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• W1980034406 endingPage "85" @default.
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• W1980034406 abstract "Recently Hirota and Kimura presented a new discretization of theEuler top with several remarkable properties. In particular thisdiscretization shares with the original continuous systemthe feature that it is an algebraicallycompletely integrable bi-Hamiltonian system in three dimensions.The Hirota-Kimura discretization scheme turns out to beequivalent to an approach to numerical integrationof quadratic vector fields that was introduced by Kahan, who applied itto the two-dimensional Lotka-Volterra system. The Euler top isnaturally written in terms of the $mathfrak{so}(3)$ Lie-Poisson algebra. Here weconsideralgebraically integrable systemsthat are associated with pairs of Lie-Poisson algebras in three dimensions,as presented by Gümral and Nutku, and construct birational mapsthat discretize them according to the scheme of Kahan and Hirota-Kimura.We show that the maps thus obtained are also bi-Hamiltonian, with pairsof compatible Poisson brackets that are one-parameter deformationsof the original Lie-Poisson algebras, and hence they are completelyintegrable. For comparison, we also present analogous discretizationsforthree bi-Hamiltonian systems that have a transcendental invariant, and finallywe analyze all of the maps obtained from the viewpoint ofHalburd'sDiophantine integrability criterion." @default.
• W1980034406 created "2016-06-24" @default.
• W1980034406 creator A5064714239 @default.
• W1980034406 creator A5089059041 @default.
• W1980034406 date "2009-01-01" @default.
• W1980034406 modified "2023-09-29" @default.
• W1980034406 title "Three-dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras" @default.
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• W1980034406 doi "https://doi.org/10.3934/jgm.2009.1.55" @default.
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