Phase-space analysis of a class of k-essence cosmology

We perform a detailed phase-space analysis of a class of k-essence cosmology. We find that critical points can be divided into three classes: points unstable but the model stable, both points and the model stable, points stable but the model unstable. Like the case of points unstable but the model stable, the case of points stable but the model unstable is not relevant from a cosmological point of view, though they can be late-time attractors for the Universe. So in order to study the possible final state of the Universe, it is important to investigate not only the stability of the critical points but also the stability of the model. The case of both points and the model stable can further be divided into two classes: points only presenting decelerating phases and points at which all decelerating, constant-speed, and accelerating phases can appear; the final state of the Universe depends on the potential.