  Using dynamical systems theory and a detailed numerical analysis, the
late-time behaviour of tilting perfect fluid Bianchi models of types IV and
VII$_h$ are investigated. In particular, vacuum plane-wave spacetimes are
studied and the important result that the only future attracting equilibrium
points for non-inflationary fluids are the plane-wave solutions in Bianchi type
VII$_h$ models is discussed. A tiny region of parameter space (the loophole) in
the Bianchi type IV model is shown to contain a closed orbit which is found to
act as an attractor (the Mussel attractor). From an extensive numerical
analysis it is found that at late times the normalised energy-density tends to
zero and the normalised variables 'freeze' into their asymptotic values. A
detailed numerical analysis of the type VII$_h$ models then shows that there is
an open set of parameter space in which solution curves approach a compact
surface that is topologically a torus.
