Coupled Quintessence and Phantom Based On a Dilaton
  Based on dilatonic dark energy model, we consider two cases: dilaton field with positive kinetic energy(coupled quintessence) and with negative kinetic energy(phantom). In the two cases, we investigate the existence of attractor solutions which correspond to an equation of state parameter $\omega=-1$ and a cosmic density parameter $\Omega_\sigma=1$. We find that the coupled term between matter and dilaton can't affect the existence of attractor solutions. In the Mexican hat potential, the attractor behaviors, the evolution of state parameter $\omega$ and cosmic density parameter $\Omega$, are shown mathematically. Finally, we show the effect of coupling term on the evolution of $X(\frac{\sigma}{\sigma_0})$ and $Y(\frac{\dot{\sigma}}{\sigma^2_0})$ with respect to $N(lna)$ numerically. 
Z. G. Huang, H. Q. Lu and W. Fang