  In this paper we study the dynamics of {\it orthogonal spatially homogeneous}
Bianchi cosmologies in $R^n$-gravity. We construct a compact state space by
dividing the state space into different sectors. We perform a detailed analysis
of the cosmological behaviour in terms of the parameter $n$, determining all
the equilibrium points, their stability and corresponding cosmological
evolution. In particular, the appropriately compactified state space allows us
to investigate static and bouncing solutions. We find no Einstein static
solutions, but there do exist cosmologies with bounce behaviours. We also
investigate the isotropisation of these models and find that all isotropic
points are flat Friedmann like.
