rewrite Eqs. (\[Fri\])-(\[dotrg\]) in the following forms: $$\begin{aligned}
\frac{dx}{dN} &=&-3x-\frac{\sqrt{6}}{2}\lambda y^2+x\left[3x^2+\frac{3\gamma}{2}\left(\frac{1}{1-z}-x^2-y^2\right)\right]\nonumber\\
&&\times(1-2z),\label{x'}\\
\frac{dy}{dN}&=&\frac{\sqrt{6}}{2}\lambda x
y+y\left[3x^2+\frac{3\gamma}{2}
\left(\frac{1}{1-z}-x^2-y^2\right)\right]\nonumber\\
&&\times(1-2z),\label{y'}\\
\frac{dz}{dN} &=&-3\gamma z-3z\left(1-z\right)\left(2x^2-\gamma x^2-\gamma y^2\right),\label{z'}\\
\frac{d\lambda}{dN}  &=&\sqrt{6}\lambda^2
x\left(\Gamma-1\right),\label{l}\end{aligned}$$ where $N=\ln a$ and $$\begin{aligned}
  \Gamma\equiv \frac{VV''}{V'^2}.\label{Gamma}\end{aligned}$$ Note that the potential $V(\phi)$ is positive in this paper, but one can also discuss a negative potential. Just as [@Heard-negative] has shown, the negative scalar potential could slow down the growth of the scale factor and cause the Universe to be in a collapsing phase. The dynamical behavior of the scalar field with the positive and negative potential in brane cosmology has been discussed by [@Copeland-scalar]. In this paper we are concerned only with an expanding universe, and both the Hubble parameter and the potential are positive.

Differentiating $\lambda$ with respect to the scalar field $\phi$, we obtain the relationship between $\lambda$ and $\Gamma$, $$\begin{aligned}
\frac{d\lambda^{-1}}{d\phi}=1-\Gamma. \label{l-G}\end{aligned}$$ If we only consider a special case of the potential, like exponential potential [@Copeland-exponential; @Hao-PRD1; @Hao-PRD2; @Samart-Phantom; @Li-O(N); @Ferreira-scaling; @Hoogen-scaling; @Billyard-interaction; @Fu-phantom], then $\lambda$ and $\Gamma$ are both constants. In this case, the four dimensional dynamical system, Eqs. (\[x’\])-(\[l\]), reduces to a 3-dimensional one, since $\lambda$ is a constant. (In the classical dynamical system, the $z$ term does not exist, 