Full investigation on the dynamics of power-law kinetic quintessence

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.