  It is shown that all contracting, spatially homogeneous, orthogonal Bianchi
cosmologies that are sourced by an ultra-stiff fluid with an arbitrary and, in
general, varying equation of state asymptote to the spatially flat and
isotropic universe in the neighbourhood of the big crunch singularity. This
result is employed to investigate the asymptotic dynamics of a collapsing
Bianchi type IX universe sourced by a scalar field rolling down a steep,
negative exponential potential. A toroidally compactified version of M*-theory
that leads to such a potential is discussed and it is shown that the isotropic
attractor solution for a collapsing Bianchi type IX universe is supersymmetric
when interpreted in an eleven-dimensional context.
