A new class of integrable Lotka-Volterra systems
aa r X i v : . [ n li n . C D ] J u l A new class of integrable Lotka–Volterra systems
H. Christodoulidi , , A.N.W. Hone , and T.E. Kouloukas Research Center for Astronomy and Applied MathematicsAcademy of Athens, Athens 11527, Greece School of Mathematics, Statistics and Actuarial Science,University of Kent, Canterbury CT2 7NF, UK School of Mathematics and Statistics,University of New South Wales, Sydney NSW 2052, AustraliaJuly 9, 2019
Abstract
A parameter-dependent class of Hamiltonian (generalized) Lotka–Volterra sys-tems is considered. We prove that this class contains Liouville integrable as well assuperintegrable cases according to particular choices of the parameters. We deter-mine sufficient conditions which result in integrable behavior, while we numericallyexplore the complementary cases, where these analytically derived conditions are notsatisfied.
The Lotka–Volterra system was introduced independently by Lotka [15] and Volterra [20]as a predator-prey model. Since then, many generalizations have been considered withapplications to several scientific disciplines. These systems in general display rich dynam-ical behavior that varies according to the parameters that define each one of them. Forexample, there are Hamiltonian and non-Hamiltonian Lotka–Volterra systems, as well asintegrable, non-integrable and chaotic ones. From the point of view of integrability, variouskinds of generalized Lotka–Volterra systems have been extensively studied in the literature,e.g. [1, 4, 5, 6, 10, 11, 12, 16, 17, 19]. A numerical study of a 4–dimensional non-integrableLotka–Volterra system can be found in [18].In this paper, we study a parametric family of (generalized) Lotka–Volterra systems ofthe form ˙ x i = x i X j>i a j x j − X j
Lotka–Volterra systems are systems of the form˙ x i = x i n X j =1 A ij x j + r i ! , i = 1 , . . . , n , (2)where A = ( A ij ) is any arbitrary n × n matrix, known as the community matrix and r = ( r , . . . , r n ) is a vector in R n .In this paper, we are going to study a particular class of Lotka–Volterra systems, with2ommunity matrix A = a a . . . a n − a a . . . a n − a − a . . . a n ... ... ... . . . − a − a − a . . . , (3)and parameters a , . . . , a n ∈ R . In this case, system (2) can be written as (1), or equiva-lently, as ˙ x i = x i n X j =1 P ij a j x j + r i ! , (4)where P is the antisymmetric matrix P ij = (cid:26) − δ ij for 1 ≤ i ≤ j ≤ n , − ≤ j < i ≤ n . (5)The special case of (1) with r = 0 was extensively studied in [13, 14], where the Liouvilleand superintegrability of the corresponding systems were proved and explicit solutions weregiven. Here, our aim is to investigate the integrability of particular cases with r = 0. Indue course we mainly restrict our attention to the case that n is even. We consider the log-canonical Poisson structure { x i , x j } = x i x j , ≤ i < j ≤ n. (6)The rank of this Poisson structure, for x . . . x n = 0, is n for even n , and n − n .In the odd case, C := x x ...x n x x ...x n − is a Casimir function. Proposition 3.1.
