An Elementary Construction of Modified Hamiltonians and Modified Measures of 2D Kahan Maps
arXiv:2309.00799 · doi:10.46298/ocnmp.12249
Abstract
We show how to construct in an elementary way the invariant of the KHK discretisation of a cubic Hamiltonian system in two dimensions. That is, we show that this invariant is expressible as the product of the ratios of affine polynomials defining the prolongation of the three parallel sides of a hexagon. On the vertices of such a hexagon lie the indeterminacy points of the KHK map. This result is obtained analysing the structure of the singular fibres of the known invariant. We apply this construction to several examples, and we prove that a similar result holds true for a case outside the hypotheses of the main theorem, leading us to conjecture that further extensions are possible.
arXiv comment30 pages, 7 figures
References in corpus (14)
- Array Programming with NumPy
- Algebraic entropy
- Geometric properties of Kahan's method
- On integrability of Hirota-Kimura type discretizations
- Algebraic entropy and the space of initial values for discrete dynamical systems
- Integrability properties of Kahan's method
- Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems
- Fiber-dependent deautonomization of integrable 2D mappings and discrete Painlevé equations
- Using discrete Darboux polynomials to detect and determine preserved measures and integrals of rational maps
- New classes of quadratic vector fields admitting integral-preserving Kahan-Hirota-Kimura discretizations
- A three-dimensional generalization of QRT maps
- Two classes of quadratic vector fields for which the Kahan discretization is integrable
- Lax pairs for the discrete reduced Nahm systems
- Space of initial values of a map with a quartic invariant