will introduce in the next section.

Further discussion of the autonomous system {#sec4}
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The dynamical behavior of the scalar field has been discussed by many authors (e.g., see [@Copeland-IJMPD; @Coley-Book; @Copeland-exponential; @Hao-PRD1; @Hao-PRD2; @Samart-Phantom; @Li-O(N); @Ferreira-scaling; @Hoogen-scaling; @Billyard-interaction; @Fu-phantom]). If one wants to get the potentials that yield the cosmological scaling solutions beyond the exponential potential, one can add a $\frac{d\phi}{dN}$ term into the autonomous system [@Nunes-scaling]. All of these methods deal with special cases of the dynamical behavior of scalar fields in backgrounds of some specific forms. By considering $\Gamma$ as a function of $\lambda$, one can treat potentials of more general forms and get the common fixed points of the general potential, as shown in [@Zhou-plb; @Fang-CQG] and in the two preceding sections. However, as is discussed in [@Fang-CQG], sometimes $\Gamma$ is not a function of $\lambda$, and then the dynamical behaviors of the scalar fields discussed above are still not general in the strict sense. For a more general discussion, we must consider the higher-order derivatives of the potential. We define $$\begin{aligned}
&&{}^{(1)}\Gamma=\frac{VV_3}{V'^2}, \quad {}^{(2)}\Gamma
=\frac{VV_4}{V'^2},\quad{}^{(3)}\Gamma=\frac{VV_5}{V'^2},\nonumber\\
&&\cdots\quad{}^{(n)}\Gamma=\frac{VV_{n+2}}{V'^2},\quad \cdots\end{aligned}$$ in which $V_{n}=\frac{d^n V}{d\phi^n},n=3,4,5,\cdots$. Then we can get $$\begin{aligned}
\frac{d\Gamma}{d N}
&=&\sqrt{6}x\left[\Gamma\lambda+{}^{(1)}\Gamma-2\lambda\Gamma^2\right],\label{G'}\\
\frac{d\left({}^{(1)}\Gamma\right)}{dN}
&=&\sqrt{6}x\left[{}^{(1)}\Gamma\lambda +{}^{(2)}\Gamma
-2\lambda\Gamma\left({}^{(1)}\Gamma\right)\right],\label{G3'}\\
\frac{d\left({}^{(2)}\Gamma\right)}{dN} &=&\sqrt{6}x\left[
