  In this work the late-time evolution of Bianchi type $VIII$ models is
discussed. These cosmological models exhibit a chaotic behaviour towards the
initial singularity and our investigations show that towards the future, far
from the initial singularity, these models have a non-chaotic evolution, even
in the case of vacuum and without inflation. These space-time solutions turn
out to exhibit a particular time asymmetry. On the other hand, investigations
of the late-time behaviour of type $VIII$ models by another author have the
result that chaos continues for ever in the far future and that these solutions
have a time symmetric behaviour: this result was obtained using the
approximation methods of Belinski, Khalatnikov and Lifshitz ($BKL$) and we try
to find out a possible reason explaining why the different approaches lead to
distinct outcomes. It will be shown that, at a heuristic level, the $BKL$
method gives a valid approximation of the late-time evolution of type $VIII$
models, agreeing with the result of our investigations.
