Geometric and Integrability Properties of Kahan's Method

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Kahan’s method was independently rediscovered by Hirota and Kimura [2, 3], and in 2011 Petrera, Pfadler and Suris [5] applied Kahan’s method to a large number of integrable quadratic vector ﬁelds and showed that the discretization in most cases preserved the integrability. In [6] and [7], we have shown that all linear and cubic integrals preserved in [5] using Kahan’s method can be explained by general propositions unrelated to complete integrability. In the present paper, Theorem 1 similarly yields the preservation of many (but not all) quadratic integrals, and building on this, Theorem 2 gives a 10-parameter family of integrable maps in R3.

2. The preservation of quadratic integrals of two variables by Kahan’s method

Let the quadratic ODE dx = f(x), x ∈ Rn (3) dt possess a quadratic integral I in 2 variables. W.l.o.g. we can choose the variables to be x1 and x2 with 1 1 I(x , x )= a x2 + a x x + a x2 + a x + a x . (4) 1 2 2 1 1 2 1 2 2 3 2 4 1 5 2 It follows that the first two components of the vector field can be written dx ∂I 1 = A(x) dt ∂x2 (5) dx ∂I 2 = −A(x) dt ∂x1 where A(x) is some affine function determined by the vector field. Theorem 1. The Kahan discretization of the vector field (3) with integral (4) possesses the modified integral 1 2 2 I(x1, x2)+ h D2(a)A(x) I˜(x) := 8 (6) 1 2 2 1+ 4 h D1(a)A(x) where

a1 a2 a4 a1 a2 D1(a)= ,D2(a)= a2 a3 a5 . (7) a2 a3 a4 a5 0

Proof.

Let the ODE dx = f(x), x ∈ Rn (8) dt possess a quadratic integral I in 2 variables. 1 1 I(x , x )= a x2 + a x x + a x2 + a x + a x (9) 1 2 2 1 1 2 1 2 2 3 2 4 1 5 2 As indicated above, it follows that the ﬁrst two components of the vector ﬁeld can be written dx 1 = A(x)I (10) dt 2 dx 2 = − A(x)I dt 1 Geometric and integrability properties of Kahan’s method: The preservation of certain quadratic integrals3 where ∂I Ii := i =1, 2. (11) ∂xi

We now discretize eqns (10) as follows: ′ x − x ′ ′ ′ 1 1 = B(x, x )I + C(x, x )I (12) h 2 2 ′ x − x ′ ′ ′ 2 2 = − B(x, x )I − C(x, x )I h 1 1 where

′ ∂I Ii := , i =1, 2. (13) ∂x ′ i x

Assuming that D1 =6 0, using (9), we obtain from (12) that

′ ′ 1 D2 2 2 ′ I(x1, x2) − 1+ h D C (x, x ) 2 D1 ≡ 1 . (14) 1 D2 2 2 ′ I x , x − 1+ h D1B (x, x ) ( 1 2) 2 D1 ′ ′ ′ Note that eq(14) is an algebraic identity, obtained without any knowledge of x3, x4,...xn.

In (14), the determinants D1 and D2 are deﬁned by: 2 D1 = a1a3 − a2 (15) 2 2 D2 =2a2a4a5 − a3a4 − a1a5 We can now consider several cases: Case(1): ′ B(x, x )= E(x) (16) ′ ′ C(x, x )= E(x ) It follows that the modiﬁed integral I˜(x) is given by

1 D2 I(x1, x2) − I˜ 2 D1 . (x)= 2 2 (17) 1+ h D1E (x) 1 This case includes Kahan’s method for E(x)= 2 A(x). Case(2): ′ ′ B(x, x )= F (x ) (18) ′ C(x, x )= F (x) It follows that the modiﬁed integral I˜(x) is given by

1 D2 2 2 I˜(x)= I(x1, x2) − [1 + h D1F (x)]. (19) 2 D1 1 This case includes the trapezoidal rule for F (x)= 2 A(x). Case(3): ′ ′ B(x, x )= C(x, x ) (20) Geometric and integrability properties of Kahan’s method: The preservation of certain quadratic integrals4

This case corresponds to the Discrete Gradient Method as applied to a quadratic integral. It follows that the discretization preserves the original integral I(x1, x2). This case has at least 2 subcases: ′ ′ 1 x+x Case(3a): B(x, x )= 2 A 2 (The midpoint rule). ′ 1 Case(3b): B(x, x )= 2 A(x). This is an almost explicit method that we have noted before.

