We investigate the behaviour of bouncing Bianchi type IX `Mixmaster'
universes in general relativity. This generalises all previous studies of the
cyclic behaviour of closed spatially homogeneous universes with and without
entropy increase. We determine the behaviour of models containing radiation by
analytic and numerical integration and show that increase of radiation entropy
leads to increasing cycle size and duration. We introduce a null energy
condition violating ghost field to create a smooth, non-singular bounce of
finite size at the end of each cycle and compute the evolution through many
cycles with and without entropy increase injected at the start of each cycle.
In the presence of increasing entropy we find that the cycles grow larger and
longer and the dynamics approach flatness, as in the isotropic case. However,
successive cycles become increasingly anisotropic at the expansion maxima which
is dominated by the general-relativistic effects of anisotropic 3-curvature.
However, it becomes positive after expansion drives the dynamics close enough
to isotropy for the curvature to become positive and for gravitational collapse
to ensue. In the presence of a positive cosmological constant, radiation and a
ghost field we show that, for a very wide range of cosmological constant
values, the growing oscillations always cease and the dynamics subsequently
approach those of the isotropic de Sitter universe at late times. This model is
not included in the scope of earlier cosmic no-hair theorems because the
3-curvature can be positive. In the case of negative cosmological constant,
radiation and an ultra-stiff field (to create non-singular bounces) we show
that a sequence of chaotic oscillations also occurs, with sensitive dependence
on initial conditions. In all cases, we follow the oscillatory evolution of the
scale factors, the shear, and the 3-curvature from cycle to cycle.