  We present a phase-space analysis of cosmology containing multiple scalar
fields with a positive or negative cross-coupling exponential potential. We
show that there exist power-law kinetic-potential-scaling solutions for a
sufficiently flat positive potential or for a steep negative potential. The
former is the unique late-time attractor, but it is difficult to yield assisted
inflation. The later is never stable in an expanding universe. Moreover, for a
steep negative potential there exists a kinetic-dominated regime in which each
solution is a late-time attractor. In the presence of ordinary matter these
scaling solutions with a negative cross-coupling potential are found unstable.
We briefly discuss the physical consequences of these results.
