  We discuss the behavior of cosmological curves and their first order infrared
approximants near critical ends of the scalar manifold $\Sigma$ and near
interior critical points of the scalar potential for tame hyperbolizable
two-field cosmological models by determining the universal forms of the
asymptotic gradient flow of the classical effective potential with respect to
the uniformizing metric near all these points and ends. We compare the
asymptotic behavior of gradient flow curves with numerical results for
cosmological curves.
