A dynamical systems approach to the tilted Bianchi models of solvable type

TL;DR

This paper analyzes the late-time behavior of tilted Bianchi cosmological models of solvable type using dynamical systems, revealing stability properties, bifurcations, and novel features such as limit cycles and torus attractors.

Contribution

It provides a detailed dynamical systems analysis of tilted Bianchi models, identifying stability, bifurcations, and new phenomena like limit cycles and torus attractors.

Findings

Vacuum plane-wave solutions are the only future attractors in Bianchi type VII_h.

Existence of closed orbits acting as attractors in certain parameter regimes.

Discovery of bifurcations leading to limit cycles and torus-shaped attractors.

Abstract

We use a dynamical systems approach to analyse the tilting spatially homogeneous Bianchi models of solvable type (e.g., types VI$_h$ and VII$_h$) with a perfect fluid and a linear barotropic $\gamma$-law equation of state. In particular, we study the late-time behaviour of tilted Bianchi models, with an emphasis on the existence of equilibrium points and their stability properties. We briefly discuss the tilting Bianchi type V models and the late-time asymptotic behaviour of irrotational Bianchi VII$_0$ models. We prove the important result that for non-inflationary Bianchi type VII$_h$ models vacuum plane-wave solutions are the only future attracting equilibrium points in the Bianchi type VII$_h$ invariant set. We then investigate the dynamics close to the plane-wave solutions in more detail, and discover some new features that arise in the dynamical behaviour of Bianchi cosmologies with the inclusion of tilt. We point out that in a tiny open set of parameter space in the type IV model (the loophole) there exists closed curves which act as attracting limit cycles. More interestingly, in the Bianchi type VII$_h$ models there is a bifurcation in which a set of equilibrium points turn into closed orbits. There is a region in which both sets of closed curves coexist, and it appears that for the type VII$_h$ models in this region the solution curves approach a compact surface which is topologically a torus.