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Intermediate accelerated solutions as generic late-time attractors in a modified Jordan-Brans-Dicke theory

We investigate a Jordan-Brans-Dicke (JBD) scalar field, $Φ$, with power-law potential in the presence of a second scalar field, $ϕ$, with an exponential potential, in both the Jordan and the Einstein frames. We present the relation of our model with the induced gravity model with power-law potential and the integrability of this kind of models is discussed when the quintessence field $ϕ$ is massless, and has a small velocity. We prove that in JBD theory, the de Sitter solution is not a natural attractor but an intermediate accelerated solution of the form $a(t)\simeq e^{α_1 t^{p_1}}$, as $t\rightarrow \infty$ where $α_1>0$ and $0<p_1<1$, for a wide range of parameters. Furthermore, in the Einstein frame we get that the attractor is also an intermediate accelerated solution of the form $\mathfrak{a}(\mathfrak{t})\simeq e^{α_2 \mathfrak{t}^{p_2}}$ as $\mathfrak{t}\rightarrow \infty$ where $α_2>0$ and $0<p_2<1$, for the same conditions on the parameters as in the Jordan frame. In the special case of a quadratic potential in the Jordan frame, or for a constant potential in the Einstein's frame, these solutions are of saddle type. Finally, we present a specific elaboration of our extension of the induced gravity model in the Jordan frame, which corresponds to a linear potential of $Φ$. The dynamical system is then reduced to a two dimensional one, and the late-time attractor is linked with the exact solution found for the induced gravity model. In this example the intermediate accelerated solution does not exist, and the attractor solution has an asymptotic de Sitter-like evolution law for the scale factor. Apart from some fine-tuned examples such as the linear, and quadratic potential ${U}(Φ)$ in the Jordan frame, it is true that intermediate accelerated solutions are generic late-time attractors in a modified JBD theory.