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

Difference schemes are considered for dynamical systems $\dot x = f (x)$ with a quadratic right-hand side, which have $t$-symmetry and are reversible. Reversibility is interpreted in the sense that the Cremona transformation is performed at each step in the calculations using a difference scheme. The inheritance of periodicity and the Painlev\'e property by the approximate solution is investigated. In the computer algebra system Sage, such values are found for the step $\Delta t$, for which the approximate solution is a sequence of points with the period $n \in \mathbb N$. Examples are given and hypotheses about the structure of the sets of initial data generating sequences with the period $n$ are formulated.