Liouville integrability and superintegrability of a generalized Lotka-Volterra system and its Kahan discretization

0.2 MB·

by  Theodoros E. Kouloukas

Liouville integrability and superintegrability of a generalized
Lotka-Volterra system and its Kahan discretization
6
1 Theodoros E. Kouloukas1,3, G. R. W. Quispel1 and Pol Vanhaecke2
0
2 1Department of Mathematics and Statistics,

n La Trobe University, Bundoora VIC 3086, Australia
a
2Laboratoire de Math´ematiques et Applications,
J
UMR 7348 du CNRS, Universit´e de Poitiers,
9
1 86962 Futuroscope Chasseneuil Cedex, France
3SMSAS, University of Kent, Canterbury, UK
]
h
p
- Abstract. We prove the Liouville and superintegrability of a generalized
h Lotka-VolterrasystemanditsKahandiscretization.
t
a
m
[

1 Contents
v
6
0 1. Introduction 1
0
5 2. A generalized Lotka-Volterra system 3
0
. 3. The Kahan discretization 10
1
0 4. Conclusion 13
6
1 References 14
:
v
i
X
r
a 1. Introduction
The Kac-van Moerbeke system is a prime example of an integrable system,
described by the differential equations
x˙i =xi(xi+1−xi−1), (i=1,...,n), (1.1)
wherex =x =0. Itwasfirstintroducedandstudied,togetherwithsomeofits
0 n+1
generalizations, by Lotka to model oscillating chemical reactions and by Volterra
to describe population evolution in a hierarchicalsystem of competing species (see
2010 Mathematics Subject Classification. 37J35,39A22.
Key words and phrases. Integrablesystems,superintegrability,Kahandiscretization.
1
2 KOULOUKAS,QUISPEL,ANDVANHAECKE
[11, 15]). Bynow,manygeneralizationsof(1.1)havebeenintroducedandstudied,
often from the point of (Liouville or algebraic) integrability [2, 8, 9] or Lie theory
[2, 5], but also in relation with other integrable systems [7, 12, 14]. In our recent
study [14], a natural generalization of (1.1) came up in the study of a class of
multi-sums of products: we considered the system
x˙i =xi xj − xj , (i=1,...,n), (1.2)
Xj>i Xj<i
 
we showedits Liouville and superintegrabilityand we used it to show the Liouville
and superintegrability (or non-commutative integrability) of the Hamiltonian sys-
tem defined by the above-mentionned class of functions. The system (1.2) has a
Hamiltonian structure, described by the Hamiltonian function and Poisson struc-
ture, which are respectively given by
n
H = x , {x ,x }=x x , (i<j). (1.3)
i i j i j
Xi=1
We consider in the present paper the case of a general linear Hamiltonian
n
H = a x , (1.4)
i i
Xi=1
with the Poisson structure still given by (1.3). The differential equations which
describe this Hamiltonian system are given by
x˙i =xi ajxj − ajxj , (i=1,...,n). (1.5)
Xj>i Xj<i
 
When all the parameters a are different from zero, a trivial rescaling (which pre-
i
serves the Poisson structure) leads us back to (1.3), so the novelty of our study is
mainly concernedwith the case where atleast one (but not all!) of the parameters
a is zero, though all results below are also valid in case all the parameters a are
i i
different from zero. By explicitly exhibiting a set of [(n+1)/2]involutive (Poisson
commuting) rational functions, which are shown to be functionally independent,
we show that (1.4) is Liouville integrable (Theorems 2.3 and 2.4). We also exhibit
n−1 functionally independent first integrals, thereby showing that (1.5) is super-
integrable (Theorem 2.5). Finally, we construct for any initial conditions explicit
solutions of (1.5) (Proposition 2.6).
In Section 3 we study the Kahan discretization (see e.g. [4]) of (1.5), which
we explicitly describe (Proposition 3.1). We also show that the map defined by
the Kahan discretization is a Poisson map (Proposition 3.2). Upon comparing the
latter map with the solutions to the continuous system (1.5), we prove that the
Kahan map is a time advance map for this Hamiltonian system, and we derive
from it that the discrete system is both Liouville and superintegrable, with the
same first integrals as the continuous system (Proposition 3.4 and Corollary 3.3).
GENERALIZED LOTKA-VOLTERRA SYSTEMS 3
We finish the paper with some comments and perspectives for future work
(Section 4).
2. A generalized Lotka-Volterra system
Let n be an arbitrary positive integer. We consider on Rn the generalized
Lotka-Volterra system
n
x˙ =x A x , (i=1,...,n), (2.1)
i i ij j
Xj=1
where A is the square matrix
0 a a ... a
2 3 n
−a1 0 a3 ... an
−a −a 0 ... a
A= 1 2 n , (2.2)
 
