Analogues of Kahan's method for higher order equations of higher degree

Abstract

Kahan introduced an explicit method of discretization for systems of first order differential equations with nonlinearities of degree at most two (quadratic vector fields). Kahan's method has attracted much interest due to the fact that it preserves many of the geometrical properties of the original continuous system. In particular, a large number of Hamiltonian systems of quadratic vector fields are known for which their Kahan discretization is a discrete integrable system. In this note, we introduce a special class of explicit order-preserving discretization schemes that are appropriate for certain systems of ordinary differential equations of higher order and higher degree.

A novel 8-parameter integrable map in R4

“…The scaling symmetry(11) is an essential ingredient in our proof of the theorem in the current Letter that the map(15) is integrable (as well as in our proof in[4] that the map (4) is integrable).…”

“…It follows that equation (15) again defines a birational map, and, importantly, it again preserves the scaling symmetry (11). [Indeed the latter is the primary reason we use the discretization (16)].…”

“…where the quartic function Δ 1 is given by ( 10) and the quadratic function Δ 2 is given by Finally, the map ( 15) is again invariant under the scaling symmetry group (11).…”

“…A special aspect of the maps we consider in this Letter is that they are an example of integrable maps arising as discretisations of ordinary differential equations (ODEs). Earlier examples of this arose using the Kahan discretisation of first-order quadratic ODEs (cf [5,10,12,15] and references therein), or by the discretisation of ODEs of order 1 and arbitrary degree using polarisation methods [4], and by the methods in [11] for the discretisation of ODEs of order o and degree o + 1, cf also [13,14].…”

A novel 8-parameter integrable map in $\mathbb{R}^4$

“…Earlier examples of this arose using the Kahan discretisation of first-order quadratic ODEs (cf. [5] and references therein), by the discretisation of ODEs of order 1 and arbitrary degree using polarisation methods [4], and by the methods in [8] for the discretisation of of ODEs of order o and degree o + 1, cf. also [9].…”