For any even n , a i , r i ∈ R and x i > , i = 1 , , . . . , n , the Lotka–Volterrasystem (1) is Hamiltonian with respect to the Poisson structure (6) and the Hamiltonianfunction H ( x ) = n X i =1 ( a i x i + k i log x i ) , where x = ( x , . . . , x n ) and k = ( k , . . . k n ) defined by k = P − r . In terms of the parameters k i , the system is written as˙ x i = x i n X j =1 P ij ( a j x j + k j ) ! . (7)3or odd n , the matrix P is not invertible. Hence, the Hamiltonian structure of Prop.3.1 does not include all the cases of (1) for arbitrary r ∈ R n . However, for any n we canrestrict our analysis to the Hamiltonian systems (7), i.e. systems (1) with r = P k .By setting u i = log x i , the Poisson bracket (6) becomes a constant one, that is { u i , u j } = P ij , and the Hamiltonian function H u = P ni =1 ( a i e u i + k i u i ). In these coordinates our systemis expressed as˙ u i = n X j =1 P ij a j e u j + r i = X j>i a j e u j − X j
The parameters a i , . . . , a n of (1) can be rescaled to a c , . . . , a n c n , by us-ing the transformation x i x i /c i , for c i >
0. This linear transformation preserves thePoisson bracket and gives rise to an equivalent Hamiltonian system with Hamiltonian H y = P ni =1 ( a i c i y i + k i log c i y i ) , in the new variables y i = x i /c i . For example, by setting c i = | a i | , all the nonzero a i can be rescaled to 1 or −
1. Hence, we can consider systemswith parameters a i ∈ {− , , } without any loss of generality.In the present work we will restrict to the even-dimensional case; however, a simi-lar approach can be considered for odd dimensions. Some additional comments on odd-dimensional cases as well as two examples, for n = 3 and n = 5, are given in the appendix. Following [14], we introduce the functions v i := a x + · · · + a i x i , i = 1 , . . . , n. If a a . . . a n = 0, the functions v i define new coordinates on R n but generally this is nottrue. Furthermore, for any even n we define the functions J m ( x ) = x x . . . x m − x x . . . x m , I m ( x ) = x m +2 x m +4 . . . x n x m +1 x m +3 . . . x n − , F m ( x ) = v m I m ( x ) , for m = 1 , . . . , n . We also set H := F n/ = v n = n X i =1 a i x i , which corresponds to the Hamiltonian function in the case of r = k = 0. So, the genericHamiltonian of (1) is written as H = H + n X i =1 k i log x i .
4n [14], it is proved that for any m, l ∈ { , . . . , n } , { J m , J l } = { F m , F l } = { F m , H } = 0 , (8)as well as the following theorem which establishes the Liouville integrability of the systemin the case of r = 0. Theorem 4.1.
Suppose that n is even. Let ℓ denote the smallest integer such that a ℓ +1 = 0 and let λ := (cid:2) ℓ (cid:3) . The n functions J , J , . . . , J λ , F λ +1 , F λ +2 , . . . , F n − , H are pairwise ininvolution and functionally independent. Here, our first goal is to determine the parameters a i and r , so that the more generalsystem (1) inherits the same integrals as the r = 0 case which ensure Liouville integrability.In the following, we always assume that n is even. Lemma 4.2.
For any m = 1 , . . . , n − { F m , H } = I m m X j =1 k j ( v j + v j − − v m ) . (9) Proof.
Since { F m , H } = { F m , H + P nj =1 k j log x j } and { F m , H } = 0 (from (8)), we get { F m , H } = n X j =1 k j { F m , log x j } = n X j =1 k j { v m I m , log x j } . (10)Now { v m , log x j } = P mi =1 { x i , x j } a i x j , so it follows that { v m , log x j } = (cid:26) v m , for 2 m < j ,v j + v j − − v m , for j ≤ m . Also, we have { I m , x j } = (cid:26) − I m x j , for 2 m < j , , for j ≤ m (this identity was proved in [14]) and { I m , log x j } = { I m , x j } x j = (cid:26) − I m , for 2 m < j , , for j ≤ m . Therefore, we see that { v m I m , log x j } = (cid:26) , for 2 m < j , ( v j + v j − − v m ) I m , for j ≤ m , and by substituting in (10) we derive (9). 5e can recast the sum that appears in (9) to derive m X j =1 k j ( v j + v j − − v m ) = m X j =1 x j a j ( − j − X i =1 k i + m X i = j +1 k i ) ;hence, from Lemma 4.2, the next proposition follows. Proposition 4.3.
