March  2009, 1(1): 55-85. doi: 10.3934/jgm.2009.1.55

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

1. 

Institute of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NF, United Kingdom

2. 

Dipartimento di Fisica, Università degli Studi Roma Tre and Sezione INFN, Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy

Received  October 2008 Published  April 2009

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 $\mathfrak{so}(3)$ 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.
Citation: Andrew N. W. Hone, Matteo Petrera. Three-dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras. Journal of Geometric Mechanics, 2009, 1 (1) : 55-85. doi: 10.3934/jgm.2009.1.55
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