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

TL;DR

This paper investigates how difference schemes for quadratic dynamical systems with symmetry properties can produce approximate solutions that preserve periodicity and Painlevé properties, with computational analysis and hypotheses on initial data structures.

Contribution

It introduces a method to analyze periodic approximate solutions of quadratic dynamical systems using symmetry-preserving difference schemes and computer algebra calculations.

Findings

Identifies step sizes where approximate solutions are periodic sequences.

Provides examples illustrating periodicity in approximate solutions.

Formulates hypotheses on initial data sets leading to periodic sequences.

Abstract

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.