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)
(The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided)
(KAR id:24243)

The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided. | |

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 |
---|---|

DOI/Identification number: | 10.3934/jgm.2009.1.55 |

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: | Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science |

Depositing User: | Andrew Hone |

Date Deposited: | 29 Jun 2011 13:54 UTC |

Last Modified: | 16 Nov 2021 10:02 UTC |

Resource URI: | https://kar.kent.ac.uk/id/eprint/24243 (The current URI for this page, for reference purposes) |

Hone, Andrew N.W.: | https://orcid.org/0000-0001-9780-7369 |

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