Dynamical Stability of the Power Law K-essence Dark Energy Model with a New Interaction

We investigate the cosmological evolution of the power law K-essence dark energy (DE) model $F(X)= -\sqrt{X} + X$ with a new interaction $Q = \alpha\rho _m\rho _{\phi }H^{-1}$ in FRWL spacetime. The evolution behavior of dark energy under this interaction is analyzed by using dynamical systems method, and ten critical points are obtained. Among those critical points, a new stable point, which we called Scaling-like dark energy(DE) solution, is very important and interesting. The cosmological meaning of this attractor is different from the Scaling solution and dark energy dominated solution. For some value of model parameters, the universe will evolve to the attractor solution with the dark energy density parameter $\Omega_{\phi}=0.682946$ and the the equation of state $w_{\phi}=-0.99$, which can be in good agreement with the observed data, and alleviate the Coincidence Problem.