We apply Kahan's discretisation method to three classes of 2-dimensional quadratic vector fields with quadratic, resp cubic, resp quartic Hamiltonians. We show that the maps obtained in this way can be geometrically understood as the composition of two involutions, one of which is a (linear) symmetry switch, and the other is a generalised Manin involution.

https://iopscience.iop.org/article/10.1088/1751-8121/aaf51e/pdf

Three classes of quadratic vector fields for which the Kahan discretisation is the root of a generalised Manin transformation

Manin transformations are maps of the plane that preserve a pencil of cubic curves. They are the composition of two involutions. Each involution is constructed in terms of an involution point that is required to be one of the base points of the pencil. We generalise this construction to explicit birational maps of the plane that preserve quadratic resp. certain quartic pencils, and show that they are measure-preserving and hence integrable. In the quartic construction the two involution points are required to be base points of the pencil of multiplicity 2. On the other hand, for the quadratic pencils the involution points can be any two distinct points in the plane (except for base points). We employ Pascal's theorem to show that the maps that preserve a quadratic pencil admit infinitely many symmetries. The full 18-parameter QRT map is obtained as a special instance of the quartic case in a limit where the two involution points go to infinity. We show by construction that each generalised Manin transformation can be brought to QRT form by a fractional affine transformation. We also specify classes of generalised Manin transformations which admit a root.

Generalised Manin transformations and QRT maps

Applying Kahan's discretization to the reduced Nahm equations, we obtain two classes of integrable mappings.

Two classes of quadratic vector fields for which the Kahan discretization is integrable

We discuss the singularity structure of Kahan discretizations of a class of quadratric vector fields and provide a classification of the parameter values such that the corresponding Kahan map is integrable, in particular, admits an invariant pencil of elliptic curves.

On the singularity structure of Kahan discretizations of a class of quadratic vector fields

We contribute to the algebraic-geometric study of discrete integrable systems generated by planar birational maps: (a) we find geometric description of Manin involutions for elliptic pencils consisting of curves of higher degree, birationally equivalent to cubic pencils (Halphen pencils of index 1), and (b) we characterize special geometry of base points ensuring that certain compositions of Manin involutions are integrable maps of low degree (quadratic Cremona maps). In particular, we identify some integrable Kahan discretizations as compositions of Manin involutions for elliptic pencils of higher degree.

/pdf/manin-involutions-for-elliptic-pencils-and-discrete-etsa5m9akm.pdf

Manin Involutions for Elliptic Pencils and Discrete Integrable Systems

We present some new families of quadratic vector fields, not necessarily integrable, for which their Kahan-Hirota-Kimura discretization exhibits the preservation of some of the characterizing features of the underlying continuous systems (conserved quantities and invariant measures).

New classes of quadratic vector fields admitting integral-preserving Kahan-Hirota-Kimura discretizations

We contribute to the algebraic-geometric study of discrete integrable systems generated by planar birational maps: 
(a) we find geometric description of Manin involutions for elliptic pencils consisting of curves of higher degree, birationally equivalent to cubic pencils (Halphen pencils of index 1), and 
(b) we characterize special geometry of base points ensuring that certain compositions of Manin involutions are integrable maps of low degree (quadratic Cremona maps). In particular, we identify some integrable Kahan discretizations as compositions of Manin involutions for elliptic pencils of higher degree.

René Zander

Papers

New classes of quadratic vector fields admitting integral-preserving Kahan-Hirota-Kimura discretizations

New classes of quadratic vector fields admitting integral-preserving Kahan-Hirota-Kimura discretizations

On the singularity structure of Kahan discretizations of a class of quadratic vector fields

Manin Involutions for Elliptic Pencils and Discrete Integrable Systems

Manin involutions for elliptic pencils and discrete integrable systems