  We investigate observational constraints on the Brans-Dicke cosmological
model using observational data coming from distant supernovae type Ia, the
Hubble function $H(z)$ measurements, information coming from the
Alcock-Paczy{\'n}ski test, and baryon acoustic oscillations. Our analysis is
based on the modified Friedmann function resulting form dynamical
investigations of Brans-Dicke cosmology in the vicinity of a de Sitter state.
The qualitative theory of dynamical systems enables us to obtain three
different behaviors in the vicinity of this state. We find for a linear
approach to the de Sitter state
$\omega_{\textrm{BD}}=-0.8606^{+0.8281}_{-0.1341}$, for an oscillatory approach
to the de Sitter state $\omega_{\textrm{BD}}=-1.1103^{+0.1872}_{-0.1729}$, and
for the transient de Sitter state represented by a saddle-type critical point
$\omega_{\textrm{BD}}=-2.3837^{+0.4588}_{-4.5459}$. We obtain the mass of the
Brans-Dicke scalar field at the present epoch as $m_{\phi}\sim H_{0}$. The
Bayesian methods of model comparison are used to discriminate between obtained
models. We show that observational data point toward vales of the
$\omega_{\textrm{BD}}$ parameter close to the value suggested by the low-energy
limit of the bosonic string theory.
