  In this paper we investigate the asymptotic dynamics of inflationary
cosmological models that are based in scalar-tensor theories of gravity. Our
main aim is to explore the global structure of the phase space in the framework
of single-field inflation models. For this purpose we make emphasis in the
adequate choice of the variables of the phase space. Our results indicate that,
although single-field inflation is generic in the sense that the corresponding
critical point in the phase space exists for a wide class of potentials, along
given phase space orbits -- representing potential cosmic histories -- the
occurrence of the inflationary stage is rather dependent on the initial
conditions. We have been able to give quantitative estimates of the relative
probability (RP) for initial conditions leading to slow-roll inflation. For the
non-minimal coupling model with the $\phi^2$-potential our rough estimates
yield to an almost vanishing relative probability: $10^{-13}\,\%\lesssim RP\ll
10^{-8}\,\%$. These bonds are greatly improved in the scalar-tensor models,
including the Brans-Dicke theory, where the relative probability $1\,\%\lesssim
RP\leq 100\,\%$. Hence slow-roll inflation is indeed a natural stage of the
cosmic expansion in Brans-Dicke models of inflation. It is confirmed as well
that the dynamics of vacuum Brans-Dicke theories with arbitrary potentials are
non-chaotic.
