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

TL;DR

This paper analyzes the complex dynamical behavior of bouncing Bianchi IX cosmologies within Hořava-Lifshitz gravity, revealing various stable, unstable, and bifurcation phenomena in different parameter regimes.

Contribution

It provides a detailed dynamical systems analysis of bouncing Bianchi IX universes in Hořava-Lifshitz gravity, identifying critical points, invariant manifolds, and bifurcations.

Findings

Existence of stable and unstable cylinders of orbits.

Homoclinic connections indicating regular dynamics.

Bifurcations related to energy variations and saddle points.

Abstract

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.