Comments: Iˆ I˜ 1 D2 (1) Note that the integral (x)= (x)+ 2 D1 is given by I(x , x )+ 1 h2D E2(x) Iˆ 1 2 2 2 (x)= 2 2 (Case(1)) (21) 1+ h D1E (x) and 1 Iˆ(x)= I(x , x )+ h2F 2(x) D I(x , x ) − D (Case(2)) (22) 1 2 1 1 2 2 2 are also defined in the case D1 = 0. We have checked that in the limit D1 → 0 these formulas are correct. (2) We have not used anywhere that the function A(x) should be affine. It follows that the above results remain true for any function A(x) (Except for the fact that in case(1) the method will not be linearly implicit if A(x) is not affine).

Some Examples: Example 1. [5]: 2D Suslov system dx 1 =2αx x (23) dt 1 2 dx 2 = − 2x2. dt 1 This system may be written

dx1 ∂I =2x1 (24) dt ∂x2 dx2 ∂I = − 2x1 dt ∂x1 with 1 1 I(x , x )= x2 + αx2. (25) 1 2 2 1 2 2 Theorem 1 explains that the Kahan discretization of (23) preserves the modiﬁed integral 1 x2 + 1 αx2 I˜ x , x 2 1 2 2 . ( 1 2)= 2 2 (26) 1+ h αx1 Geometric and integrability properties of Kahan’s method: The preservation of certain quadratic integrals5

Example 2. [5]: Zhukovsky-Volterra system with vanishing β3 dx 1 = αx x − β x (27) dt 2 3 2 3 dx 2 = β x (28) dt 1 3 dx 3 = − αx x − β x + β x (29) dt 1 2 1 2 2 1 Equations (27) and (28) may be written

dx1 ∂I = x3 (30) dt ∂x2 dx2 ∂I = − x3 (31) dt ∂x1 with 1 I(x , x )= αx2 − β x − β x . (32) 1 2 2 2 1 1 2 2 Theorem 1 explains that the Kahan discretization of (27, 28, 29) preserves the modiﬁed integral 1 1 I˜(x , x , x )= αx2 − β x − β x − h2αβ2x2. (33) 1 2 3 2 2 1 1 2 2 8 1 3

Example 3. [5]: Two coupled Euler tops dx 1 = α x x (34) dt 1 2 3 dx 2 = α x x (35) dt 2 3 1 dx 3 = α x x + α x x (36) dt 3 1 2 4 4 5 dx 4 = α x x (37) dt 5 5 3 dx 5 = α x x (38) dt 6 3 4 Equations (34) and (35) may be written

dx1 ∂I1 = x3 (39) dt ∂x2 dx2 ∂I1 = − x3 dt ∂x1 with α α I (x , x )= 1 x2 − 2 x2. (40) 1 1 2 2 2 2 1 Moreover (37) and (38) may be written

dx4 ∂I2 = x3 (41) dt ∂x5 dx5 ∂I2 = −x3 dt ∂x4 Geometric and integrability properties of Kahan’s method: The preservation of certain quadratic integrals6 with α α I (x , x )= 5 x2 − 6 x2. (42) 2 4 5 2 5 2 6 Theorem 1 explains that the Kahan discretisation of (34-38) preserves the modiﬁed integrals 1 2 1 2 1 2 1 2 2 α1x2 − 2 α2x1 2 α5x5 − 2 α6x6 I˜ (x , x , x )= 2 ; I˜ (x , x , x )= 2 (43) 1 1 2 3 h 2 2 3 4 5 h 2 1 − 4 α1α2x3 1 − 4 α5α6x3 If the super-integrability condition

α1α2 = α5α6 (44) holds, eqs (34-38) may be written d(x + x ) 1 4 = x (α x + α x ) (45) dt 3 1 2 5 5 d(α x + α x ) 1 2 5 5 = x α α (x + x ) dt 3 1 2 1 4 Deﬁning

X := x1 + x4 (46)

Y := α1x2 + α5x5 eq(45) becomes dX ∂I = x 3 (47) dt 3 dY dY ∂I = −x 3 dt 3 dX with 1 α α I (X, Y )= Y 2 − 1 2 X2. (48) 3 2 2 i.e. a quadratic function of the two variables X and Y . Since the Kahan discretisation is the restriction of a Runge-Kutta method to quadratic vector fields, and since all Runge-Kutta methods commute with all affine transformations (and hence with the transformation (46)), Theorem 1 also explains why the Kahan discretisation preserves the modified integral 1 2 1 2 2 (α1x2 + α5x5) − 2 α1α2(x1 + x4) I˜ (x , x , x , x , x )= 2 . (49) 3 1 2 3 4 5 h 2 1 − 4 α1α2x3 Geometric and integrability properties of Kahan’s method: The preservation of certain quadratic integrals7

