How one can repair non-integrable Kahan discretizations. II. A planar system with invariant curves of degree 6

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.

arXiv comment15 pages, 4 figures

References in corpus (14)

• Algebraic entropy
• Symetric Monopoles
• Geometric properties of Kahan's method
• Symmetric Monopoles
• On integrability of Hirota-Kimura type discretizations
• Integrability properties of Kahan's method
• Discretization of polynomial vector fields by polarization
• Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems
• New classes of quadratic vector fields admitting integral-preserving Kahan-Hirota-Kimura discretizations
• On the construction of elliptic solutions of integrable birational maps
• Manin involutions for elliptic pencils and discrete integrable systems
• On the singularity structure of Kahan discretizations of a class of quadratic vector fields
• New results on integrability of the Kahan-Hirota-Kimura discretizations
• A construction of a large family of commuting pairs of integrable symplectic birational 4-dimensional maps

Cited by in corpus (1)

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