A novel 8-parameter integrable map in R<sup>4</sup>

Quispel, G.; McLaren, David; Kamp, Peter H. van der

doi:10.1088/1751-8121/aba3df

Cited by 2 publications

(1 citation statement)

References 16 publications

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“…Note that some further examples of discretizations with coefficients depending on the stepsize can be found in [22, Sect. 6.5], while an example of a system where polarization yields an integrable discretization in the homogeneous case, but must be adjusted to achieve an integrable discretization in the inhomogeneous case, was given in [34].…”

Section: Discussionmentioning

confidence: 99%

How One can Repair Non-integrable Kahan Discretizations. II. A Planar System with Invariant Curves of Degree 6

Schmalian

Suris

Tumarkin

2021

Math Phys Anal Geom

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We find a novel one-parameter family of integrable quadratic Cremona maps of the plane preserving a pencil of curves of degree 6 and of genus 1. They turn out to serve as Kahan-type discretizations of a novel family of quadratic vector fields possessing a polynomial integral of degree 6 whose level curves are of genus 1, as well. These vector fields are non-homogeneous generalizations of reduced Nahm systems for magnetic monopoles with icosahedral symmetry, introduced by Hitchin, Manton and Murray. The straightforward Kahan discretization of these novel non-homogeneous systems is non-integrable. However, this drawback is repaired by introducing adjustments of order $$O(\epsilon ^2)$$ O ( ϵ 2 ) in the coefficients of the discretization, where $$\epsilon $$ ϵ is the stepsize.

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Section: Discussionmentioning

confidence: 99%

How One can Repair Non-integrable Kahan Discretizations. II. A Planar System with Invariant Curves of Degree 6

Schmalian

Suris

Tumarkin

2021

Math Phys Anal Geom

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Birational maps from polarization and the preservation of measure and integrals

McLachlan

McLaren

Quispel

2023

J. Phys. A: Math. Theor.

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The main result of this paper is the discretization of second-order Hamiltonian systems of the form ẍ = –K ∇W (x), where K is a constant symmetric matrix and W : ℝn → ℝ is a polynomial of degree d = 4 in any number of variables n. The discretization uses the method of polarization and preserves both the energy and the invariant measure of the diﬀerential equation, as well as the dimension of the phase space. This generalises earlier work for discretizations of ﬁrst order systems with d = 3, and of second order systems with d = 4 and n = 1.

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A novel 8-parameter integrable map in R⁴

Abstract: We present a novel 8-parameter integrable map in R 4 . The map is measurepreserving and possesses two functionally independent 2-integrals, as well as a measure-preserving 2-symmetry.

Cited by 2 publications

References 16 publications

How One can Repair Non-integrable Kahan Discretizations. II. A Planar System with Invariant Curves of Degree 6

How One can Repair Non-integrable Kahan Discretizations. II. A Planar System with Invariant Curves of Degree 6

Birational maps from polarization and the preservation of measure and integrals

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A novel 8-parameter integrable map in R4

Abstract: We present a novel 8-parameter integrable map in R 4 . The map is measurepreserving and possesses two functionally independent 2-integrals, as well as a measure-preserving 2-symmetry.

Cited by 2 publications

References 16 publications

How One can Repair Non-integrable Kahan Discretizations. II. A Planar System with Invariant Curves of Degree 6

How One can Repair Non-integrable Kahan Discretizations. II. A Planar System with Invariant Curves of Degree 6

Birational maps from polarization and the preservation of measure and integrals

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A novel 8-parameter integrable map in R⁴