Multidimensional cosmological-type model with n Einstein factor spaces in the theory with l scalar fields and multiple exponential potential is considered .
The dynamics of the model near the singularity is reduced to a billiard on the ( N - 1 ) -dimensional Lobachevsky space H ^ { N - 1 } , N = n + l .
It is shown that for n > 1 the oscillating behaviour near the singularity is absent and solutions have an asymptotical Kasner-like behavior .
For the case of one scale factor ( n = 1 ) billiards with finite volumes ( e.g .
coinciding with that of the Bianchi-IX model ) are described and oscillating behaviour of scalar fields near the singularity is obtained .