  We study numerically the evolution of Bose-Einstein condensate in the Sinai
oscillator trap described by the Gross-Pitaevskii equation in two dimensions.
In the absence of interactions this trap mimics the properties of Sinai
billiards where the classical dynamics is chaotic and the quantum evolution is
described by generic properties of quantum chaos and random matrix theory. We
show that, above a certain border, the nonlinear interactions between atoms
lead to the emergence of dynamical thermalization which generates the
statistical Bose-Einstein distribution over eigenmodes of the system without
interactions. Below the thermalization border the evolution remains
quasi-integrable. Such a Sinai oscillator trap, formed by the oscillator
potential and a repulsive disk located in the vicinity of the center, had been
already realized in rst experiments with the Bose-Einstein condensate formation
by Ketterle group in 1995 and we argue that it can form a convenient test bed
for experimental investigations of dynamical of thermalization. Possible links
and implications for Kolmogorov turbulence in absence of noise are also
discussed.