3. A family of integrable maps in R3

Deﬁne 1 1 H(x, y) = a x2 + a xy + a y2 + a x + a y 2 1 2 2 3 4 5 (50) 1 1 K(y,z) = b y2 + b yz + b z2 + b y + b z 2 1 2 2 3 4 5 Consider the integrable (and divergence-free) Nambu system

∂K ∂H ∂z ∂y dx   = ∇H × ∇K = − ∂K ∂H (51) dt  ∂z ∂x   ∂H ∂K   ∂x ∂y    Theorem 2. The Kahan discretization of vector field (51) with integrals (50) possesses the modified integrals 1 2 2 H(x, y)+ h D2(a)(b2y + b3z + b5) H˜ (x,y,z)= 8 (52) 1 2 2 1+ 4 h D1(a)(b2y + b3z + b5) 1 2 2 K(y,z)+ h D2(b)(a1x + a2y + a4) K˜ (x,y,z)= 8 (53) 1 2 2 1+ 4 h D1(b)(a1x + a2y + a4) and preserves the modified measure g(x,y,z) dx ∧ dy ∧ dz (54) with 1 −1 1 −1 g(x,y,z)= 1+ h2D (a)(b y+b z+b )2 1+ h2D (b)(a x+a y+a )2 (55) 4 1 2 3 5 4 1 1 2 4 where D1(a), D2(a) are defined in (7) and D1(b), D2(b) similarly. It follows that the Kahan discretization of (51) is completely integrable. Proof.

(i) The preservation of the two integrals (52) and (53) follows using Theorem 1 and ∂K ∂H = b y + b z + b , resp = a x + a y + a (56) ∂z 2 3 5 ∂x 1 2 4 (ii) By deﬁnition, the measure g dx ∧ dy ∧ dz is preserved if ′ ′ ′ ′ ′ ′ g(x,y,z) dx ∧ dy ∧ dz = g(x ,y ,z ) dx ∧ dy ∧ dz , (57) Z Z hence if the so-called density g satisﬁes ∂x′ ∂x′ ∂x′ ∂x ∂y ∂z ′ ′ ′ ′ ′ ′ g(x,y,z)= g(x ,y ,z ) ∂y ∂y ∂y . (58) ∂x ∂y ∂z

∂z′ ∂z′ ∂z′ ∂x ∂y ∂z

′ ′ ′ In the case at hand, the map (x,y,z) 7→ (x ,y ,z ) is given by Kahan’s map, and the identity (58) has been veriﬁed using Maple, after substituting (55). (iii) By deﬁnition, integrability of a three-dimensional map follows directly from the preservation of two integrals plus a preserved measure. Geometric and integrability properties of Kahan’s method: The preservation of certain quadratic integrals8

Comments:

(i) We note that the functional form of the preserved integrals (52) and (53), and density (55) is not unique, because any function of the integrals is an integral, and the product of the density with any integral will be a preserved density.

In particular, if D1(a) and D1(b) do not vanish, alternative discrete integrals are given by H x, y − 1 D2(a) ( ) D1 a Hˆ (x,y,z)= 2 ( ) (59) 1 2 2 1+ 4 h D1(a)(b2y + b3z + b5) K y,z − 1 D2(b) ( ) D1 b Kˆ (x,y,z)= 2 ( ) , (60) 1 2 2 1+ 4 h D1(b)(a1x + a2y + a4) and an alternative preserved density is given by − − 1 D (a) 1 1 D (b) 1 g(x,y,z)= H(x, y) − 2 K(y,z) − 2 . (61) 2 D1(a) 2 D1(b) (ii) Note that the density (61) does not depend on the timestep h, and therefore is also preserved by the ODE (51). (iii) Of course, if the reader so chooses, it is possible to introduce normal forms for this family of maps by applying appropriate affine transformations to the coordinates. t (iv) The ODE (5) is invariant under I → αI, A → βA, t → αβ . Similarly, the modified integral h (6) preserved by the Kahan discretization is covariant under I → αI, A → βA, h → αβ I. (v) For some examples of other integrable families of maps in R3 published in the literature, the reader is referred to [7–12]. The maps in [7] are closest to the maps in the current paper. Nevertheless they are different: the integrals of the ODE in [7] are essentially homogeneous, whereas the integrals of the ODE (51) are generically inhomogeneous.

Acknowledgements

This research was supported by the Australian Research Council and by the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 691070, and by The Research Council of Norway. GRWQ is grateful to Jason Frank for valuable discussions, and to Khaled Hamad for Maple assistance with the proof of Theorem 1.

References

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