  In this paper we seek for relevant information on the asymptotic cosmological
dynamics of the Brans--Dicke theory of gravity for several self-interaction
potentials. By means of the simplest tools of the dynamical systems theory, it
is shown that the general relativity de Sitter solution is an attractor of the
Jordan frame (dilatonic) Brans--Dicke theory only for the exponential potential
$U(\vphi)\propto\exp\vphi$, which corresponds to the quadratic potential
$V(\phi)\propto\phi^2$ in terms of the original Brans--Dicke field
$\phi=\exp\vphi$, or for potentials which asymptote to $\exp\vphi$. At the
stable de Sitter critical point, as well as at the stiff-matter equilibrium
configurations, the dilaton is necessarily massless. We find bounds on the
Brans--Dicke coupling constant $\omega_\textsc{bd}$, which are consistent with
well-known results.
