On periodic approximate solutions of dynamical systems with a quadratic right-hand side

We consider difference schemes for dynamical systems $\dot x = f (x)$ with a quadratic right-hand side that have $t$-symmetry and are reversible. Reversibility is interpreted in the sense that the Cremona transformation is performed at each step of the calculations using a difference scheme. The inheritance of periodicity and the Painlevé property by the approximate solution is investigated. In the computer algebra system Sage, values are found for the step $\Delta t$ for which the approximate solution is a sequence of points with period $n \in \mathbb N$. Examples are given, and conjectures about the structure of the sets of initial data generating sequences with period $n$ are formulated.