  We study the first order infared behavior of tame hyperbolizable two-field
cosmological models, defined as those classical two-field models whose scalar
manifold is a connected, oriented and topologically finite hyperbolizable
Riemann surface $(\Sigma,\mathcal{G})$ and whose scalar potential $\Phi$ admits
a positive and Morse extension to the end compactification of $\Sigma$. We
achieve this by determining the universal forms of the asymptotic gradient flow
of the classical effective potential $V$ with respect to the uniformizing
metric $G$ near all interior critical points and ends of $\Sigma$, finding that
some of the latter act like fictitious but exotic stationary points of the
gradient flow. We also compare these results with numerical studies of
cosmological orbits. For critical cusp ends, we find that cosmological curves
have transient quasiperiodic behavior but are eventually attracted or repelled
by the cusp along principal geodesic orbits determined by the extended
effective potential. This behavior is approximated in the infrared by that of
gradient flow curves near the cusp.
