  We present the full set of evolution equations for the spatially homogeneous
cosmologies of type VI$_h$ filled with a tilted perfect fluid and we provide
the corresponding equilibrium points of the resulting dynamical state space. It
is found that only when the group parameter satisfies $h>-1$ a self-similar
solution exists. In particular we show that for $h>-{\frac 19}$ there exists a
self-similar equilibrium point provided that $\gamma \in
({\frac{2(3+\sqrt{-h})}{5+3\sqrt{-h}}},{\frac 32}) $ whereas for $h<-{\frac
19}$ the state parameter belongs to the interval $\gamma \in
(1,{\frac{2(3+\sqrt{-h})}{5+3\sqrt{-h}}}) $. This family of new exact
self-similar solutions belongs to the subclass $n_\alpha ^\alpha =0$ having
non-zero vorticity. In both cases the equilibrium points have a five
dimensional stable manifold and may act as future attractors at least for the
models satisfying $n_\alpha ^\alpha =0$. Also we give the exact form of the
self-similar metrics in terms of the state and group parameters. As an
illustrative example we provide the explicit form of the corresponding
self-similar radiation model ($\gamma={\frac 43}$), parametrised by the group
parameter $h$. Finally we show that there are no tilted self-similar models of
type III and irrotational models of type VI$_h$.
