  We give a full investigation on the dynamics of power-law kinetic
quintessence $L(X, \phi)=V(\phi)(-X+X^2)$ by considering the potential related
parameter $\Gamma$($=\frac{V V''}{V'^2}$) as a function of another potential
parameter $\lambda$($=\frac{V'}{\kappa V^{3/2}}$), which correspondingly
extends the analysis of the dynamical system of our universe from two-dimension
to three-dimension. Beside the critical points found in previous papers, we
find a new de-Sitter-like dominant attractor(\textit{cp$6$}) and give its
stable condition using the center manifold theorem. For the dark energy
dominant solution(\textit{cp$6$} and \textit{cp$7$}), it could be distinguished
from canonical quintessence and tachyon models since the sound speed $c_s^2=0$
or $c_s^2\ll 1$. For the scaling solution (\textit{cp$8$}), it is very
interesting that the sound speed $c_s^2=1/5$ while it behaves as ordinary
matter. We therefore point out that the power-law kinetic quintessence should
have different signatures on cold dark matter power spectrum and cosmic
microwave background both at early time when this scalar field is an early dark
energy with $\Omega_\phi$ being non-negligible at high redshift and at late
time when it drives the accelerating expansion. We even do not know whether
there are any degeneracies of the impacts between these two epoches. They are
expected to be investigated in future.
