equal $1$. But, if $y_*>1$, then $z_c=1-1/y_*^2<1$, and this means that the fixed point is located in the quantum-dominated regions. Although the stability of this point in the quantum regions may depend on $f(\lambda)$, it is not necessary to analyze its dynamical properties, since it does not have any physical meanings. The reason is the following: If the Universe is stable, it will not evolve to today’s pictures. If the Universe is unstable, it will always be unstable. We will just focus on point $P_3$ staying in the classical regions. Then $y_c=y_*=1,z_c=1-1/y_*^2=0$, i.e., for the classical cosmology region, $\rho\ll\rho_c$.

3.  Since the adiabatic index $\gamma$ satisfies $0<\gamma< 2$ (in particular, for radiation $\gamma=\frac43$ and for dust $\gamma=1$), all the terms that contain $\gamma$ should not change sign. A more general situation of $\gamma$ is $0\leq \gamma\leq 2$ [@Billyard-scaling]. If $\gamma=0$ or $\gamma=2$, the eigenvalues corresponding to points $P_{1,2,4,5,9}$ will have some zero elements and some negative ones. To analyze the stability of these points, we need to resort to other more complex methods, just as we do in the Appendix for the dynamical properties of point $P_3$. In this paper, we just consider the barotropic fluid which includes radiation and 