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We use a dynamical systems approach to analyse the tilting spatiallyhomogeneous 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. Inparticular, we study the late-time behaviour of tilted Bianchi models, with anemphasis on the existence of equilibrium points and their stability properties.We briefly discuss the tilting Bianchi type V models and the late-timeasymptotic behaviour of irrotational Bianchi VII$_0$ models. We prove theimportant result that for non-inflationary Bianchi type VII$_h$ models vacuumplane-wave solutions are the only future attracting equilibrium points in theBianchi type VII$_h$ invariant set. We then investigate the dynamics close tothe plane-wave solutions in more detail, and discover some new features thatarise in the dynamical behaviour of Bianchi cosmologies with the inclusion oftilt. We point out that in a tiny open set of parameter space in the type IVmodel (the loophole) there exists closed curves which act as attracting limitcycles. More interestingly, in the Bianchi type VII$_h$ models there is abifurcation in which a set of equilibrium points turn into closed orbits. Thereis a region in which both sets of closed curves coexist, and it appears thatfor the type VII$_h$ models in this region the solution curves approach acompact surface which is topologically a torus.Comment: 29 page

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