If one wants to study as much as possible the dynamical properties of this infinite-dimensional autonomous system, one has to consider a truncation that sets ${}^{(M)}\Gamma=\rm{Const.}$, with $M$ above a certain positive integer. Then the infinite-dimensional system can be reduced to $M+4)$ dimensions. And we find that there is just the scalar-field-dominated scaling solution for this autonomous system. We only give out the basic properties of this infinite-dimensional autonomous system in this paper, and will continue the discussion in the paper in [@Xiao-scaling].

We only get the scalar-field-dominated scaling solutions, whether we consider $\Gamma$ as a function of $\lambda$ or consider the higher-order derivatives of the potential. This conclusion is very different from the autonomous system which is just described by a scalar field with an exponential potential [@Samart-Phantom]. This is an interesting problem that awaits further analysis.

K. Xiao thanks Professor X. Li for his help with the center manifold theorem. This work was supported by the National Natural Science Foundation of China, under Grant No. 10875012 and the Fundamental Research Funds for the Central Universities.

The stability properties of the Point $P_3$
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In Sec. \[sec3\], we point out that it is impossible to get the stability properties 