  A detailed analysis of dynamics of cosmological models based on $R^{n}$
gravity is presented. We show that the cosmological equations can be written as
a first order autonomous system and analyzed using the standard techniques of
dynamical system theory. In absence of perfect fluid matter, we find exact
solutions whose behavior and stability are analyzed in terms of the values of
the parameter $n$. When matter is introduced, the nature of the (non-minimal)
coupling between matter and higher order gravity induces restrictions on the
allowed values of $n$. Selecting such intervals of values and following the
same procedure used in the vacuum case, we present exact solutions and analyze
their stability for a generic value of the parameter $n$. From this analysis
emerges the result that for a large set of initial conditions an accelerated
expansion is an attractor for the evolution of the $R^n$ cosmology. When matter
is present a transient almost-Friedman phase can also be present before the
transition to an accelerated expansion.
