  Cyclic universes with bouncing solutions are candidates for solving the big
bang initial singularity problem. Here we seek bouncing solutions in a modified
Gauss-Bonnet gravity theory, of the type $R+f(G)$, where $R$ is the Ricci
scalar, $G$ is the Gauss-Bonnet term, and $f$ some function of it. In finding
such a bouncing solution we resort to a technique that reduces the order of the
differential equations of the $R+f(G)$ theory to second order equations. As
general relativity is a theory whose equations are of second order, this order
reduction technique enables one to find solutions which are perturbatively
close to general relativity. We also build the covariant action of the order
reduced theory.
