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

Abstract

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

We apply Kahan's discretisation method to three classes of 2D 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. Applications to 2D Suslov and reduced Nahm equations are included.