Networks of cosmological histories, crossing of the phantom divide line and potentials with cusps

We discuss the phenomenon of the smooth dynamical gravity induced crossing of the phantom divide line in a framework of simple cosmological models where it appears to occur rather naturally, provided the potential of the unique scalar field has some kind of cusp. The behavior of cosmological trajectories in the vicinity of the cusp is studied in some detail and a simple mechanical analogy is presented. The phenomenon of certain complementarity between the smoothness of the spacetime geometry and matter equations of motion is elucidated. We introduce a network of cosmological histories and qualitatively describe some of its properties.