Ancient Dynamics in Bianchi Models: Approach to Periodic Cycles

TL;DR

This paper analyzes the behavior of Bianchi cosmological models near the big-bang singularity, proving the existence of solutions approaching heteroclinic cycles around the Kasner circle, extending to complex heteroclinic chains.

Contribution

It demonstrates the existence of a family of solutions approaching heteroclinic cycles in Bianchi models near singularities, extending previous results to arbitrary bounded heteroclinic chains.

Findings

Existence of solutions approaching heteroclinic 3-cycles near the Kasner circle.

Extension of theory to arbitrary heteroclinic chains away from Taub points.

Applicability to all closed heteroclinic cycles of the Kasner map.

Abstract

We consider cosmological models of Bianchi type. In particular, we are interested in the alpha-limit dynamics near the Kasner circle of equilibria for Bianchi classes VIII and IX. They correspond to cosmological models close to the big-bang singularity. We prove the existence of a codimension-one family of solutions that limit, for t to negative infinity, onto a heteroclinic 3-cycle to the Kasner circle of equilibria. The theory extends to arbitrary heteroclinic chains that are uniformly bounded away from the three critical Taub points on the Kasner circle, in particular to all closed heteroclinic cycles of the Kasner map.