Suppose that n is even. For every m = 1 , . . . , n − , { F m , H } = 0 ifand only if S jm = 0 , for every j = 1 , . . . , m , where S jm = a j ( − j − X i =1 k i + m X i = j +1 k i ) . (11)Solutions of the system S jm = 0, for m = 1 , . . . , n − j = 1 , . . . , m , provideconditions on the parameters a i and k i ensuring that the functions F , F , . . . , F n − arefirst integrals of the system. Moreover, according to (8), these integrals are pairwise ininvolution. Therefore, in the case where F m = 0, for all m = 1 , . . . , n −
1, these conditionson the parameters provide Liouville integrability. For example, in the particular casewhere a j = 0, for every j = 1 , . . . , n , the corresponding system implies the unique solution k = k = · · · = k n − = 0. Corollary 4.4.
For a a . . . a n = 0 , k = k = · · · = k n − = 0 , k n − , k n ∈ R and r = P k = ( k n − + k n , k n − + k n , . . . , k n , − k n − ) , the Hamiltonian system (1) is Liouvilleintegrable with first integrals H, F , F , . . . , F n − . Now, let a = a = · · · = a ℓ = 0, a ℓ +1 = 0 and λ := (cid:2) ℓ (cid:3) . In such a case, F = · · · = F λ =0. So, for any choice of parameters there are not enough F -type integrals to ensure theintegrability of the system. However, Theorem 4.1 suggests that we could probably replacethe first λ missing F -integrals by λ J -integrals. Hence next, we are going to determine theconditions on the parameters to ensure that { J m , H } = 0, for m = 1 , . . . , λ . Lemma 4.5.
Let a = a = · · · = a ℓ = 0 , a ℓ +1 = 0 and λ := (cid:2) ℓ (cid:3) . Then, { J m , H } = J m ( k + k + · · · + k m ) , (12) for m = 1 , . . . , λ .Proof. We consider m ∈ { , . . . , λ } . From Theorem 4.1, it follows that { J m , H } = 0. So, { J m , H } = n X j =1 k j { J m , log x j } = m X j =1 k j { J m , log x j } + n X j =2 m +1 k j { J m , log x j } . (13) The proof of the functional independence of the integrals is given in Prop. 4.7. For m > λ , J m cannot be an integral of the system i.e. { J m , H } 6 = 0. So, the total number of F and J integrals cannot exceed n − { J m , x j } = n X i =1 { x i , x j } ∂J m ∂x i = j − X i =1 x i x j ∂J m ∂x i − n X i = j +1 x i x j ∂J m ∂x i and x i ∂J m ∂x i = (cid:26) ( − i +1 J m , for 1 ≤ i ≤ m , , for 2 m < i ≤ n . Consequently, after some calculations we obtain { J m , x j } = (cid:26) J m x j , for j ≤ m , , for j > m and { J m , log x j } = (cid:26) J m , for j ≤ m , , for j > m . Substituting this into (13), we derive (12).Finally, if we combine Lemma 4.5 with Prop. 4.3 we come up with the following theorem.
Theorem 4.6.
Suppose that n is even. Let ℓ denote the smallest integer such that a ℓ +1 = 0 and let λ = (cid:2) ℓ (cid:3) . The n functions J , J , . . . , J λ , F λ +1 , F λ +2 , . . . , F n − , H are pairwise ininvolution if and only if k i = − k i − , for i = 1 , . . . , λ , and S jm = 0 , for m = λ +1 , . . . , n − , j = ℓ + 1 , . . . , m .Proof. Let a = a = · · · = a ℓ = 0, a ℓ +1 = 0 and λ = (cid:2) ℓ (cid:3) . From Lemma 4.5, we concludethat { J , H } = { J , H } = · · · = { J λ , H } = 0 if and only if k + k + · · · + k i = 0, for all i = 1 , . . . , λ , which is equivalent to k i = − k i − , for i = 1 , . . . , λ . Also, from Prop. 4.3,we derive that for m = λ + 1 , . . . , n − { F m , H } = 0 if and only if S jm = 0, for every j = ℓ + 1 , . . . , m (for j = 1 , . . . ℓ , S jm = 0, since a = · · · = a l = 0). Finally, Theorem 4.1shows that all the other pairs of functions are in involution too.We will close this section by proving the functional independence of the integrals. Proposition 4.7.
