On periodic approximate solutions of ordinary differential equations

Shiwei Wang^a, A. V. Zorin^a, M. A. Konyaeva^a, M. D. Malykh^ab, L. A. Sevastianov<sup>ab

a</sup> Peoples' Friendship University of Russia named after Patrice Lumumba, Moscow
^b Joint Institute for Nuclear Research, Dubna, Moscow region

Abstract: The issue of inheriting periodicity of an exact solution of а dynamic system by а difference scheme is considered. It is shown that some difference schemes (midpoint scheme, Kahan scheme) in some special cases provide approximate solutions of differential equations, which are periodic sequences. Such solutions are called periodic. A purely algebraic method of finding such solutions is developed. It is shown that midpoint scheme inherits periodicity not only in case of linear oscillator, but also in case of nonlinear oscillator, integrable in elliptic functions.

Key words and phrases: dynamic systems, gyroscope, elliptic functions, finite differences, periodic solutions.

UDC: 517.962.1, 517.583

Received: 29.09.2025