Detecting and determining preserved measures and integrals of birational maps

2019/02/13 by Elena Celledoni, Charalambos Evripidou, Celledoni, Elena +9
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Nonlinear Waves and Solitons #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.1902.04685

openalex publication_date 2019/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we use the method of discrete Darboux polynomials to calculate preserved measures and integrals of rational maps. The approach is based on the use of cofactors and Darboux polynomials and relies on the use of symbolic algebra tools. Given sufficient computing power, most, if not all, rational preserved integrals can be found (and even some non-rational ones). We show, in a number of examples, how it is possible to use this method to both determine and detect preserved measures and integrals of the considered rational maps. Many of the examples arise from the Kahan-Hirota-Kimura discretization of completely integrable systems of ordinary differential equations.

Related

• Using aromas to search for preserved measures and integrals in Kahan's method
• Deducing properties of ODEs from their discretization
• Integrals of rational symmetric functions, two matrix models and biorthogonal polynomials
• Collet, Eckmann and the bifurcation measure
• Dynamical degrees of birational maps from indices of polynomials with respect to blow-ups II. 3D examples