  We study cosmological solutions in $R + \beta R^{N}$-gravity for an isotropic
Universe filled with ordinary matter with the equation of state parameter
$\gamma$. Using the Bogolyubov-Krylov-Mitropol'skii averaging method we find
asymptotic oscillatory solutions in terms of new functions, which have been
specially introduced by us for this problem and appeared as a natural
generalization of the usual sine and cosine. It is shown that the late-time
behaviour of the Universe in the model under investigation is determined by the
sign of the difference $\gamma-\gamma_{crit}$ where $\gamma_{crit}=2N/(3N-2)$.
If $\gamma < \gamma_{crit}$, the Universe reaches the regime of small
oscillations near values of Hubble parameter and matter density, corresponding
to General Relativity solution. Otherwise higher-curvature corrections become
important at late times. We also study numerically basins of attraction for the
oscillatory and phantom solutions, which are present in the theory for $N>2$.
Some important differences between $N=2$ and $N>2$ cases are discussed.
