We investigate the behaviour of bouncing Bianchi type IX `Mixmaster'universes in general relativity. This generalises all previous studies of thecyclic behaviour of closed spatially homogeneous universes with and withoutentropy increase. We determine the behaviour of models containing radiation byanalytic and numerical integration and show that increase of radiation entropyleads to increasing cycle size and duration. We introduce a null energycondition violating ghost field to create a smooth, non-singular bounce offinite size at the end of each cycle and compute the evolution through manycycles 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 andlonger and the dynamics approach flatness, as in the isotropic case. However,successive cycles become increasingly anisotropic at the expansion maxima whichis dominated by the general-relativistic effects of anisotropic 3-curvature.However, it becomes positive after expansion drives the dynamics close enoughto isotropy for the curvature to become positive and for gravitational collapseto ensue. In the presence of a positive cosmological constant, radiation and aghost field we show that, for a very wide range of cosmological constantvalues, the growing oscillations always cease and the dynamics subsequentlyapproach those of the isotropic de Sitter universe at late times. This model isnot included in the scope of earlier cosmic no-hair theorems because the3-curvature can be positive. In the case of negative cosmological constant,radiation and an ultra-stiff field (to create non-singular bounces) we showthat a sequence of chaotic oscillations also occurs, with sensitive dependenceon initial conditions. In all cases, we follow the oscillatory evolution of thescale factors, the shear, and the 3-curvature from cycle to cycle.
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