  We investigate the dynamical properties of a class of spatially homogeneous
and isotropic cosmological models containing a barotropic perfect fluid and
multiple scalar fields with independent exponential potentials. We show that
the assisted inflationary scaling solution is the global late-time attractor
for the parameter values for which the model is inflationary, even when
curvature and barotropic matter are included. For all other parameter values
the multi-field curvature scaling solution is the global late-time attractor
(in these solutions asymptotically the curvature is not dynamically
negligible). Consequently, we find that in general all of the scalar fields in
multi-field models with exponential potentials are non-negligible in late-time
behaviour, contrary to what is commonly believed. The early-time and
intermediate behaviour of the models is also studied. In particular, n-scalar
field models are investigated and the structure of the saddle equilibrium
points corresponding to inflationary m-field scaling solutions and
non-inflationary m-field matter scaling solutions are also studied (where m<n),
leading to interesting transient dynamical behaviour with new physical
scenarios of potential importance.
