Abstract
We discuss the singularity structure of Kahan discretizations of a class of quadratic 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.
Highlights
The Kahan discretization scheme was introduced in the unpublished notes [13] as a method applicable to any system of ordinary differential equations in Rn with a quadratic vector field f (x) = Q(x) + Bx + c, x ∈ Rn, where each component of Q : Rn → Rn is a quadratic form, while B ∈ Rn×n and c ∈ Rn
We study the singularity structure of the Kahan discretization as a birational quadratic map φ : CP2 → CP2
Diller and Favre showed that for any birational map φ : X → X of a smooth projective surface we can construct by a finite number of successive blow-ups a surface X such that φ lifts to an analytically stable birational map φ : X → X
Summary
Integrability of the Kahan maps φ : C2 → C2 was established for several cases of parameters (γ1, γ2, γ3): If (γ1, γ2, γ3) = (1, 1, 1), (2) is a canonical Hamiltonian system on R2 with homogeneous cubic Hamiltonian. If (γ1, γ2, γ3) = (1, 1, 0), (2) is a Hamiltonian vector field on R2 with linear Poisson tensor and homogeneous quadratic Hamiltonian In this case, a rational integral for the Kahan map φ was found in [5]. We study the singularity structure of the Kahan discretization as a birational quadratic map φ : CP2 → CP2. The sequence of degrees d(m) of iterates φm grows exponentially, so that the map φ is non-integrable, except for the following cases:. Some of the integrable cases are discussed in further detail in Sects. 4, 5, 6, 7 and 8
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