 .. .. .. .. 
 . . . . 
 
−a −a −a ... 0
1 2 3
 
and(a ,...,a )∈Rn\{(0,...,0)}. LikemostLotka-Volterrasystem,ithasalinear
1 n
function as Hamiltonian, to wit H :=a x +a x +···+a x ; the corresponding
1 1 2 2 n n
(quadratic) Poisson structure is defined by the brackets {x ,x } := x x , for 1 6
i j i j
i < j 6 n. The following elementary lemma, which will play a key rˆole in the
proof of Theorem 2.3 below, shows that rescaling the parameters a by non-zero
i
constants leads to isomorphic Hamiltonian systems.
Lemma 2.1. Let c ,...,c be arbitrary non-zero real constants. Then the lin-
1 n
ear change of coordinates x 7→ x /c transforms the generalized Lotka-Volterra
i i i
system with parameters a ,...,a into the generalized Lotka-Volterra system with
1 n
parameters a c ,...,a c .
1 1 n n
Proof. Let y := x /c . Then {y ,y } = {x ,x }/(c c ) = xixj = y y , for
i i i i j i j i j cicj i j
anyi<j,whichshowsthatthechangeofvariablespreservesthePoissonstructure.
Clearly, in terms of the new variables, the Hamiltonian reads H = a c y +···+
1 1 1
a c y , which is the Hamiltonian of the generalized Lotka-Volterra system with
n n n
constants a c . (cid:3)
i i
As an application of the lemma, we have that when the parameters a are all
i
non-zero, we can rescale them all to 1, and (2.1) becomes (1.2) (which is system
(3.5)in[14]). Inthiscase,thematrixAisskew-symmetricandso(2.1)isagenuine
Lotka-Volterrasystem,whoseLiouvilleandsuperintegrabilityhaveextensivelybeen
studied in [14]. When some of the parameters a are zero, we get new (non-
i
isomorphic)systems. Aswewillshowinthissection,allthesesystemsareLiouville
and superintegrable.
4 KOULOUKAS,QUISPEL,ANDVANHAECKE
For the study of the general case, it is convenient to introduce the functions
v := a x +···+a x , for i = 1,...,n; we also set v := 0. In terms of these
i 1 1 i i 0
functions, H =v and the system (2.1) can equivalently be written as
n
x˙i =xi(H −vi−vi−1), (i=1,...,n). (2.3)
For i < j, one has {v ,x } = v x , and so the Poisson brackets of the functions v
i j i j i
are given by
{v ,v }=v (v −v ), (i<j). (2.4)
i j i j i
