can find that points $P_{1-10}$ are just special cases of the fixed points of an infinite-dimensional autonomous systems. Considering the definition of $\Gamma$ (see Eq. (\[Gamma\])), the simplest potential is an exponential potential when $\Gamma_c=1$. The properties of these fixed points have been discussed by many authors [@Copeland-exponential; @Hao-PRD1; @Hao-PRD2; @Samart-Phantom; @Li-O(N); @Ferreira-scaling; @Hoogen-scaling; @Billyard-interaction; @Fu-phantom]. If $x_c=0$ and $y_c=0$, this corresponds to a fluid-dominated universe, which we do not consider here. If $x_c=\pm 1$, $\Gamma_c=0$ and ${}^{(n)}\Gamma_c=0$, we do not need to consider the $\Gamma$ and the $^{(n)}\Gamma$ terms. Then the stability properties of these points are the same as points $P_{4,5}$ in Table **I**, and there are unstable points. The last case is $(x_c,y_c,z_c,\lambda_c)=(0,y_*,1-1/y_*^2,0)$ and $\Gamma,{}^{(n)}\Gamma$ can be any value. To analyze the dynamical properties of this autonomous system, we need to consider the ${}^{(n)}\Gamma_c$ terms. We will get an infinite series. In order to solve this infinite series, we must truncate it by setting a sufficiently high-order ${}^{(M)}\Gamma$ to be a constant, for a positive integer $M$, so that $d\left({}^{(M)}\Gamma\right)/dN=0$. Thus we can get an $(M+4)$-dimensional autonomous system. One example is the quadratic potential $V=\frac12m^2\phi^2$ with some positive constant $m$ that gives a five dimensional autonomous system, 