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Half a century ago, Belinsky and Khalatnikov proposed a generic solution ofthe Einstein equations near their cosmological singularity, based on ageneralization of the homogeneous model of Bianchi type IX. Consideration ofthe evolution of the most general non-diagonal case of this model is greatlysimplified if it is assumed that, when approaching the singularity t=0, itreduces to the so-called asymptotic dynamics, at which inequality (6) holds. Ithas been suggested that this inequality continues to be true from the moment ofits first fulfilment up to the singularity of space-time. We analyze thisassumption and show that it is incorrect in the general case. However, it isshown that there is always a time t_0, after which this assumption becomestrue. The value of t_0 is the smaller the less is the degree of non-diagonalityof the model. Some details of the behaviour of the non-diagonal homogeneousmodel of Bianchi type IX are considered at the stage of asymptotic dynamics ofapproaching the singularity.

The dynamics of general Bianchi IX model near the cosmological singularity