We use a dynamical systems approach to analyse the tilting spatially
homogeneous Bianchi models of solvable type (e.g., types VIh and VIIh)
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 VII0 models. We prove the
important result that for non-inflationary Bianchi type VIIh models vacuum
plane-wave solutions are the only future attracting equilibrium points in the
Bianchi type VIIh 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 VIIh 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 VIIh models in this region the solution curves approach a
compact surface which is topologically a torus.Comment: 29 page