We investigate the cosmological evolution of the power law k-essence dark energy (DE) model with interaction in FRWL spacetime with the Lagrangian that contains a kinetic function $F(X)=-\sqrt{X}+X$. Concretely, the cosmological evolution in this model are discussed by the autonomous dynamical system and its critical points, together with the corresponding cosmological quantities, such as ${\mathsf{\Omega }}_{\varphi }$, $w_{\varphi }$, $c_s^2$, and q, are calculated at each critical point. The evolutionary trajectories are drawn in order to show the dynamical process on the phases plan around the critical points. The result that we obtained indicates that there are four dynamical attractors, and all of them correspond to an accelerating expansion of universe for certain potential parameter and coupling parameter. Besides that, the geometrical diagnostic by the statefinder hierarchy $S_3^{(1)}$ and $S_4^{(1)}$ of this scalar field model are numerically obtained by the phase components, as an extended null diagnostic for the cosmological constant. This diagnostic shows that both the potential parameter $\lambda$ and interaction parameter $\alpha$ play important roles in the evolution of the statefinder hierarchy.

中文翻译：

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相互作用的幂律K-本质暗能量模型的动力稳定性和几何诊断

我们研究了幂律k本质暗能量（DE）模型在FRWL时空与包含动力学函数的拉格朗日相互作用中的宇宙演化 $F（X）=-\sqrt{X}+X$。具体而言，通过自主动力系统及其临界点，以及相应的宇宙学量（例如：${\mathsf{\Omega }}_{\varphi }$， $w_{\varphi }$， $C_s^2$和q在每个关键点计算。绘制了演化轨迹，以显示关键点周围相计划上的动力学过程。我们得到的结果表明，存在四个动态吸引子，它们都对应于某些势能参数和耦合参数的宇宙加速膨胀。除此之外，状态查找器层次结构的几何诊断${\mathrm{小号}}_3^{（\mathrm{1个}）}$ 和 ${\mathrm{小号}}_4^{（\mathrm{1个}）}$通过相分量在数值上获得该标量场模型的特征，作为宇宙常数的扩展零诊断。此诊断表明，两个潜在参数$\lambda$ 和交互参数 $\alpha$ 在statefinder层次结构的演变中扮演重要角色。