  We consider spatially homogeneous Ho\v{r}ava-Lifshitz (HL) models that
perturb General Relativity (GR) by a parameter $v\in (0,1)$ such that GR occurs
at $v=1/2$. We describe the dynamics for the extremal case $v=0$, which possess
the usual Bianchi hierarchy: type $\mathrm{I}$ (Kasner circle of equilibria),
type $\mathrm{II}$ (heteroclinics that induce the Kasner map) and type
$\mathrm{VI_0},\mathrm{VII_0}$ (further heteroclinics). For type
$\mathrm{VIII}$ and $\mathrm{IX}$, we use a computer-assisted approach to prove
the existence of periodic orbits which are far from the Mixmaster attractor and
thereby we obtain a new behaviour which is not described by the BKL picture of
bouncing Kasner-like states.
