On the trajectories of dynamical systems with quadratic right sides, calculated by reversible difference schemes

E. A. Ayryan ; M. M. Gambaryan ; M. D. Malykh ; L. A. Sevastyanov

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXIV, Tome 517 (2022), pp. 17-35

Résumé

Approximate trajectories of dynamic systems described by ordinary differential equations with quadratic right sides, found by reversible schemes, are considered. These schemes are notable for the fact that the transition from layer to layer is described by Creomna transformations, which gives a large set of algebraic properties. On the way of generalizing the theory of Lagutinski determinants, the necessary and sufficient condition of belonging of approximate trajectories to the hypersurfaces of given linear systems was found. When approximating classical oscillators integrated in elliptic functions, the points of the approximate solution line up on phase space in some lines that are ellipitic curves. Their equations are written explicitly for the Jacobi oscillator. In the case of the Volterra-Lotka system, these points line up in lines that are not algebraic. For the Kowalevski case of solid motion, it has been proven that the points of the approximate solution cannot lie even on hypersurfaces of the 4th order.