to three dimensions. It is easy to check that the values $(x_{ec},y_{ec},z_{ec})$ of the fixed points are just the values $(x_c,y_c,z_c)$ of points $P_{6-10}$ in Table **I**. We focus on the two special fixed points: $$\begin{aligned}
&& F_1: (x_{ec},y_{ec},z_{ec})=(-\lambda/\sqrt{6}, \sqrt{1-\lambda^2/6}, 0),\nonumber\\
&&F_2: (x_{ec},y_{ec},z_{ec})=(-\sqrt{3/2}\gamma/\lambda,
\sqrt{3\gamma(2-\gamma)/(2\lambda^2)}, 0).\nonumber\end{aligned}$$ Using Lyapunov’s linearization method, we can find that $F_2$ is unstable and $F_1$ is stable if $\lambda<3\gamma$. It is easy to check that $\rho_\gamma=H^2[1/(1-z_{ec})-x_{ec}^2-y_{ec}^2]=0$ when $(x_{ec},y_{ec},z_{ec})=(-\lambda/\sqrt{6}, \sqrt{1-\lambda^2/6},
0)$. From the above analysis, we find that there is just the scalar-field-dominated scaling solution when we consider the autonomous system to be described by a self-interacting scalar field coupled with a barotropic fluid in the LQC scenario.

Conclusions {#sec5}
===========

The aim of this paper is two-fold. We discuss the dynamical behavior of scalar field in the LQC scenario following [@Fang-CQG; @Zhou-plb]. To further analyze the dynamical properties of scalar field with more general potential, we introduce an infinite-dimensional autonomous system.

To discuss the dynamical properties of scalar field in the LQC scenario, we take $\Gamma$ as a function of $\lambda$, and extend the autonomous system from three dimensions to four dimensions. We find this extended autonomous system has more fixed points than the three dimensional one 