  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.
