Hone, Andrew N.W.,
Petrera, Matteo
(2009)
*
Three-dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras.
*
Journal of Geometric Mechanics,
1
(1).
pp. 55-85.
ISSN 1941-4889.
(doi:10.3934/jgm.2009.1.55)
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(KAR id:24243)

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Official URL http://dx.doi.org/10.3934/jgm.2009.1.55 |

## Abstract

Recently Hirota and Kimura presented a new discretization of the Euler top with several remarkable properties. In particular this discretization shares with the original continuous system the feature that it is an algebraically completely integrable bi-Hamiltonian system in three dimensions. The Hirota-Kimura discretization scheme turns out to be equivalent to an approach to numerical integration of quadratic vector fields that was introduced by Kahan, who applied it to the two-dimensional Lotka-Volterra system.

The Euler top is naturally written in terms of the so(3) Lie-Poisson algebra. Here we consider algebraically integrable systems that are associated with pairs of Lie-Poisson algebras in three dimensions, as presented by Gümral and Nutku, and construct birational maps that discretize them according to the scheme of Kahan and Hirota-Kimura. We show that the maps thus obtained are also bi-Hamiltonian, with pairs of compatible Poisson brackets that are one-parameter deformations of the original Lie-Poisson algebras, and hence they are completely integrable. For comparison, we also present analogous discretizations for three bi-Hamiltonian systems that have a transcendental invariant, and finally we analyze all of the maps obtained from the viewpoint of Halburd's Diophantine integrability criterion.

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