  We investigate the general asymptotic behaviour of Friedman-Robertson-Walker
(FRW) models with an inflaton field, scalar-tensor FRW cosmological models and
diagonal Bianchi-IX models by means of Liapunov's method. This method provides
information not only about the asymptotic stability of a given equilibrium
point but also about its basin of attraction. This cannot be obtained by the
usual methods found in the literature, such as linear stability analysis or
first order perturbation techniques. Moreover, Liapunov's method is also
applicable to non-autonomous systems. We use this advantadge to investigate the
mechanism of reheating for the inflaton field in FRW models.
