  We study the Kaluza-Klein inspired Brans-Dicke model with barotropic matter.
Following from our previous work, the traditional Kaluza-Klein gravity action
is introduced with an additional scalar field and 2 gauge fields. The
compactification process results in a Brans-Dicke model with a dilaton coupled
to the tower of scalar fields whereas a gauge field from 5-dimensional metric
forms a set of mutually orthogonal vectors with 2 additional gauge fields. The
barotropic matter is then introduced to complete a realistic set up. To
demonstrate the analytical solutions of the model, we consider the case in
which only 2 lowest modes becoming relevant for physics at low scale. After
derivation, equations of motion and Einstein field equations form a set of
autonomous system. The dynamical system is analysed to obtain various critical
points. Interestingly, by only inclusion of barotropic matter, the model
provides us the critical points which capable of determining the presences of
dark matter, dark energy and phantom dark energy.
