the effective adiabatic index $\gamma_\phi=(\rho_\phi+p_\phi)/\rho_\phi=0$, which means that the scalar field is an effective cosmological constant. For point $P_{9}$, $\gamma_\phi=\lambda^2_a/2$. This describes a scaling solution that, as the universe evolves, the kinetic energy and the potential energy of the scalar field scale together. And we can see that there is not any barotropic fluid coupled with the scalar-field-dominated scaling solution. This is different from the dynamical behavior of scalar field with exponential potential $V=V_0
\exp(-\lambda\kappa\phi)$ in classical cosmology [@Copeland-exponential; @Hao-PRD1; @Hao-PRD2; @Samart-Phantom; @Li-O(N); @Ferreira-scaling; @Hoogen-scaling; @Billyard-interaction; @Fu-phantom], and also is different from the properties of the scalar field in brane cosmology [@Copeland-scalar], in which $\lambda=\rm{const.}$ (notice that the definition of $\lambda$ in [@Copeland-scalar] is different from the one in this paper) and $\Gamma$ is a function of $L(\rho(a))$ and $|V|$. In these models, the Universe may enter a stage dominated by a scalar field coupled with fluid when $\lambda,\gamma$ satisfy some conditions [@Copeland-exponential; @Copeland-scalar].

We discuss the dynamical behavior of the scalar field by considering $\Gamma$ as a function of $\lambda$ in this and the preceding sections. But $\Gamma$ can not always be treated as a function of $\lambda$. We need to consider a more general autonomous system, which we 