global
attractor are capable of uniformly approximating all trajectories within the
accuracy in the natural strong metric after sufficiently large time when the
observed dissipative system is asymptotically compact. Moreover, we obtain the
strong equicontinuity of all the complete trajectories on the global attractor.
These results follow by proving the existence of a strongly compact strong
trajectory attractor. The notion of a trajectory attractor was previously
constructed for a family of auxiliary systems including the originally
considered one without uniqueness. Recently, Cheskidov and the author developed
a new framework called evolutionary system, with which a (weak) trajectory
attractor can be actually defined for the original system. In this paper, the
theory of trajectory attractors is further developed in the natural strong
metric for our purpose. We then apply it to both the 2D and the 3D
Navier-Stokes equations and a general nonautonomous reaction-diffusion system.
