Abstract
We study the bounce and cyclicity realization in the framework of new gravitational scalar-tensor theories. In these theories, the Lagrangian contains the Ricci scalar and its first and second derivatives, in a specific combination that makes them free of ghosts, and transformed into the Einstein frame, they are proved to be a subclass of biscalar extensions of general relativity. We present analytical expressions for the bounce requirements, and we examine the necessary qualitative behavior of the involved functions that can give rise to a given scale factor. Having in mind these qualitative forms, we reverse the procedure, and we construct suitable simple Lagrangian functions that can give rise to a bounce or cyclic scale factor.