Mixmaster behavior can be studied in the context of Hamiltonian dynamics with a semi-bounded potential (figure 2) arising from the spatial scalar curvature . A solution of this Hamiltonian system is an infinite sequence of Kasner epochs with parameters that change when the phase space trajectories bounce off the potential walls, which become exponentially steep as the system evolves towards the singularity. Several numerical calculations of the Liapunov exponents indicated that the Mixmaster system is nonchaotic and in apparent contradiction with the discrete maps [19, 35]. However, it has since been suggested that the usual definition of the Liapunov exponents is not appropriate in this case as it is not invariant under time reparametrizations, and that with a more natural time variable one obtains positive exponents [15, 30], confirming the chaotic nature of the anisotropy bounces.
BLK conjectured that local Mixmaster oscillations might be the generic behavior for singularities in more general classes of spacetimes . However, this conjecture has yet to be established.
Considering inhomogeneous spacetimes, Isenberg and Moncrief  proved that the singularity in the polarized Gowdy model is AVTD type. This has also been conjectured to be the case for more general Gowdy models, and numerical simulations of one-dimensional plane symmetric Gowdy spacetimes support the notion. Furthermore, a symplectic numerical method [16, 50] has been applied to investigate the AVTD conjecture in even more general spacetimes, namely spacetimes with U (1) symmetry . Although there are concerns about the solutions due to difficulties in resolving the steep spatial gradients near the singularity, the numerical calculations find no evidence of BLK oscillations. Berger  attributes this to several possibilities: 1) the BLK conjecture is false; 2) the simulations have not been run long enough; 3) Mixmaster behavior is present but hidden in the variables; or 4) the initial data is insufficiently generic. In any case, further investigations are needed to confirm either the BLK or AVTD conjectures.
|Physical and Relativistic Numerical Cosmology
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