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

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.


INTRODUCTION
The Kahan discretization scheme was introduced in the unpublished notes [13] as a method applicable to any system of ordinary differential equations in R n with a quadratic vector field where each component of Q : R n → R n is a quadratic form, while B ∈ R n×n and c ∈ R n .Kahan's discretizations reads as where is the symmetric bilinear form corresponding to the quadratic form Q. Equation ( 1) is linear with respect to x and therefore defines a rational map x = φ ε (x).Since equation (1) remains invariant under the interchange x ↔ x with the simultaneous sign inversion ε → −ε, one has the reversibility property φ −1 ε (x) = φ −ε (x).In particular, the map φ ε is birational.
The sequence of degrees d(m) of iterates φ m grows exponentially, so that the map φ is non-integrable, except for the following cases: (1,2,3), the sequence d(m) of degrees grows quadratically.The map φ admits an invariant pencil of elliptic curves.The degree of a generic curve of the pencil is 3, 4, 6, respectively.
grows linearly.The map φ admits an invariant pencil of rational curves.

Birational maps of surfaces.
Definition 2.1.Let φ be a birational map of a smooth projective surface X.The dynamical degree of the map φ is defined as where (φ m ) * denote the induced pullback maps on the Picard group Pic(X).
Diller and Favre provide the following classification for birational maps with λ 1 = 1: Theorem 2.2 (Diller, Favre [10], Theorem 0.2).Let φ : X → X be a birational map of a smooth projective surface with λ 1 = 1.Up to birational conjugacy, exactly one of the following holds.
(i) The sequence (φ m ) * is bounded, and φ m is an automorphism isotopic to the identity for some m.
(ii) The sequence (φ m ) * grows linearly, and φ preserves a rational fibration.In this case, φ cannot be conjugated to an automorphism.(iii) The sequence (φ m ) * grows quadratically, and φ is an automorphism preserving an elliptic fibration.
One says that φ : X → X is analytically stable (AS) if (φ * ) m = (φ m ) * on Pic(X).This relates the dynamical degree λ 1 to the spectral radius of the induced pullback φ * : Pic(X) → Pic(X).Equivalently, analytic stability is characterized by the condition that there is no curve V ⊂ X such that φ n (V) ∈ I(φ) for some integer n ≥ 0, where I(φ) is the indeterminacy set of φ (see [10], Theorem 1.14).Therefore, the notion of analytic stability is closely related to singularity confinement (see [15]).Indeed, a singularity confinement pattern for a map φ : X → X involves a curve V ⊂ X such that φ(V) = P is a point (so that P ∈ I(φ −1 )) and φ n−1 (P) ∈ I(φ), so that φ n (P) is a curve again for some positive integer n ∈ N.Such a singularity confinement pattern can be resolved by blowing up the orbit of P. Upon resolving all singularity confinement patterns, one lifts φ to an AS map φ : X → X .
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 (see [10], Theorem 0.1).