For every even n , the functions J , J , . . . , J λ , F λ +1 , F λ +2 , . . . , F n − , H , are functionally independent.Proof. For k = 0, J , . . . , J λ , F λ +1 , . . . , F n − , H are functionally independent. This followsfrom Theorem 4.1, since in this case H coincides with H . Hence, by continuity the samefunctions remain functionally independent for parameters k in a sufficiently small openneighborhood U of k = 0. Now, let us consider any k = ( k , . . . , k n ) ∈ R n . Then there is µ > k ′ = ( k ′ . . . , k ′ n ) ∈ U , such that k = µ k ′ . Also, in view of Remark 3.2, we canrescale the a i parameters to µa i , by setting y i = x i /µ . The Hamiltonian function in thenew y -coordinates then becomes H y ( y ) = n X i =1 ( a i µy i + k i log µy i ) = µ n X i =1 ( a i y i + k ′ i log µy i ) . dH y = µdH ′ , where H ′ ( y ) = P ni =1 ( a i y i + k ′ i log y i ), i.e. the Hamiltonian of thecorresponding system with parameters a i and k ′ i . Therefore, from the functional inde-pendence of J , . . . , J λ , F λ +1 , . . . , F n − , H ′ that we proved, the functional independenceof J , . . . , J λ , F λ +1 , . . . , F n − , H y follows and consequently the functional independence of J , . . . , J λ , F λ +1 , . . . , F n − , H for all parameters a i and k i . In [13, 14], a second set of first integrals in involution has been introduced for the caseof r = 0. By considering this set of integrals we can derive more integrable cases of oursystem. The main observation to accomplish this is that system (7) remains invariant underthe transformation x i x n +1 − i and the reparametrization a i
7→ − a n +1 − i , k i
7→ − k n +1 − i ,for i = 1 , . . . , n . Let us now consider the involution ι ( x , x , . . . , x n ) ( x n , x n − , . . . , x )and the functions ˜ J m = J ◦ ι , ˜ I m = I ◦ ι , ˜ F m = ˜ v m ˜ I m , where ˜ v i := a n +1 − i x n +1 − i + a n +2 − i x n +2 − i + · · · + a n x n , for i = 1 , . . . , n . Then, by Theorem4.6 and the described symmetry of the system we derive the next theorem. Theorem 5.1.
Suppose that n is even. Let d denote the smallest integer such that a n − d = 0 and let δ = (cid:2) d (cid:3) . The n functions ˜ J , ˜ J , . . . , ˜ J δ , ˜ F δ +1 , ˜ F δ +2 , . . . , ˜ F n − , H are pairwise ininvolution if and only if k n +1 − i = − k n +2 − i , for i = 1 , . . . , δ , and ˜ S jm = 0 , for m = δ + 1 , . . . , n − , j = d + 1 , . . . , m , where ˜ S jm = a n +1 − j ( − j − X i =1 k n +1 − i + m X i = j +1 k n +1 − i ) . Theorem 5.1, determines different values of the parameters of the system that lead to in-tegrability. Furthermore, a combination of Theorems 4.6-5.1 provide some superintegrablecases. For any ℓ, d ∈ { , , . . . , n − } , we consider the following two sets of parameters:Σ ℓ = { ( a , k ) ∈ R n : a = a = . . . a ℓ = 0 , a ℓ +1 = 0 , k i + k i − = S jm = 0 , for i = 1 , . . . , (cid:20) ℓ (cid:21) , m = (cid:20) ℓ (cid:21) + 1 , . . . , n − , j = ℓ + 1 , . . . , m } , ˜Σ d = { ( a , k ) ∈ R n : a n = a n − = · · · = a n − d +1 = 0 , a n − d = 0 ,k n +1 − i + k n +2 − i = ˜ S jm = 0 , for i = 1 , . . . , (cid:20) d (cid:21) , m = (cid:20) d (cid:21) + 1 , . . . , n − ,j = d + 1 , ..., m } , where ( a , k ) := ( a , . . . , a n , k , . . . , k n ). Then we conclude with the following theorem.8 heorem 5.2. If ( a , k ) ∈ Σ ℓ ∪ ˜Σ d for some ℓ, d ∈ { , , . . . , n − } , then for every even n system (7) with parameters a , k is Liouville integrable. If ( a , k ) ∈ Σ ℓ ∩ ˜Σ d , then thecorresponding system (7) is superintegrable, i.e. it admits the following n − functionallyindependent integrals: J , J , . . . , J λ , F λ +1 , F λ +2 , . . . , F n − , ˜ J , ˜ J , . . . , ˜ J δ , ˜ F δ +1 , ˜ F δ +2 , . . . , ˜ F n − , H . Example 5.3.
