the expression here due to the complexity of the formula. Making the redshift $z={1/a}-1$ vary in an enough large range involving far future and far past, e.g. from $-1$ to 5, one can easily get the evolution trajectories in the statefinder $s-r$ plane of this model. Selected curves of $r(s)$ are plotted in Fig.5 and Fig.6. In Fig.5, we fix $\alpha=0.5$ and vary $w_X$ as $-0.8$, $-1$, and $-1.2$, respectively. In Fig.6, we fix $w_X=-1.2$ and vary $\alpha$ as 0, $\pm0.2$, $\pm0.5$, and $\pm0.8$, respectively. Other parameters are taken as the same as previous figures. In these two figures, dots locate the today’s values of the statefinder parameters $(s_0, r_0)$ and arrows denote the evolution directions of the statefinder trajectories $r(s)$. The $\Lambda$CDM model locates at $(0, 1)$ in the $s-r$ plane also denoted as a dot.

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The statefinder diagnostic can discriminate between various dark energy models effectively. Different cosmological models involving dark energy exhibit qualitatively different evolution trajectories in the $s-r$ plane. For example, the $\Lambda$CDM scenario corresponds to the fixed point $s=0,~r=1$ as shown in (\[lcdm\]), and the SCDM (standard cold dark matter) scenario corresponds to the point $s=1,~r=1$. For the “quiessence" (XCDM) models, the trajectories 