Future asymptotic behaviour of tilted Bianchi models of type IV and VIIh

Using dynamical systems theory and a detailed numerical analysis, the late-time behaviour of tilting perfect fluid Bianchi models of types IV and VIIh is investigated. In particular, vacuum plane-wave spacetimes are studied and the important result that the only future attracting equilibrium points for noninflationary fluids are the plane-wave solutions in Bianchi type VIIh 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 normalized energy density tends to zero and the normalized variables 'freeze' into their asymptotic values. A detailed numerical analysis of the type VIIh 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.