On the dynamics of k-essence models

We investigate cosmological dynamics of models with higher-order corrections to the canonical (second-order) kinetic lagrangian for a scalar field, which have been termed k-essence. We study the qualitative dynamics of simple purely kinetic k-essence models and find that the simplest attempts to construct non-singular cosmological models by including higher-order terms in the kinetic lagrangian fail because of a di.erent type of singularity where the scalar field theory becomes ill-defined.