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How one can repair non-integrable Kahan discretizations. II. A planar system with invariant curves of degree 6
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TLDR
In this paper, a one-parameter family of integrable quadratic Cremona maps of the plane preserving a pencil of curves of degree 6 and of genus 1 is presented.Abstract:
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 $O(\epsilon^2)$ in the coefficients of the discretization, where $\epsilon$ is the stepsize.read more
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BookDOI
Discrete Integrable Systems
TL;DR: In this paper, the authors give a complete treatment not only of the basic facts about QRT maps, but also the background theory on which these maps are based, assuming Theorem 3.7.