Periodic orbits in Hořava–Lifshitz cosmologies

Abstract

We consider spatially homogeneous Hořava–Lifshitz 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 \(\textrm{I}\) (Kasner circle of equilibria), type \(\textrm{II}\) (heteroclinics that induce the Kasner map) and type \(\mathrm {VI_0},\mathrm {VII_0}\) (further heteroclinics). For type \(\textrm{VIII}\) and \(\textrm{IX}\), we use a computer-assisted approach to prove the existence of periodic orbits which are far from the Mixmaster attractor. Therefore we obtain a new behaviour which is not described by the BKL picture of bouncing Kasner-like states.