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

Hone, A.N.W. and Petrera, M. (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. (The full text of this publication is not available from this repository)

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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.

Item Type: Article
Uncontrolled keywords: Integrable discretizations, Lie-Poisson algebras, Diophantine integrability.
Subjects: Q Science > QA Mathematics (inc Computing science) > QA150 Algebra
Q Science > QA Mathematics (inc Computing science) > QA801 Analytic mechanics
Q Science > QA Mathematics (inc Computing science) > QA252 Lie algebras
Q Science > QA Mathematics (inc Computing science) > QA252 Lie algebras
Divisions: Faculties > Science Technology and Medical Studies > School of Mathematics Statistics and Actuarial Science > Applied Mathematics
Faculties > Science Technology and Medical Studies > School of Mathematics Statistics and Actuarial Science
Faculties > Science Technology and Medical Studies > School of Mathematics Statistics and Actuarial Science > Pure Mathematics
Depositing User: Andrew N W Hone
Date Deposited: 29 Jun 2011 13:54
Last Modified: 15 Oct 2012 13:53
Resource URI: http://kar.kent.ac.uk/id/eprint/24243 (The current URI for this page, for reference purposes)
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