Abstract

A few years ago, Hirota and Kimura found a new completely integrable discretization of the Euler top. The method of discretization that they used had already appeared in the numerical analysis literature, in the work of Kahan, who found an unconventional integration scheme for the Lotka-Volterra predator-prey system. Kahan’s approach, as rediscovered by Hirota and Kimura, applies to any system of quadratic vector fields, and is consistent with a general methodology for nonstandard discretizations developed earlier by Mickens. Some new examples of integrable maps have recently been found using this method. Here we describe the results of applying this approach to integrable bi-Hamiltonian vector fields associated with pairs of compatible Lie-Poisson algebras in three dimensions, and mention some other examples (including maps from the QRT family, and discrete Painleve equations). This is joint work with Matteo Petrera and Kim Towler.