Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation

2018/06/15 by Peter H. van der Kamp, van der Kamp, Peter H., Elena Celledoni +9
Mathematics · #37K10 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1806.05917

openalex publication_date 2018/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We apply Kahan's discretisation method to three classes of 2-dimensional quadratic vector fields with quadratic, resp cubic, resp quartic Hamiltonians. We show that the maps obtained in this way can be geometrically understood as the composition of two involutions, one of which is a (linear) symmetry switch, and the other is a generalised Manin involution.

Related

• Two classes of quadratic vector fields for which the Kahan\n discretization is integrable
• Geometric and integrability properties of Kahan's method
• Kahan-Hirota-Kimura maps preserving original cubic hamiltonians
• Analogues of Kahan's method for higher order equations of higher degree
• Generalised Manin transformations and QRT maps