Cosmological dynamics in the theory of gravity both with non-minimal and non-minimal derivative coupling

This paper explores cosmological scenarios in a scalar-tensor theory of gravity, including both a non-minimal coupling with scalar curvature of the form $R\phi^2$ and a non-minimal derivative coupling of the form $G^{\mu\nu}\phi_{,\mu}\phi_{,\nu}$ in the presence of a scalar field potential with the monomial dependence $V(\phi) = V_0\phi^n$. Critical points of the system were obtained and analyzed. In the absence of a scalar field potential, stability conditions for these points were determined. Using methods of dynamical systems theory, the asymptotic behavior of the model was analyzed. It was shown that in the case of $V(\phi)\equiv0$ or $n < 2$, a quasi-de Sitter asymptotic behavior exists, corresponding to an early inflationary universe. This asymptotic behavior in the approximation $V_0 \rightarrow 0,\ \xi \rightarrow 0$ coincides with the value $H = \frac{1}{\sqrt{9|\eta|}}$ obtained in works devoted to cosmological models with non-minimal kinetic coupling. For $|\xi|\ \rightarrow \infty$, this asymptotic behavior tends to the value $H = \frac{1}{\sqrt{3|\eta|}}$. Moreover, unstable regimes with phantom expansion $w_{eff} < -1$ were found for the early dynamics of the model. For the late dynamics, the following stable asymptotic regimes were obtained: a power-law expansion with $w_{eff} \ge 1$, an expansion with $w_{eff} =\frac{1}{3}$ ($V(\phi)\equiv0$), at which the effective Planck mass tends to zero, and an exponential expansion with $w_{eff} = 0$ as $n = 2$. In this case, the asymptotic value of the Hubble parameter depends only on $V_0 = \frac{1}{2}m^2$ and $\xi$. Numerical integration of the model dynamics was performed for specific values of the theory parameters. The results are presented as phase portraits.