2.2.
Birational quadratic maps of CP 2 .As shown, e.g., in [1], every quadratic birational map φ : CP 2 → CP 2 can be represented as φ = A 1 • q i • A 2 , where A 1 , A 2 are linear projective transformations of CP 2 and q i is one of the three standard quadratic involutions: In these three cases, the indeterminacy set I(φ) consists of three, respectively two, one (distinct) singularities.The last two cases correspond to a coalescence of singularities.Therefore, the first case is the generic one.
In the present work, we only consider the first case: φ = A 1 • q 1 • A 2 .In this case, I(φ) = {B − }.Suppose that the map admits s singularity confinement patterns (0 ≤ s ≤ 3).That means there are positive integers n 1 , . . ., n s ∈ N and (σ 1 , . . ., σ s ) such that + for i = 1, . . .s.We assume that the n i are taken to be minimal and, for simplicity, we also assume that φ k (B (i) − ) for any k, l ≥ 0 and i j.As shown by Bedford and Kim [3] one can resolve the singularity confinement patterns by blowing up the finite sequences Those sequences are also called singular orbits.In this paper, we only encounter the situation that the orbits of different B (i) − are disjoint.As shown in [3], one can adjust the procedure to the more general situation.
Let H ∈ Pic(X) be the pullback of the divisor class of a generic line in CP 2 .Let E i,n ∈ Pic(X), for i ≤ s and 0 ≤ n ≤ n i − 1, be the divisor class of the exceptional divisor associated to the blow-up of the point The induced pullback φ * : Pic(X) → Pic(X) is determined by (see Bedford, Kim, [3] and Diller, [9]) The induced pushforward φ * : Pic(X) → Pic(X) is determined by The maps φ * , φ * are adjoint w.r.t. the intersection product (see [10], Proposition 1.1), i.e., ( Bedford and Kim have computed the characteristic polynomial χ(λ) = det( φ * − λid) explicitly for any given orbit data (see [3], Theorem 3.3).
Let C(m) = ( φ * ) m (H) ∈ Pic(X) be the class of the m-th iterate of a generic line.Set so that d(m) is the algebraic degree of the m-th iterate of the map φ.Set The expression on the right-hand side indeed depends on i and m + j only: using that the maps φ * , φ * are adjoint w.r.t. the intersection product and the relations (7), we find In particular, µ i (m) = (C(m), E i,0 ) can be interpreted as the multiplicity of − on the m-th iterate of a generic line.
The sequence of degrees d(m) of iterates of the map φ satisfies a system of linear recurrence relations.
Corollary 2.4 (Generating functions).Consider the generating functions d(z), µ i (z) for the sequences from theorem 2.3.They are rational functions which can be definded as solutions of the functional equations (11) with initial conditions as in theorem 2.3.
3. THE (γ 1 , γ 2 , γ 3 )-CLASS The class of quadratic differential equations we want to consider is a generalization of the two-dimensional reduced Nahm systems introduced in [11 Such systems can be explicitly integrated in terms of elliptic functions and they admit integrals of motion given respectively by Note that the curves {H i (x, y) = λ} are of genus 1. Systems (12) have been discussed in [11] and discretized by means of the Kahan method in [17].Integrability of Kahan discretizations was shown in [17].They have been studied in the context of minimization of rational elliptic surfaces in [5].
The following generalization of reduced Nahm systems has been introduced in [8,20]: We use the notation x = (x, y) ∈ C 2 .Consider the two-dimensional quadratic differential equations where 13) has the function (14) as an integral of motion and an invariant measure form The Kahan discretization of (13) reads It was shown in [20] that the Kahan map admits (15) as invariant measure form.Now, multiplying (16) from the left by the vectors ∇ T i , i = 1, 2, 3, we obtain where From equations (17)(18)(19) it follows that the Kahan map leaves the lines { i (x) = 0}, i = 1, 2, 3, invariant.Explicitly, the Kahan discretization of (13) as map φ + : CP 2 → CP 2 is as follows: with with homogeneous polynomials of deg ≤ 2 where denotes the sum over all cyclic permutations of (i, j, k) of (1, 2, 3).
Proof.This is the result of straightforward computations.
In the following, we assume that d 12 , d 23 , d 31 0, i.e., that the lines { i (x, y) = 0} are pairwise distinct.Also, we consider C 2 as affine part of CP 2 consisting of the points [x, y, z] ∈ CP 2 with z 0. We identify the point (x, y) ∈ C 2 with the point [x, y, 1] ∈ CP 2 .

Proposition 3.2. The singularities B (i)
+ , i = 1, 2, 3, of the Kahan map φ + and B (i) where (i, j, k) is a cyclic permutation of (1, 2, 3).Let L ∓ denote the line through the points B (j) ± .Then we have Proof.Substituting B (i) Suppose that nγ i γ j + γ k , for 0 ≤ n < N. Then we have where (i, j, k) is a cyclic permutation of (1, 2, 3).In particular, we have for a positive integer n i ∈ N. (ii) The only orbit data with exactly three singular orbits that can be realized is (σ 1 , σ 2 , σ 3 ) = (1, 2, 3) and (up to permutation) (iii) The only orbit data with exactly two singular orbits that can be realized is (σ 1 , σ 2 ) = (1, 2) and The only orbit data with exactly one singular orbit that can be realized is σ 1 = 1 and n 1 ∈ N arbitrary.

