On periodic approximate solutions of ordinary differential equations

TL;DR

This paper investigates how certain difference schemes can produce approximate periodic solutions to ordinary differential equations, including nonlinear oscillators, and introduces an algebraic method to find such solutions.

Contribution

It demonstrates that specific schemes like the midpoint and Kahan can inherit periodicity and develops an algebraic approach to identify these solutions.

Findings

Midpoint scheme inherits periodicity for linear and certain nonlinear oscillators.

Kahan scheme can produce approximate periodic solutions.

An algebraic method for finding periodic solutions is proposed.

Abstract

The issue of inheriting periodicity of an exact solution of a dynamic system by a 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 for 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 into elliptic functions.