We study a gravitational model with curvature-squared R2 and curvature-quartic R4 nonlinearities. The effective scalar degree of freedom $\phi${} (the scalaron) has a multivalued potential U($\phi${}) consisting of a number of branches. These branches are fitted with each other in the branching and monotonic points. In the case of four-dimensional space-time, we show that the monotonic points are penetrable for the scalaron, while in the vicinity of the branching points, the scalaron has the bouncing behavior and cannot cross these points. Moreover, there are branching points where the scalaron bounces an infinite number of times with decreasing amplitude, and the Universe asymptotically approaches the de~Sitter stage. Such accelerating behavior we call bouncing inflation. For this accelerating expansion, there is no need for original potential U($\phi${}) to have a minimum or to check the slow-roll conditions. A necessary condition for such inflation is the existence of the branching points. This is a new type of inflation. We show that bouncing inflation takes place both in the Einstein and Brans-Dicke frames.

Inhomogeneous multidimensional cosmological models with a higher-dimensional space-time manifold ${M=M}_0\times {\prod }_{i=1\mathrm{}}^n{M}_i$ $(n>~1)$ are investigated under dimensional reduction to $D_0$-dimensional effective models. In the Einstein conformal frame, small excitations of the scale factors of the internal spaces near minima of an effective potential can be observed as massive scalar fields in the external space-time. Parameters of models that ensure minima of the effective potentials are obtained for particular cases and masses of gravitational excitons are estimated.

We consider multidimensional gravitational models with a nonlinear scalar curvature term and form fields in the action functional. In our scenario it is assumed that the higher dimensional spacetime undergoes a spontaneous compactification to a warped product manifold. Particular attention is paid to models with quadratic scalar curvature terms and a Freund-Rubin-like ansatz for solitonic form fields. It is shown that for certain parameter ranges the extra dimensions are stabilized. In particular, stabilization is possible for any sign of the internal space curvature, the bulk cosmological constant, and of the effective four-dimensional cosmological constant. Moreover, the effective cosmological constant can satisfy the observable limit on the dark energy density. Finally, we discuss the restrictions on the parameters of the considered nonlinear models and how they follow from the connection between the D-dimensional and the four-dimensional fundamental mass scales.