  We perform a detailed analysis for the asymptotic behaviour for the
multi-scalar field Chiral cosmological scenario. We present the asymptotic
behaviour for the one-field, two-fields and three-fields Chiral models. From
these results, and deriving conserved quantities, we present a Theorem for the
$N$-fields model for the Chiral model with $N$--fields. We find that the
maximum number of scalar fields which provide interesting physical results is
two-fields, while for $N>2$ the new stationary points are only of mathematical
interest since they do not describe new exact solutions different from those
recovered for $N=2$.
