    December  2019, 6(2): 401-408. doi: 10.3934/jcd.2019020

## Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor

 Institut für Mathematik, MA 7-1, Technische Universität Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany

* Corresponding author: Yuri B. Suris

Received  February 2019 Published  November 2019

Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic Hamiltonian vector field with a linear Poisson tensor and with a quadratic Hamilton function, this map is known to be integrable and to preserve a pencil of conics. In the paper "Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation" by P. van der Kamp et al. , it was shown that the Kahan discretization can be represented as a composition of two involutions on the pencil of conics. In the present note, which can be considered as a comment to that paper, we show that this result can be reversed. For a linear form $\ell(x,y)$, let $B_1,B_2$ be any two distinct points on the line $\ell(x,y) = -c$, and let $B_3,B_4$ be any two distinct points on the line $\ell(x,y) = c$. Set $B_0 = \tfrac{1}{2}(B_1+B_3)$ and $B_5 = \tfrac{1}{2}(B_2+B_4)$; these points lie on the line $\ell(x,y) = 0$. Finally, let $B_\infty$ be the point at infinity on this line. Let $\mathfrak E$ be the pencil of conics with the base points $B_1,B_2,B_3,B_4$. Then the composition of the $B_\infty$-switch and of the $B_0$-switch on the pencil $\mathfrak E$ is the Kahan discretization of a Hamiltonian vector field $f = \ell(x,y)\begin{pmatrix}\partial H/\partial y \\ -\partial H/\partial x \end{pmatrix}$ with a quadratic Hamilton function $H(x,y)$. This birational map $\Phi_f:\mathbb C P^2\dashrightarrow\mathbb C P^2$ has three singular points $B_0,B_2,B_4$, while the inverse map $\Phi_f^{-1}$ has three singular points $B_1,B_3,B_5$.

Citation: Matteo Petrera, Yuri B. Suris. Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor. Journal of Computational Dynamics, 2019, 6 (2) : 401-408. doi: 10.3934/jcd.2019020
##### References:
  E. Celledoni, R. I. McLachlan, B. Owren and G. R. W. Quispel, Geometric properties of Kahan's method, J. Phys. A, 46 (2013), 025201, 12 pp. doi: 10.1088/1751-8113/46/2/025201.   E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren and G. R. W. Quispel, Integrability properties of Kahan's method, J. Phys. A, 47 (2014), 365202, 20 pp. doi: 10.1088/1751-8113/47/36/365202.   E. Celledoni, D. I. McLaren, B. Owren and G. R. W. Quispel, Geometric and integrability properties of Kahan's method: The preservation of certain quadratic integrals, J. Phys. A, 52 (2019), 065201, 9 pp. doi: 10.1088/1751-8121/aafb1e.   P. H. van der Kamp, D. I. McLaren and G. R. W. Quispel, Generalised Manin transformations and QRT maps, preprint, arXiv: 1806.05340. P. H. van der Kamp, E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren and G. R. W. Quispel, Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation, J. Phys. A, 52 (2019), 045204. W. Kahan, Unconventional Numerical Methods for Trajectory Calculations, Unpublished lecture notes, 1993. M. Petrera, A. Pfadler and Y. B. Suris, On integrability of Hirota-Kimura-type discretizations: Experimental study of the discrete Clebsch system, Exp. Math., 18 (2009), 223-247.  doi: 10.1080/10586458.2009.10128900.   M. Petrera, A. Pfadler and Y. B. Suris, On integrability of Hirota-Kimura type discretizations, Regular Chaotic Dyn., 16 (2011), 245-289.  doi: 10.1134/S1560354711030051.   M. Petrera, J. Smirin and Y. B. Suris, Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems, Proc. Royal Soc. A, 475 (2019), 20180761, 13 pp.  M. Petrera and Y. B. Suris, On the Hamiltonian structure of Hirota-Kimura discretization of the Euler top, Math. Nachr., 283 (2010), 1654-1663.  doi: 10.1002/mana.200711162.   show all references

##### References:
  E. Celledoni, R. I. McLachlan, B. Owren and G. R. W. Quispel, Geometric properties of Kahan's method, J. Phys. A, 46 (2013), 025201, 12 pp. doi: 10.1088/1751-8113/46/2/025201.   E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren and G. R. W. Quispel, Integrability properties of Kahan's method, J. Phys. A, 47 (2014), 365202, 20 pp. doi: 10.1088/1751-8113/47/36/365202.   E. Celledoni, D. I. McLaren, B. Owren and G. R. W. Quispel, Geometric and integrability properties of Kahan's method: The preservation of certain quadratic integrals, J. Phys. A, 52 (2019), 065201, 9 pp. doi: 10.1088/1751-8121/aafb1e.   P. H. van der Kamp, D. I. McLaren and G. R. W. Quispel, Generalised Manin transformations and QRT maps, preprint, arXiv: 1806.05340. P. H. van der Kamp, E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren and G. R. W. Quispel, Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation, J. Phys. A, 52 (2019), 045204. W. Kahan, Unconventional Numerical Methods for Trajectory Calculations, Unpublished lecture notes, 1993. M. Petrera, A. Pfadler and Y. B. Suris, On integrability of Hirota-Kimura-type discretizations: Experimental study of the discrete Clebsch system, Exp. Math., 18 (2009), 223-247.  doi: 10.1080/10586458.2009.10128900.   M. Petrera, A. Pfadler and Y. B. Suris, On integrability of Hirota-Kimura type discretizations, Regular Chaotic Dyn., 16 (2011), 245-289.  doi: 10.1134/S1560354711030051.   M. Petrera, J. Smirin and Y. B. Suris, Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems, Proc. Royal Soc. A, 475 (2019), 20180761, 13 pp.  M. Petrera and Y. B. Suris, On the Hamiltonian structure of Hirota-Kimura discretization of the Euler top, Math. Nachr., 283 (2010), 1654-1663.  doi: 10.1002/mana.200711162.    The black curve is the conic $C = 0$, the red lines represent the reducible conic $D = 0$. The finite base points are $B_1, \ldots, B_4$. The points $B_0$, $B_5$ are singular points of the Kanan map $\Phi_f$, resp. of $\Phi_f^{-1}$, for the following data: $\ell(x,y) = 3x-y$, $H(x,y) = -x^2+(2/5)xy+(1/2)y^2-4x+4y$
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