Ho\v{r}ava-Lifshitz Bouncing Bianchi IX Universes: A Dynamical System Analysis

We examine the Hamiltonian dynamics of bouncing Bianchi IX cosmologies in Ho\v{r}ava-Lifshitz gravity. The $6$-dim phase space presents two critical points, one asymptotic de Sitter attractor at infinity and a $2$-dim invariant plane. We identified four distinct parameter domains $A$, $B$, $C$ and $D$ for which the pair of critical points engenders distinct features in the dynamics. In the domain $A$ the dynamics consists basically of periodic bouncing orbits, or oscillatory orbits with a finite number of bounces. The center with multiplicity two engenders in its neighborhood the topology of stable and unstable cylinders $R \times S^3$ of orbits. We show that the stable and unstable cylinders coalesce realizing a smooth homoclinic connection to the center manifold, a rare event of regular/non-chaotic dynamics in bouncing Bianchi IX cosmologies. The presence of a saddle of multiplicity two in the domain $B$ engenders a high instability in the dynamics so that the cylinders emerging from the center manifold towards the bounce have four distinct attractors: the center manifold itself, the de Sitter attractor at infinity and two further momentum-dominated attractors with infinite anisotropy. In the domain $C$ we examine the features of invariant manifolds of orbits about a saddle of multiplicity two. The presence of the saddle of multiplicity two engenders bifurcations of the invariant manifold as the energy $E_0$ of the system increases relative to the energy $E_{cr}$ providing structures that were not yet observed in the literature. The domain $D$ is not examined as most of its features are present already in the previous domains.