Brans-Dicke theory and the emergence of $\Lambda \mathrm{CDM}$ model

The dynamics of the Brans-Dicke theory with a scalar field potential function is investigated. We show that the system with a barotropic matter content can be reduced to an autonomous three-dimensional dynamical system. For an arbitrary potential function, we found the values of the Brans-Dicke parameter for which a global attractor in the phase space representing the de Sitter state exists. Using linearized solutions in the vicinity of this critical point, we show that the evolution of the Universe mimics the $\Lambda \mathrm{CDM}$ model. From the recent Planck satellite data, we obtain constraints on the variability of the effective gravitational coupling constant as well as the lower limit of the mass of the Brans-Dicke scalar field at the de Sitter state.