In particular, remembering that H =v ,
n
v˙ ={v ,H}=v (H −v ), (2.5)
i i i i
for i = 1,...,n. If a a ...a 6= 0, the functions v define new coordinates on Rn,
1 2 n i
since then xk =(vk−vk−1)/ak for k =1,...,n; moreover,the system (2.1) totally
decouplesintermsofthesecoordinatessinceittakesthesimpleformv˙ =v (H−v ),
i i i
for i=1,...,n. However, the functions v do not define coordinates when at least
i
one of the ak is zero, because if ak =0 then vk =vk−1.
With a view to proving Liouville integrability, we define for k =1,..., n the
2
functions (cid:2) (cid:3)
x1x3...x2k−1
J := , (2.6)
k
x x ...x
2 4 2k
and for k =1,..., n+1 the functions
2
(cid:2) (cid:3)
x x ...x
2k+1 2k+3 n
v2k−1 if n is odd,
 x2kx2k+2...xn−1
F := (2.7)
k  x2k+2x2k+4...xn
v if n is even.
2k
 x2k+1x2k+3...xn−1
Notice that F = v = H, the Hamiltonian (1.4). For odd n, we also
[(n+1)/2] n
introduce the function
x x ...x
1 3 n
C := .
x2x4...xn−1
Proposition 2.2. For any k,l ∈{1,..., n },
2
(cid:2) (cid:3)
{J ,J }={F ,F}={F ,H}=0. (2.8)
k l k l k
Moreover, when n is odd, C is a Casimir function of the Poisson bracket {·,·}.
Proof. First, we notice that for any k =1,..., n
2
(cid:2) (cid:3)
∂J (−1)i+1J for 16i62k,
x k = k (2.9)
i∂x (cid:26) 0 for 2k <i6n.
i
GENERALIZED LOTKA-VOLTERRA SYSTEMS 5
It follows that, for k <l∈{1,..., n }, we have
2
(cid:2) (cid:3)
∂J ∂J ∂J ∂J
k l k l
{J ,J } = x x −
k l i j
(cid:18)∂x ∂x ∂x ∂x (cid:19)
16Xi<j6n i j j i
= [(−1)i+1J (−1)j+1J −(−1)j+1J (−1)i+1J ]
k l k l
16iX<j62k
+ (−1)i+1J (−1)j+1J =0.
k l
16i6X2k<j62l
This shows the first equality of (2.8). We show the two other equalities of (2.8) for
even n. To do this, it suffices to show that {F ,F} =0 for 16 k <l 6n/2 since
k l
F =H. We set F =v I , i.e., we define I by
n/2 k 2k k k
x x ...x
2k+2 2k+4 n
I := .
k
x2k+1x2k+3...xn−1
As in (2.9), we have that
∂Ik 0 for 16i62k ,
x = (2.10)
i∂xi (cid:26) (−1)iIk for 2k<i6n,
from which it follows, as above, that {I ,I } = 0 and that {I ,J } = 0 for all
k l k l
k,l∈{1,...,n/2}. Also, for any j ∈{1,...,n}
n
∂I ∂I ∂I
k k k
{Ik,xj}= {xi,xj}= xi − xi xj
∂x ∂x ∂x
Xi=1 i 16Xi<j i j<Xi6n i
 