The simplest interesting case is n = 4 (for n = 2 the system is alwaysintegrable since it is Hamiltonian). In this case we have,Σ = { ( a , k ) ∈ R : a = 0 , k = a k = 0 } , Σ = { ( a , k ) ∈ R : a = 0 , a = 0 , k = 0 } , Σ = { ( a , k ) ∈ R : a = a = 0 , a = 0 , k = − k } , Σ = { ( a , k ) ∈ R : a = a = a = 0 , a = 0 , k = − k } , ˜Σ = { ( a , k ) ∈ R : a = 0 , k = a k = 0 } , ˜Σ = { ( a , k ) ∈ R : a = 0 , a = 0 , k = 0 } , ˜Σ = { ( a , k ) ∈ R : a = a = 0 , a = 0 , k = − k } , ˜Σ = { ( a , k ) ∈ R : a = a = a = 0 , a = 0 , k = − k } , where ( a , k ) = ( a , . . . , a , k , . . . , k ). Now, using Theorem 5.2 we can detect differentintegrable and superintegrable cases. So for example, when a . . . a n = 0, from Σ ∪ ˜Σ we come up with two integrable cases, for k = (0 , , k , k ) and k = ( k , k , , ∩ ˜Σ is when k = 0. On the otherhand, for a = 0 and a a a = 0, we derive the integrable cases with k = ( k , , k , k )and k = ( k , k , , k = ( k , , , n = 4 The purpose of this section is to explore numerically the behavior of 4-dimensional Lotka–Volterra systems of the form (1) and investigate their integrability in cases that are notdescribed in the previous sections. In the rest of the paper we will restrict to the case of a = a = a = a = 1 and we vary only the k i values. We perform a series of numericalcalculations for the system˙ x i = x i X j =1 P ij ( x j + k j ) ! , i = 1 , . . . , , (14)with different k , . . . , k values, which are complementary to the two integrable casesdescribed by Theorem 5.2. We numerically integrate the system’s equations of motion to-gether with its variational equations to compute the value of the largest Lyapunov exponent λ . The variational equations of the system (14) are δ ˙ x = [ J · ∇ H ( x ( t ))] · δ x , (15)9 x x x x x x x x Figure 1: The Poincar´e surface of section x = 1 , x > a i = 1 and E = 6 for various k i , i = 1 , , , k , k , k , k ) = ( − , − , − , − k , k , k , k ) = ( − , − , − , − k , k , k , k ) = ( − , − , − , − k , k , k , k ) = ( − , − , − , − x , x , x plane for the system with ( k , k , k , k ) =( − , − , − , −
1) and for initial conditions: (a) close to a fixed point of Fig.1(d) ( E = 6),(b) on an ellipse around the fixed point ( E = 6), (c) randomly chosen from Fig.1(d) ( E = 6)and (d) randomly chosen at a higher total energy ( E = 20) exhibiting chaotic behavior.11here δ x = ( δx , δx , δx , δx ) is a vector which evolves on the tangent space of the system(14) and ∇ H denotes the Hessian matrix of the Hamiltonian function H calculated alongthe reference orbit x ( t ) of the system (14). In particular, we used the classical Runge–Kutta forth-order scheme with time-step τ = 10 − for the numerical integration of thesystems (14) and (15), which conserved the energy E = H ( x ) of the system (14) withaccuracy of more than 8 significant figures during integration times of the order of a fewthousand. The indicator which controls of the relative energy error is RE = log (cid:12)(cid:12)(cid:12)(cid:12) E ( t ) − E E (cid:12)(cid:12)(cid:12)(cid:12) , where E is the initial energy of the system and E ( t ) the actual energy during the numericalintegration.For k i < i = 1 , . . .
