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Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation
Peter H. van der Kamp, Elena Celledoni, Robert I. McLachlan +3
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 obtain…
Generalised Manin transformations and QRT maps
Peter H. van der Kamp, David I. McLaren, G. R. W. Quispel
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 invol…
Geometric and integrability properties of Kahan's method
Elena Celledoni, David McLaren, Brynjulf Owren +1
Given a quadratic vector field on \mathbb{R}^n possessing a quadratic first integral depending on two of the independent variables, we give a constructive proof that Kahan's discre…