and using (2.10) we derive that
0 for j 62k , 0 for j 62k ,
{I ,x }= and {I ,v }=
k j (cid:26) −Ikxj for 2k<j , k j (cid:26) −Ik(vj −v2k) for 2k <j .
(2.11)
It follows from (2.4) and (2.11) that, for any k <l6n/2,
{F ,F } = {v I ,v I }=v I {I ,v }+v I {v ,I }+I I {v ,v }
k l 2k k 2l l 2k l k 2l 2l k 2k l k l 2k 2l
= −v I I (v −v )+0+v I I (v −v )=0.
2k k l 2l 2k 2k k l 2l 2k
This shows the second half of (2.8) for n even; for n odd, the proof is very similar
(in this case, H =F and one proves as above that {F ,F }=0 for 16k <
(n+1)/2 k l
l6(n+1)/2). Finally we show that C is a Casimir function (when n is odd). For
j =1,...,n,
n
∂C ∂C ∂C
{C,xj} = {xi,xj}= xi − xi xj
∂x ∂x ∂x
Xi=1 i 16Xi<j i j<Xi6n i
 
= (−1)i+1Cx − (−1)i+1Cx =0,
j j
16Xi<j j<Xi6n
which shows our claim. (cid:3)
Theorem 2.3. Suppose that n is even. Let ℓ denote the smallest integer such
that a 6=0 (in particular, ℓ=0 when a 6=0) and let λ:= ℓ . The n functions
ℓ+1 1 2 2
J1,J2,...,Jλ,H,Fλ+1,Fλ+2,...,Fn2−1 are pairwise in involu(cid:2)tio(cid:3)n and functionally
independent, hence they define a Liouville integrable system on (Rn,{·,·}).
6 KOULOUKAS,QUISPEL,ANDVANHAECKE
Proof. We know alreadyfromProposition2.2that the functions J are pair-
k
wise in involution,and also the functions F (recall that F =H). We show that
l n/2
{J ,F }=0 for k =1,...,λ and l =λ+1,...,n. To do this, we use the following
k l 2
analog of (2.11), which is easily obtained from (2.9):
J v for j 62k ,
{J ,v }= k j
k j (cid:26) Jkv2k for 2k<j .
It follows that, for the above values of k,l, which satisfy k 6 λ < l, one has
{J ,v } = J v = 0 (the last equality follows from 2k 6 2λ 6 ℓ and v = 0 for
k 2l k 2k i
i6ℓ), and so
{J ,F }={J ,v I }=v {J ,I }+I {J ,v }=0;
k l k 2l l 2k k l l k 2l
in the last step we also used that the functions I and J are in involution (see the
i j
proof of Proposition 2.2). This shows that the n functions
2
J1,J2,...,Jλ,H,Fλ+1,Fλ+2,...,Fn2−1 (2.12)
are pairwise in involution.
We now show that these functions are functionally independent. We first do
this when all a are zero, except for a which we may suppose to be equal to 1;
i ℓ+1
then v = x = H for i > ℓ and v = 0 for i 6 ℓ. The Jacobian matrix of the
i ℓ+1 i
above functions (2.12) with respect to x ,...,x (in that order) is easily seen to
1 n
have the following block form:
A 0 0
Jac=0 1 0 ,
0 ⋆ B
 
where A has size λ×ℓ and B has size (n −λ−1)×(n−ℓ−1). We show that
2
this matrix has full rank n/2 (which is equal to the number of rows of Jac). To
do this, it is sufficient to show that A has full rank λ and that B has full rank
n/2−λ−1 (the value of the column vector ⋆ is irrelevant). Consider the square
′
submatrixA ofAconsistingonlyofitseven-numberedcolumns. Fork <lwehave
′
A = A = ∂J /∂x = 0, since J only depends on x ,...,x . It follows that
kl k,2l k 2l k 1 2k
′ ′ ′
A is a lower triangular matrix. Moreover, A =A =∂J /∂x 6=0, hence A
kk k,2k k 2k
′
is non-singular. This shows that rank(A) = rank(A) = λ. Similarly, we extract
′
fromB asquaresubmatrixB byselectingfromB its even-numbered(respectively
odd-numbered) columns when ℓ is even (respectively odd). For k > l we have
′
B = ∂F /∂x = ∂(v I )/∂x = ∂(x I )/∂x =
kl λ+k 2λ+1+2l 2λ+2k λ+k 2λ+1+2l ℓ+1 λ+k 2λ+1+2l
x ∂I /∂x = 0, since I is independent of x ,...,x . How-
ℓ+1 λ+k 2λ+1+2l λ+k 1 2λ+2k
′
ever, B = x ∂I /∂x 6= 0, because I does depend on x .
kk ℓ+1 λ+k 2λ+1+2k λ+k 2λ+1+2k
′
This shows that B is a non-singular upper triangular matrix, hence rank(B) =
′
rank(B ) = n/2− λ − 1. We have thereby shown that if H = x , then the
ℓ+1
n/2 functions in (2.12) are functionally independent; since the rank of the Poisson
structure {·,·} is n, we have shown Liouville integrability in this case.
GENERALIZED LOTKA-VOLTERRA SYSTEMS 7
We now consider the general case, where several of the a may be non-zero.
i
We may still suppose that a = 1; as above, a = ··· = a = 0. Let us view
ℓ+1 1 ℓ
a ,...,a as arbitrary parameters and consider the matrix
ℓ+2 n
′
A 0 0
′
Jac :=0 1 0 ,
′
0 ⋆ B
 
