  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 bi-scalar 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.
