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