Asymptotic behavior of FRW cosmologies with many scalar fields

The dynamical evolution of FRW cosmologies conformally coupled to a finite set of scalar fields is studied. The scalar fields interact between them and we also consider a self-interaction term. lt is proved that when $k=-1$ and $k=0\mathrm{}$ only one physical asymptotic behavior is possible: the radius of the universe expands forever. When $k=1\mathrm{}$ either we have the just described behavior or the universe will ultimately collapse in a finite time. But we also give evidence that an oscillatory chaotic behavior is possible in this case if a fine-tuning between the initial conditions is satisfied.