  We investigate whether a self-interacting Brans-Dicke theory in $d=5$ without
matter and with a time-dependent metric can describe, after dimensional
reduction to $d=4$, the FLRW model with accelerated expansion and
non-relativistic matter. By rewriting the effective 4-dimensional theory as an
autonomous three-dimensional dynamical system and studying its critical points,
we show that the $\Lambda$CDM cosmology cannot emerge from such a model. This
result suggests that a richer structure in $d=5$ may be needed to obtain the
accelerated expansion as well as the matter content of the 4-dimensional
universe.
