The dynamics of general Bianchi IX model near the cosmological singularity

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