Three-dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras

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dc.creatorHone, A. N. W.
dc.creatorPetrera, M.
dc.date2008-10-30
dc.date.accessioned2026-07-07T10:14:12Z
dc.date.available2026-07-07T10:14:12Z
dc.descriptionRecently 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 $\mathfrak{so}(3)$ 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.
dc.description32 pages
dc.identifierhttps://arxiv.org/abs/0810.5490
dc.identifierhttp://arxiv.org/abs/0810.5490
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/172796
dc.subjectExactly Solvable and Integrable Systems
dc.titleThree-dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras
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