′ ′
where A and B are square matrices which are constructed as in the previous
paragraph. It depends polynomially on the parameters a ,...,a and we have
ℓ+2 n
′
shown that the determinant of Jac is non-zero when we set all the parameters
a ,...,a equal to zero. By continuity, the determinant remains non-zero when
ℓ+2 n
theparametersa ,...,a aresufficientlyclosetozero,whichprovesthatthen/2
ℓ+2 n
functions in (2.12) are functionally independent for such values of the parameters.
InviewofLemma2.1,anynon-zerorescalingoftheparametersleadstoisomorphic
systems, so for any values of a ,...,a , the functions in (2.12) are functionally
ℓ+2 n
independent. This shows Liouville integrability for any values of the parameters
a ,...,a . (cid:3)
1 n
When nis odd,the rankofthe Poissonstructure{·,·}is n−1,soforLiouville
integrability we need (n+1)/2 functionally independent functions in involution.
RecallfromProposition2.2thatinthiscaseC isaCasimirfunction. TheLiouville
integrability is in this case given by the following theorem, whose proof is omitted
because it is very similar to the proof of Theorem 2.3.
Theorem 2.4. Suppose that n is odd. As before, let ℓ denote the smallest inte-
ger such that a 6=0 and let λ:= ℓ . The n+1 functions J ,J ,...,J ,H,F ,
ℓ+1 2 2 1 2 λ λ+2
Fλ+3,...,Fn−1,C are pairwise in i(cid:2)nv(cid:3)olution and functionally independent, hence
2
define a Liouville integrable system on (Rn,{·,·}).
We show inthe following theoremthatthe Hamiltonianvector fielddefined by
H is also superintegrable.
Theorem 2.5. The Hamiltonian system (1.5) has n−1 functionally indepen-
dent first integrals, hence is superintegrable.
Proof. We denote, as before, by ℓ the smallest integer such that a 6= 0
ℓ+1
(in particular, ℓ = 0 when a 6= 0). Suppose first that a is the only a which
1 ℓ+1 i
is different from zero; by a simple rescaling, we may assume a = 1, so that
ℓ+1
H =x . Then the equations of motion (1.5) take the following simple form:
ℓ+1
x H i6ℓ,
i
x˙ = 0 i=ℓ+1, (2.13)
i
 −x H i>ℓ+1.
i

