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.