Dynamical system approach to scalar-vector-tensor cosmology

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 $\omega=0.27647$.