8 KOULOUKAS,QUISPEL,ANDVANHAECKE
Whenℓ=0,acomplete setofn−1independentfirstintegralsof(2.13)isgivenby
H =x andx /x , (i=3,...,n). Whenℓ6=0,wecantakebesidestheHamiltonian
1 i 2
H =x the functions x /x , (i=2,...,ℓ) and x x , (i=ℓ+2,...,n).
ℓ+1 i 1 1 i
In the general case, we partition the set {1,2,...,n} into three subsets (A or
C may be empty):
A := {1,2,...,ℓ},
B := {i|a 6=0},
i
C := {i|i>ℓ+1 and a =0}.
i
Since we have treated the case #B = 1, we may henceforth assume that #B > 2.
Notice that each function v (and in particular H) depends only on the variables
i
x with i∈B. It follows that the differential equations (2.3),
i
x˙i =xi(H −vi−vi−1), (i∈B),
involveonlythevariablesx withj ∈B,sotheyformasubsystemwhichisthesame
j
as the original system, but now of dimension m := #B, and with all parameters
a , i∈B different fromzero. As explained above(see Lemma 2.1 andthe remarks
i
which follow its proof) this subsystem is by a simple rescaling isomorphic to the
system (1.2), for which we know from [14] that it is superintegrable, with m−1
first integrals which we denote here by G1,...,Gm−1. We do not need here the
preciseformulasforthesefunctions,butonlythefactthattheydependonlyonthe
variables xj with j ∈B; this obvious fact implies that the functions G1,...,Gm−1
are first integrals of the full system (1.5) as well. Consider, for i ∈ A∪ C the
following rational function:
(H −a x )x
ℓ+1 ℓ+1 i
, i∈A,
 x
K := ℓ+1
i  (H −aℓ+1xℓ+1)vi2 , i∈C .
x x
i ℓ+1
Notice that H −a x is different from zero, because #B > 2. For i ∈ A, we
ℓ+1 ℓ+1
have that
· · ·
(lnK ) = (ln(H −a x )) +(ln(x /x ))
i ℓ+1 ℓ+1 i ℓ+1
a x˙
ℓ+1 ℓ+1
= − +a x =0.
ℓ+1 ℓ+1
H −a x
ℓ+1 ℓ+1
Indeed, x˙ = x (H −v −v ) = x (H −a x ). Similarly, for i ∈ C,
ℓ+1 ℓ+1 ℓ+1 ℓ ℓ+1 ℓ+1 ℓ+1
we have from (2.3) and (2.5) that
· · · ·
(lnK ) = (ln(H −a x )) +2(lnv ) −(ln(x x ))
i ℓ+1 ℓ+1 i i ℓ+1
= −a x +2(H −v )−(H −2v )−(H −a x )=0.
ℓ+1 ℓ+1 i i ℓ+1 ℓ+1
This shows that the n−1 functions G1,...,Gm−1 and Ki, i ∈ A∪C, are first
integralsof(1.5). RecallthatthefunctionallyindependentfunctionsG1,...,Gm−1
depend onx with i∈B only andnotice that fori∈A∪C the variablex appears
i i
GENERALIZED LOTKA-VOLTERRA SYSTEMS 9
only in K . It follows that these n − 1 first integrals of (1.5) are functionally
i
independent, hence (1.5) is superintegrable. (cid:3)
Finally, we compute the solution x(t) of (2.1) which corresponds to any given
initial condition x(0) = (x(0),...,x(0)). We also introduce the derived functions
1 n
v (t) = a x (t)+···+a x (t), for i = 1,...,n. We denote by h the value of the
i 1 1 i i 0
Hamiltonian H at the initial condition x(0) and we denote v(0) :=v (0). It follows
i i
from (2.3) and (2.5) that we need to solve
dx
i
(t)=xi(t)(h0−vi(t)−vi−1(t)), (i=1,...,n), (2.14)
dt
where
dv
i
(t)=v (t)(h −v (t)), (i=1,...,n). (2.15)
i 0 i
dt
When v(0) = 0, the latter equation has v (t) = 0 as its unique solution; otherwise
i i
(2.15) is easily integrated by a separation of variables, giving
1 1
v (t)= , or v (t)= , (2.16)
i h10 +Cie−h0t i t+Ci′
′
depending on whether h 6= 0 or h = 0. The integrating constants C and C are
0 0 i i
computed from v (0)=v(0), which leads to
i i
1 1 1
′
C = − , and C = .
i v(0) h0 i v(0)
i i
The functions v (t) in (2.16) have very simple primitives, to wit
i
eh0t
′
v (t)dt=ln +C , or v (t)dt=ln(t+C ). (2.17)
Z i (cid:18) h i(cid:19) Z i i
0
Substituted in (2.14), which we write now as dldntxi(t) = h0 −vi(t)−vi−1(t), we
obtainby integrationandby usingthe primitives (2.17)(or v (t)dt=constantin
