Integrability properties of Kahan's method

Abstract

We present several novel examples of integrable quadratic vector fields for which Kahan's discretization method preserves integrability. Our examples include generalized Suslov and Ishii systems, Nambu systems, Riccati systems, and the first Painlevé equation. We also discuss how Manin transformations arise in Kahan discretizations of certain vector fields.

Cited by in corpus (15)

• Discretization of polynomial vector fields by polarization
• Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems
• Bi-rational maps in four dimensions with two invariants
• 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
• On the construction of elliptic solutions of integrable birational maps
• A three-dimensional generalization of QRT maps
• On the singularity structure of Kahan discretizations of a class of quadratic vector fields
• An Elementary Construction of Modified Hamiltonians and Modified Measures of 2D Kahan Maps
• Lax pairs for the discrete reduced Nahm systems
• A construction of a large family of commuting pairs of integrable symplectic birational 4-dimensional maps
• Explicit isomorphisms for the symmetry algebras of continuous and discrete isotropic oscillators
• A new approach to integrals of discretizations by polarization
• How one can repair non-integrable Kahan discretizations. II. A planar system with invariant curves of degree 6
• Kahan-Hirota-Kimura maps preserving original cubic hamiltonians