Reversible differential schemes for elliptical oscillators

For classical nonlinear oscillators, a comparison between the classical continuous theory of integration in elliptic functions and the discrete theory based on reversible difference schemes was made. These schemes are notable for the fact that the transition from layer to layer is described by Cremona transformations, which gives a large set of algebraic properties. Several properties are shown for the example of the Jacobi oscillator: 1). points of approximate trajectories fall on elliptic curves, 2). difference scheme can be written using quadrature, 3). the approximate solution is periodic. Explicit formulas to calculate the time step for which the approximate solution is a periodic sequence were found.