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 equiv
alent 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.
Kahan discretization is applicable to any system of ordinary differential equations on $mathbb R^n$ with a quadratic vector field, $dot{x}=f(x)=Q(x)+Bx+c$, and produces a birational map $xmapsto widetilde{x}$ according to the formula $(widetilde{x}-x
)/epsilon=Q(x,widetilde{x})+B(x+widetilde{x})/2+c$, where $Q(x,widetilde{x})$ is the symmetric bilinear form corresponding to the quadratic form $Q(x)$. When applied to integrable systems, Kahan discretization preserves integrability much more frequently than one would expect a priori, however not always. We show that in some cases where the original recipe fails to preserve integrability, one can adjust coefficients of the Kahan discretization to ensure its integrability.
We study the problem of preservation of canard connections for time discretized fast-slow systems with canard fold points. In order to ensure such preservation, certain favorable structure preserving properties of the discretization scheme are requir
ed. Conventional schemes do not possess such properties. We perform a detailed analysis for an unconventional discretization scheme due to Kahan. The analysis uses the blow-up method to deal with the loss of normal hyperbolicity at the canard point. We show that the structure preserving properties of the Kahan discretization imply a similar result as in continuous time, guaranteeing the occurrence of canard connections between attracting and repelling slow manifolds upon variation of a bifurcation parameter. The proof is based on a non-canonical Melnikov computation along an invariant separating curve, which organizes the dynamics of the map similarly to the ODE problem.
Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic Hamiltonian vector field with a linear Poisson tensor and with a quadratic Hamilt
on function, this map is known to be integrable and to preserve a pencil of conics. In the paper `Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation by P. van der Kamp et al., it was shown that the Kahan discretization can be represented as a composition of two involutions on the pencil of conics. In the present note, which can be considered as a comment to that paper, we show that this result can be reversed. For a linear form $ell(x,y)$, let $B_1,B_2$ be any two distinct points on the line $ell(x,y)=-c$, and let $B_3,B_4$ be any two distinct points on the line $ell(x,y)=c$. Set $B_0=tfrac{1}{2}(B_1+B_3)$ and $B_5=tfrac{1}{2}(B_2+B_4)$; these points lie on the line $ell(x,y)=0$. Finally, let $B_infty$ be the point at infinity on this line. Let $mathfrak E$ be the pencil of conics with the base points $B_1,B_2,B_3,B_4$. Then the composition of the $B_infty$-switch and of the $B_0$-switch on the pencil $mathfrak E$ is the Kahan discretization of a Hamiltonian vector field $f=ell(x,y)begin{pmatrix}partial H/partial y -partial H/partial x end{pmatrix}$ with a quadratic Hamilton function $H(x,y)$. This birational map $Phi_f:mathbb C P^2dashrightarrowmathbb C P^2$ has three singular points $B_0,B_2,B_4$, while the inverse map $Phi_f^{-1}$ has three singular points $B_1,B_3,B_5$.
Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic Hamiltonian vector field, this map is known to be integrable and to preserve a pe
ncil of cubic curves. Generically, the nine base points of this pencil include three points at infinity (corresponding to the asymptotic directions of cubic curves) and six finite points lying on a conic. We show that the Kahan discretization map can be represented in six different ways as a composition of two Manin involutions, corresponding to an infinite base point and to a finite base point. As a consequence, the finite base points can be ordered so that the resulting hexagon has three pairs of parallel sides which pass through the three base points at infinity. Moreover, this geometric condition on the base points turns out to be characteristic: if it is satisfied, then the cubic curves of the corresponding pencil are invariant under the Kahan discretization of a planar quadratic Hamiltonian vector field.
R. Hirota and K. Kimura discovered integrable discretizations of the Euler and the Lagrange tops, given by birational maps. Their method is a specialization to the integrable context of a general discretization scheme introduced by W. Kahan and appli
cable to any vector field with a quadratic dependence on phase variables. We report several novel observations regarding integrability of the Kahan-Hirota-Kimura discretization. For several of the most complicated cases for which integrability is known (Clebsch system, Kirchhoff system, and Lagrange top), - we give nice compact formulas for some of the more complicated integrals of motion and for the density of the invariant measure, and - we establish the existence of higher order Wronskian Hirota-Kimura bases, generating the full set of integrals of motion. While the first set of results admits nice algebraic proofs, the second one relies on computer algebra.
We analyze the relation of the notion of a pluri-Lagrangian system, which recently emerged in the theory of integrable systems, to the classical notion of variational symmetry, due to E. Noether. We treat classical mechanical systems and show that, f
or any Lagrangian system with $m$ commuting variational symmetries, one can construct a pluri-Lagrangian 1-form in the $(m+1)$-dimensional time, whose multi-time Euler-Lagrange equations coincide with the original system supplied with $m$ commuting evolutionary flows corresponding to the variational symmetries. We also give a Hamiltonian counterpart of this construction, leading, for any system of commuting Hamiltonian flows, to a pluri-Lagrangian 1-form with coefficients depending on functions in the phase space.