Proof.
(i) We show (28) by induction on n.For n = 0 the claim is true by Proposition 3.2.In the induction step (from n < N to n + 1) with (21-23) and (24-27) we find that Since nγ i γ j + γ k , we find that This proves the claim.
(ii) From conditions (29), for i = 1, 2, 3, we obtain the linear system This system has nontrivial solutions if and only if n 1 n 2 n 3 − n 1 n 2 − n 1 n 3 − n 2 n 3 = 0.This yields the proof.
(iii) From conditions (29), for i = 1, 2, we obtain the linear system Note that we have to exclude those values (n 1 , n 2 ) ∈ N 2 for which the solutions (γ 1 , γ 2 , γ 3 ) correspond to orbit data with three singular orbits.This yields the proof.
(iv) From conditions (29), for i = 1, we obtain the linear equation This yields the proof.
We arrive at the following classification result (compare Theorem 1.1): Theorem 3.4.The sequence of degrees d(m) of iterates φ m + grows exponentially, so that the map φ + is non-integrable, except for the following cases: (1,2,3), the sequence d(m) of degrees grows quadratically.The map φ + admits an invariant pencil of elliptic curves.The degree of a generic curve of the pencil is 3, 4, 6, respectively.
, by Theorem 3.3 and (32) we have the orbit data σ 1 = 1, n 1 = 1.With Theorem 2.3 we find that the sequence d(m) grows linearly.The claim about the existence of an invariant pencil of rational curves follows from Theorem 2.2.With (32) we find that all other cases with one singular orbit have orbit data σ 1 = 1, n 1 > 1.With Theorem 3.3 in [3] and Theorem 5.1 in [4] it follows that in those cases λ 1 > 1, i.e., the sequence d(m) grows exponentially.s = 0. We have λ 1 = 2.The sequence d(m) grows exponentially.

THE CASE
By Theorem 3.3 this case corresponds to the orbit data (n In this case, we consider the Kahan map φ + : C 2 → C 2 corresponding to a quadratic vector field of the form ẋ = J∇H(x), H(x) = 1 (x) 2 (x) 3 (x).
The Kahan map φ + : C 2 → C 2 admits an integral of motion (see [6,18]): where The geometry of the Kahan discretization has been studied in [18].The phase space of φ + : C 2 → C 2 is foliated by the one-parameter family (pencil) of invariant curves We consider C 2 as an affine part of CP 2 consisting of the points [x, y, z] ∈ CP 2 with z 0. We define the projective curves E λ as projective completion on E λ : where we set P(x, y, z) = z 2 P 2 (x/z, y/z).
(We have H(x, y, z) = z 3 H(x/z, y/z) = H(x, y) since H(x, y) is homogeneous of degree three.)The pencil contains two reducible curves , and consisting of the conic {P(x, y, z) = 0} and the line at infinity {z = 0}.All curves E λ pass through the set of base points which is defined as E 0 ∩ E ∞ .According to the Bézout theorem, there are 9 base points, counted with multiplicities.on the exceptional divisors according to the scheme (compare with (37))

L
(1) where L (i) ± denotes the proper transform of the line L (i) ± .We compute the induced pullback map on the Picard group φ * + : Pic(X) → Pic(X).Let H ∈ Pic(X) be the pullback of the class of a generic line in CP 2 .Let E i,n ∈ Pic(X), for i ≤ 3 and 0 ≤ n ≤ 2, be the class of E i,n .Then the Picard group is The rank of the Picard group is 10.The induced pullback φ * + : Pic(X) → Pic(X) is determined by (6).With Theorem 2.3 we arrive at the system of recurrence relations for the degree d(m): The generating functions of the solution to this system of recurrence relations are given by: The sequence d(m) grows quadratically.
We compute the induced pullback map on the Picard group φ * + : Pic(X) → Pic(X).Let H ∈ Pic(X) be the pullback of the class of a generic line in CP 2 .Let E i,n ∈ Pic(X), for i ≤ 3 and 0 ≤ n ≤ n i − 1, be the class of E i,n .Then the Picard group is The rank of the Picard group is 11.The induced pullback φ * + : Pic(X) → Pic(X) is determined by (6).With Theorem 2.3 we arrive at the system of recurrence relations for the degree d(m): and µ 3 (m) = 0, for m = 0, 1.The generating functions of the solution to this system of recurrence relations are given by: The sequence d(m) grows quadratically.
The sequence d(m) grows quadratically.
lines are exceptional in the sense that they are blown down by φ to points: φ(L(i) − ) = B (i)− .The inverse map is also quadratic with set of indeterminacy points I(φ −1 ) = {B

−
into equations (21)-(23) with ε replaced by −ε, and using (24-27) the first claim follows immediately.The second claim is the result of a straightforward (symbolic) computation using Maple.The map φ + blows down the lines L (i) − to the points B (i) − and blows up the points B