i
case v(0) =0) and the initial condition x (0)=x(0), the folloRwing result:
i i i
Proposition 2.6. The solution x(t) of (2.1) which corresponds to the initial
condition x(0) =(x(0),...,x(0)) is given by
1 n
(1−f(t)h )(1+f(t)h )
(0) 0 0
x (t)=x , (i=1,...,n),
i i 1−f(t)h +2f(t)v(0) 1−f(t)h +2f(t)v(0)
0 i−1 0 i
(cid:16) (cid:17)(cid:16) (cid:17)
(2.18)
where f(t)= eh0t−1 = 1 tanh(h0t)when h (thevalue of H at x(0)) is different
(eh0t+1)h0 h0 2 0
(0) (0) (0)
from zero and f(t)=t/2 otherwise. Also, v =a x +···+a x .
i 1 1 i i
Notice that when h 6=0, (2.18) can be rewritten as
0
x(0)eth0h2
x (t)= i 0 , (i=1,...,n).
i
h0+(eth0 −1)vi(−0)1 h0+(eth0 −1)vi(0)
(cid:16) (cid:17)(cid:16) (cid:17)
10 KOULOUKAS,QUISPEL,ANDVANHAECKE
Remark 2.7. When several of the parameters a in the Hamiltonian function
i
H areequaltozero,sothatH isindependentofthecorrespondingvariablesx ,the
i
vectorfield(1.5)isaHamiltonianvectorfieldwithrespecttoafamilyofcompatible
Poisson structures, always with the same Hamiltonian H. Indeed, suppose that
a =a =0,withi<j. Then,inthecomputationofthevectorfieldx˙ ={x ,H},
i j k k
k = 1,...,n, the Poisson brackets {x ,x } = −{x ,x } are not used, so we may
i j j i
replace{x ,x }=−{x ,x } by anarbitraryfunction f of x ,...,x without any
i j j i ij 1 n
effect on the vector field. However, in order for the new bracket to be a Poisson
bracket, it has to satisfy the Jacobi identity, which puts several restrictions on the
function f . One way to satisfy this restriction is to take f :=a x x , where a
ij ij ij i j ij
is an arbitrary constant. In fact, replacing {x ,x } = x x by {x ,x } = a x x
i j i j i j ij i j
foralli<j forwhicha =a =0,thenewbracketswillstillbe ofthegeneralform
i j
{x ,x } = b x x , known in the literature as diagonal brackets; such brackets are
i j ij i j
known to automatically satisfy the Jacobi identity [10, Example 8.14] so they are
Poissonbrackets. Clearly,anylinearcombinationofthesediagonalPoissonbrackets
is again a diagonal Poisson bracket, hence all these brackets are compatible. The
upshot is that when k > 2 parameters are equal to zero, then (1.5) has a multi-
Hamiltonian structure: it is Hamiltonian with respect to a k -dimensional family
2
of Poisson brackets. (cid:0) (cid:1)
3. The Kahan discretization
In this section we consider the Kahan discretization of the system (2.1). Let
us recallquickly the constructionof the Kahandiscretizationof a quadratic vector
field x˙ = Q (x) (see e.g. [4]). Let Φ (y,z) denote the symmetric bilinear form
i i i
whichisassociatedtothe quadraticformQ andletǫ denote apositiveparameter,
i
which should be thought of as being small. Then the Kahan discretization with
step size ǫ is the map1 x 7→x˜ , implicitly defined by
i i
x˜ −x =ǫΦ (x,x˜). (3.1)
i i i
We refer to this map as the Kahan map (associated to x˙ = Q (x)). It is well
i i
known that the Kahan map preserves the linear integrals of the initial continuous
system (quadratic vector field). So, in our case of the generalized Lotka-Volterra
system, its Hamiltonian function H = a x +a x +···+a x is an invariant of
1 1 2 2 n n
the Kahan map. As we are going to show in this section the Kahan map (of this
system) preservesthe Poissonstructure as well; we will also see in the next section
that all constants of motion, in particular the ones that appear in Theorems 2.3,
2.4 and 2.5, are also invariants of the Kahan map.
1When the map which is defined by the discretization is iterated, one often writes it as
xi(m)7→xi(m+1).