dust, and $\gamma\neq 0,2$. Notice that if one considers $\gamma=0$, the barotropic fluid describes the dark energy. This is an interesting topic, but will not be considered here for the sake of simplicity.

4.  $-\sqrt{6}<\lambda_a<\sqrt{6},\lambda_a\neq 0$ should hold for point $P_9$, hence $-3+\frac12\lambda^2_a<0$.

5.  $\lambda_a>0$ should hold, since $y_c>0$ for point $P_{10}$. The eigenvalue of this point is $$\begin{aligned}
    \label{p10}
    \mathbf{M}=\begin{pmatrix}
    -3\gamma\\-3\lambda_a\gamma f_1(\lambda_a)\\-\frac32+\frac34\gamma+\frac{3}{4\lambda_a}\sqrt{(2-\gamma)(\lambda^2_a(2-\gamma)+8\gamma+24\gamma^2)}\\
    -\frac32+\frac34\gamma-\frac{3}{4\lambda_a}\sqrt{(2-\gamma)(\lambda^2_a(2-\gamma)+8\gamma+24\gamma^2)}
    \end{pmatrix}.\nonumber\\\end{aligned}$$ Since we just consider $0<\gamma<2$ in this paper, it is easy to check that $(2-\gamma)(\lambda^2_a(2-\gamma)+8\gamma+24\gamma^2)>0$ is always satisfied. And this point is unstable with $f_1(\lambda_a)=\left.\frac{df(\lambda)}{d\lambda}\right|_{\lambda=\lambda_a}$ being either negative or positive, since $-\frac32+\frac34\gamma+\frac{3}{4\lambda_a}\sqrt{(2-\gamma)(\lambda^2_a(2-\gamma)+8\gamma+24\gamma^2)}$ is always positive.

Based on Table **I** and the related remarks above, we have the folloing conclusions:

1.  Points $P_{1,2}$: The related critical values, eigenvalues and stability properties do not depend on the specific form of the potential, since $\lambda_c=0$ or $\lambda$ can be any value $\lambda_*$.

2.  Point $P_3$: The related stability properties depend on $f_1(0)=\left.\frac{df(\lambda)}{d\lambda}\right|_{\lambda=0}$.

3.  Points $P_{4,5}$: The related eigenvalues and stability properties do not depend on the form of the potential, but the critical values of these points should satisfy $\lambda^2 f(\lambda)=0$ since $x_c\neq 0$.

4.  Point $P_6$: It is a special case of $P_2$, but $f(\lambda_a)=0$ should be 