Contents

We study cosmological solutions in R+βRN-gravity for an isotropic Universe filled with ordinary matter with the equation of state parameter γ. 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 ap-peared 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 γ−γcrit where γcrit = 2N/(3N −2). If γ < γcrit, the Universe reaches the regime of small oscillations near values of Hubble parameter and matter density, corresponding to General Relativity solution. Otherwise higher-curvature cor-rections 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