  We analyze the qualitative behaviors of scalar-tensor cosmologies with an
arbitrary monotonic $\omega(\Phi)$ function. In particular, we are interested
on scalar-tensor theories distinguishable at early epochs from General
Relativity (GR) but leading to predictions compatible with solar-system
experiments. After extending the method developed by Lorentz-Petzold and
Barrow, we establish the conditions required for convergence towards GR at
$t\rightarrow\infty$. Then, we obtain all the asymptotic analytical solutions
at early times which are possible in the framework of these theories. The
subsequent qualitative evolution, from these asymptotic solutions until their
later convergence towards GR, has been then analyzed by means of numerical
computations. From this analysis, we have been able to establish a
classification of the different qualitative behaviors of scalar-tensor
cosmological models with an arbitrary monotonic $\omega(\Phi)$ function.
