  Using scalar-vector-tensor Brans Dicke (VBD) gravity [3] in presence of self
interaction BD potential $V(\phi)$ and perfect fluid matter field action we
solve corresponding field equations via dynamical system approach for flat
Friedmann Robertson Walker metric (FRW). We obtained 3 type critical points for
$\Lambda CDM$ vacuum de Sitter era where stability of our solutions are
depended to choose particular values of BD parameter $\omega.$ One of these
fixed points is supported by a constant potential which is stable for
$\omega<0$ and behaves as saddle (quasi stable) for $\omega\geq0.$ Two other
ones are supported by a linear potential $V(\phi)\sim\phi$ which one of them is
stable for $\omega=0.27647.$ For a fixed value of $\omega$ there is at least 2
out of 3 critical points reaching to a unique critical point. Namely for
$\omega=-0.16856(-0.56038)$ the second (third) critical point become unique
with the first critical point. In dust and radiation eras we obtained 1
critical point which never become unique fixed point. In the latter case
coordinates of fixed points are also depended to $\omega.$ To determine
stability of our solutions we calculate eigenvalues of Jacobi matrix of 4D
phase space dynamical field equations for de Sitter, dust and radiation eras.
We should be point also potentials which support dust and radiation eras must
be similar to $V(\phi)\sim\phi^{-\frac{1}{2}}$ and $V(\phi)\sim\phi^{-1}$
respectively. In short our study predicts that radiation and dust eras of our
VBD-FRW cosmology transmit to stable de Sitter state via non-constant potential
(effective variable cosmological parameter) by choosing $\omega=0.27647$.
