Late time evolution of a nonminimally coupled scalar field system

We revisit the dynamics of a nonminimally coupled scalar field model in case of $F(\varphi )R$ coupling with $F(\varphi )=1-\xi {\varphi }^2$, and the potentials $V(\varphi )=V_0(1+{\varphi }^p{)}^2$, $V(\varphi )=V_0{e}^{\lambda {\varphi }^2}$. We use an autonomous system to bring out new asymptotic regimes, and find stable de-Sitter solution. Under the chosen functional form of $F(\varphi )$ and steep exponential potentials, a true de-Sitter solution is trivially satisfied for which the equation of state $w_{\varphi }\simeq -1$, the effective gravitational constant $G_{eff}$ and field $\varphi$ are constant that has been missed in the power law case and our previous study.