Hořava-Lifshitz bouncing Bianchi IX universes: A dynamical system analysis

We examine the Hamiltonian dynamics of bouncing Bianchi IX cosmologies with three scale factors in Hořava-Lifshitz (HL) gravity. We assume a positive cosmological constant plus noninteracting dust and radiation as the matter content of the models. In this framework the modified field equations contain additional terms which turn the dynamics nonsingular. The six-dimensional phase space presents (i) two critical points in a finite region of the phase space, (ii) one asymptotic de Sitter attractor at infinity and (iii) a two-dimensional invariant plane containing the critical points; together they organize the dynamics of the phase space. We identify four distinct parameter domains A, B, C and D for which the pair of critical points engenders distinct features in the dynamics, connected to the presence of centers of multiplicity 2 and saddles of multiplicity 2. In the domain A the dynamics consists basically of periodic bouncing orbits, or oscillatory orbits with a finite number of bounces before escaping to the de Sitter attractor. The center with multiplicity 2 engenders in its neighborhood the topology of stable and unstable cylinders R×S3 of orbits, where R is a saddle direction and S3 is the center manifold of unstable periodic orbits. We show that the stable and unstable cylinders coalesce, realizing a smooth homoclinic connection to the center manifold, a rare event of regular/nonchaotic dynamics in bouncing Bianchi IX cosmologies. The presence of a saddle of multiplicity 2 in the domain B engenders a high instability in the dynamics so that the cylinders emerging from the center manifold about P2 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 2 P2. The presence of the saddle of multiplicity 2 engenders bifurcations of the invariant manifold as the energy E0 of the system increases relative to the energy Ecr2 of P2: (i) for E0Ecr2 the center manifold bifurcates into a 3-torus; (iv) for E0 sufficiently large the 3-torus bifurcates into three S3, an invariant manifold multiply connected. Such structures were not yet