Stability of a modified Jordan-Brans-Dicke theory in the dilatonic frame

We investigate the Jordan-Brans-Dicke action in the cosmological scenario of FLRW spacetime with zero spatially curvature and with an extra scalar field minimally coupled to gravity as matter source. The field equations are studied in two ways. The method of group invariant transformations, i.e., symmetries of differential equations apply in order to constraint the free functions of the theory and determine conservation laws for the gravitational field equations. The second method that we apply for the study of the evolution of the field equations is the stability analysis of equilibrium points. Particularly, we find solutions with $w_{\text{tot}}=-1$, and we study their stability by means of the Center Manifold Theorem. We show this solution is an attractor in the dilatonic frame but it is an intermediate accelerated solution $a \simeq e^{A t^p}, p:=\frac{2}{2+l}, \quad \frac{32}{57+6 \omega_0}<p<\frac{2}{3}, \;\text{as}\; t\rightarrow \infty$, and not de Sitter solution. The exponent $p$ is reduced, in a particular case, to the exponent already found for the Jordan's and Einstein's frames by A. Cid, G. Leon and Y. Leyva, JCAP 1602, no. 02, 027 (2016). We obtain some equilibrium points which represent stiff solutions. Additionally, we find solutions that can be a phantom solution, a solution with $w_{\text{tot}}=-1$ or a quintessence solution. Other equilibrium points mimic a standard dark matter source ($0<w_{\text{tot}}<1$), radiation ($w_{\text{tot}}=\frac{1}{3}$), among other interesting features. For the dynamical system analysis we develop an extension of the method of $F$-devisers. The new approach relies upon two arbitrary functions $h(\lambda, s)$ and $F(s)$. The main advantage of this procedure allows us to perform a phase-space analysis of the cosmological model without the need for specifying the potentials, revealing the full capabilities of the model.