  The asymptotic behavior of expanding, generic, $T^2$-Symmetric, vacuum
spacetimes is examined via numerical simulations. After validation of the
numerical methods, the properties of these generic spacetimes are explored and
compared to non-generic subfamilies where proven results exist.
  The non-generic subfamilies within this class, including the Kasner, the
Gowdy, the pseudo-homogeneous, and the $B=0$ spacetimes, all have known
asymptotic behaviors in the expanding direction which have been determined
either from the explicit solutions or using analytic methods. For the $B\ne 0$
spacetimes, the generic case within the $T^2$-Symmetric vacuum solutions, the
asymptotic behavior has not been determined analytically. In this work, we use
numerical simulations to explore the asymptotic behavior of the $B\ne 0$
spacetimes. Our results indicate that, for these generic spacetimes, the
asymptotic behavior in the expanding direction differs from that seen in the
non-generic subfamilies.
  In addition to differences in asymptotic power laws, an apparent
quasi-periodic exchange of energy from one gravitational mode to the other for
the generic non-polarized solutions is observed.
