  We study a gravitational model with curvature-squared $R^2$ and
curvature-quartic $R^4$ nonlinearities. The effective scalar degree of freedom
$\phi$ (scalaron) has a multi-valued 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 scalaron while in the vicinity of the
branching points scalaron has the bouncing behavior and cannot cross these
points. Moreover, there are branching points where 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.
