and another example is the Polynomial (concave) potential $V=M^{4-n}\phi^n$ [@Lindle-potential] that gives an $(n+3)$-dimensional autonomous system. Following the method we used in the two preceding sections, we can get the dynamical behavior of such finite-dimensional systems.

In the remainder of this section, we discuss whether this autonomous system has a scaling solution.

If $x_c=0$, then $\Gamma_c\neq 0,{}^{(n)}\Gamma_c\neq 0$, and the stability of the fixed points may depend on the truncation. As an example, if we choose ${}^{(2)}\Gamma=0$, then we can get a six dimensional autonomous system. The eigenvalues for the fixed point $(x_c,y_c,z_c,\lambda_c,\Gamma_c,{}^{(1)}\Gamma_c)=(0,0,0,\lambda_b,\Gamma_*,{}^{(1)}\Gamma_*)$, where $\lambda_b=0$ or $\lambda_b=\lambda_*$, is $$\begin{aligned}
\mathbf{M}^T=(0,0,0,\frac32\gamma,-3\gamma,-3+\frac32\gamma).\nonumber\end{aligned}$$ Obviously, this is an unstable point, and it has no scaling solution. The eigenvalues for the fixed point $(x_c,y_c,z_c,\lambda_c,\Gamma_c,{}^{(1)}\Gamma_c)=(0,1,0,0,\Gamma_*,{}^{(1)}\Gamma_*)$ is $$\begin{aligned}
\mathbf{M}^T=(0,0,0,0,-3\gamma,-3-3\gamma).\nonumber\end{aligned}$$ According to the center manifold theorem (see Chapter 8 of [@Khalil-non], or [@DynamicalReduction]), there are two nonzero eigenvalues, and we need to reduce the dynamical system to two dimensions to get the stability properties of the autonomous system. This point may have scaling solution, but we need more complex mathematical method. We discuss this problem in another paper [@Xiao-scaling].

We discuss the last case. If $\Gamma_c=1$, we can consider an exponential potential. Then the autonomous system is reduced 