On hyperbolicity violations in cosmological models with vector fields

Cosmological models with vector fields received much at in recent years. Unfortunately, roost of them are plagued with severe instabilities or other problems. In particular, it was noted in ref. ill that the models with a non-linear function of the kinetic term do always imply violations of hyperbolicity somewhere in the phase space. In this work we make this statement more precise in several respects and show that those violations may not be present around spatially homogeneous configurations of the vector field.