Late time evolution of a nonminimally coupled scalar field system

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