Stable manifolds of heteroclinic chains in Hořava-Lifshitz cosmology

We consider vacuum spatially homogeneous models in Hořava-Lifshitz gravity, which are expected to describe generic singularities for these modified gravity theories. These models perturb the well-known Bianchi models in general relativity (GR) by a parameter $v\in (0,1)$, in which GR occurs at $v=1\mathrm{/}2$. We show that all periodic heteroclinic chains of Bianchi type II orbits are robust for small perturbations of GR, i.e. they exist for any $v\approx 1\mathrm{/}2$. Moreover, we prove the existence of a stable manifold for each heteroclinic chain within a Cantor set for the supercritical case, $v\in (1\mathrm{/}2,1)$. In particular, we discuss the heteroclinic chains of periods three and four in detail.