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Multidimensional model describing the cosmological evolution of n Einsteinspaces in the theory with l scalar fields and forms is considered. Whenelectro-magnetic composite p-brane ansatz is adopted, and certain restrictionson the parameters of the model are imposed, the dynamics of the model near thesingularity is reduced to a billiard on the (N-1)-dimensional Lobachevskyspace, N = n+l. The geometrical criterion for the finiteness of the billiardvolume and its compactness is used. This criterion reduces the problem to theproblem of illumination of (N-2)-dimensional sphere by point-like sources. Someexamples with billiards of finite volume and hence oscillating behaviour nearthe singularity are considered. Among them examples with square and triangle2-dimensional billiards (e.g. that of the Bianchi-IX model) and a 4-dimensionalbilliard in ``truncated'' D = 11 supergravity model (without the Chern-Simonsterm) are considered. It is shown that the inclusion of the Chern-Simons termdestroys the confining of a billiard.Comment: 27 pages Latex, 3 figs., submit. to Class. Quantum Gra

Billiard representation for multidimensional cosmology with intersecting p-branes near the singularity