are the common properties of scalar field from those that are just related to the special potentials [@Fang-CQG]. If we consider a more general $\lambda$, then we can get the more general stability properties of scalar field in the LQC scenario. We will continue the discussion of this topic in Sec. \[sec4\]. In this case, Eq. (\[l\]) becomes $$\begin{aligned}
\frac{d\lambda}{dN}&=&\sqrt{6}\lambda^2 xf(\lambda).\label{l'}\end{aligned}$$ Hereafter, Eqs. (\[x’\])-(\[z’\]) along with Eq. (\[l’\]) definitely describe a dynamical system. We will discuss the stability of this system in the following section.

Properties of the autonomous system {#sec3}
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Obviously, the terms on the right-hand side of Eqs. (\[x’\])-(\[z’\]) and (\[l’\]) only depend on $x,y,z,\lambda$, but not on $N$ or other variables. Such a dynamical system is usually called an autonomous system. For simplicity, we define $
\frac{dx}{dN}=F_1(x,y,z,\lambda)\equiv F_1,
\frac{dy}{dN}=F_2(x,y,z,\lambda)\equiv F_2,
\frac{dz}{dN}=F_3(x,y,z,\lambda)\equiv F_3$, and $\frac{d\lambda}{dN}=F_4(x,y,z,\lambda)\equiv F_4.$ The fixed points $(x_c,y_c,z_c,\lambda_c)$ satisfy $F_i=0, i=1,2,3,4$. From Eq. (\[l’\]), it is straightforward to see that $x=0, \lambda=0$ or $f(\lambda)=0$ can make $F_4(x,y,z,\lambda)=0$. Also, we must consider $\lambda^2f(\lambda)=0$. Just as [@Fang-CQG] argued, it is possible that $\lambda^2 f(\lambda)\neq0$ and $\frac{d\lambda}{dN}\neq0$ when $\lambda=0$. Thus the necessary conditionfor the existence of the fixed points with $x\neq 0$ is $\lambda^2f(\lambda)=0$. Taking into account these 