{}^{(2)}\Gamma\lambda+{}^{(3)}\Gamma
-2 \lambda\Gamma\left({}^{(2)}\Gamma\right)\right],\label{G4'}\\
\frac{d\left({}^{(3)}\Gamma\right)}{dN}
&=&\sqrt{6}x\left[{}^{(3)}\Gamma\lambda+{}^{(4)}\Gamma
-2\lambda\Gamma\left({}^{(3)}\Gamma\right)\right],\label{G5'}\\
&&  \cdots\cdots\nonumber\\
\frac{d\left({}^{(n)}\Gamma\right)}{dN}
&=&\sqrt{6}x\left[{}^{(n)}\Gamma\lambda+{}^{(n+1)}\Gamma
-2\lambda\Gamma \left({}^{(n)}\Gamma\right)\right],\label{Gn'}\\
&& \cdots\cdots\nonumber\end{aligned}$$ To discuss the dynamical behavior of scalar field with more general potential, e.g., when neither $\lambda$ nor $\Gamma$ is constant, we need to consider a dynamical system described by Eqs. (\[x’\])-(\[l\]) coupled with Eqs. (\[G’\])-(\[Gn’\]). It is easy to see that this dynamical system is also an autonomous one. We can discuss the values of the fixed points of this autonomous system. Considering Eq. (\[l\]), we can see that the values of fixed points should satisfy $x_c=0$, $\lambda_c=0$, or $\Gamma_c=1$. Then, we can get the fixed points of this infinite-dimensional autonomous system.

1.  If $x_c=0$, considering Eqs. (\[x’\])-(\[z’\]), one can get $(y_c,z_c,\lambda_c)=(0,0,0)$ or $(y_c,z_c,\lambda_c)=(0,0,\lambda_*)$, and $\Gamma_c,{}^{(n)}\Gamma_c$ can be any values.

2.  If $\lambda_c=0$, considering Eqs. (\[x’\])-(\[z’\]), one can see that the fixed points of $(x,y,z)$ are $(x_c,y_c,z_c)=(0,y_*,1-1/y_*^2)$and $(x_c,y_c,z_c)=(\pm1,0,0)$. If $x_c=0$, $\Gamma_c$ and ${}^{(n)}\Gamma_c$ can be any values, and if $x_c=\pm 1$, ${}^{(n)}\Gamma_c=0$.

3.  If $\Gamma_c=1$, considering Eqs. (\[x’\])-(\[z’\]), one can get that the fixed points of $(x,y,z,\lambda)$ are $(x_c,y_c,z_c,\lambda_c)=(0,0,0,\lambda_*)$ and $(x_c,y_c,z_c,\lambda_c)=(\pm 1,0,0,\lambda_*)$. And ${}^{(n)}\Gamma_c$ should satisfy ${}^{(n)}\Gamma_c=\lambda_*^n$. There are other fixed points, which will be discussed below.

Based on the above analysis and Table **I**, one 