  We study the behaviour of Bianchi class A universes containing an ultra-stiff
isotropic ghost field and a fluid with anisotropic pressures which is also
ultra-stiff on the average. This allows us to investigate whether cyclic
universe scenarios, like the ekpyrotic model, do indeed lead to isotropisation
on approach to a singularity (or bounce) in the presence of dominant
ultra-stiff pressure anisotropies. We specialise to consider the closed Bianchi
type IX universe and show that when the anisotropic pressures are stiffer on
average than any isotropic ultra-stiff fluid then, if they dominate on approach
to the singularity, it will be anisotropic. We include an isotropic ultra-stiff
ghost fluid with negative energy density in order to create a cosmological
bounce at finite volume in the absence of the anisotropic fluid. When the
dominant anisotropic fluid is present it leads to an anisotropic cosmological
singularity rather than an isotropic bounce. The inclusion of anisotropic
stresses generated by collisionless particles in an anisotropically expanding
universe is therefore essential for a full analysis of the consequences of a
cosmological bounce or singularity in cyclic universes.