4, the point x = ( − k , − k , − k , − k ) is an elliptic fixed pointof the system. Furthermore, in this case H ( x ) admits a global minimum at x and all theorbits of the system are bounded.We start our numerical study with examples of bounded motion, which correspond tonegative values for all k i . In Fig.1 some Poincar´e surfaces of section x = 1, x > k i < E = 6. However, at this energy level all of themexhibit regular behavior. These Poincar´e surfaces of section are constructed for a grid ofinitial conditions on the x , x plane, with x = 1 and x found numerically by Newton’smethod requiring that H ( x ) = E . We find a rich morphology consisting of periodic andquasiperiodic trajectories, island chains as well as separatrices. Each fixed point on thePoincar´e surface represents a periodic orbit, while the ellipse-like curves correspond toquasiperiodic trajectories lying on tori. Fig.2 presents different trajectories projected onthe x , x , x plane for the system with ( k , k , k , k ) = ( − , − , − , −
1) which correspondsto Fig.1(d). The first three panels of Fig.2 correspond to E = 6 and the last one to E = 20.We find qualitatively similar behavior to the examples of Fig.1 for k = · · · = k = − x = 1, x > E = 4 .
2, there is no evidence of chaotic behavior. We verify thisresult in Fig.4(a) by computing the largest Lyapunov exponent λ , which approximatelydecays as 1 /t for randomly chosen initial conditions. Similarly with the well-known H´enon–Heiles model [9], chaotic dynamics in the Lotka–Volterra system (14) for k i < k i = − E is gradually increased, we observe a gradual transformation of fixed points andellipses–like curves, while at energies of the order of E = 30 (Fig.3(d)) the chaotic motion isnot only evident but also prevails over the ordered motion. The largest Lyapunov exponentat this energy, which is plotted in Fig.4(b), converges to a positive value λ ≃ . n = 4, a = (1 , , , k = (0 , , k , k ), k , k ∈ R or k = ( k , k , , k , k ∈ R .We choose ( k , k , k , k ) = (0 , , − , − x + x ) x /x is preservedbesides the Hamiltonian. Fig.5(a) displays the evolution of the four variables log x i in timefor a random choice of initial conditions. It turns out that x decays asymptotically tozero, approximately like e − . t , while the rest variables x , x , x asymptotically approach12 .5 1.0 1.50.51.01.5 x x x x x x x x Figure 3: The Poincar´e surface of section x = 1 , x > a i = 1 and k i = − i = 1 , , , E = 4 .
2, (b) E = 6, (c) E = 8,(d) E = 29. -6 -5 -4 -3 -2 -1 t -6 -5 -4 -3 -2 -1 t Figure 4: The largest Lyapunov exponent λ for the Lotka–Volterra system with a i = 1 and k i = − i = 1 , , , E = 4 . E = 29.13
20 40 60 80 100-60-50-40-30-20-10010 logx logx logx logx t logx logx logx t Figure 5: The evolution in time of the phase space variables for the integrable cases: (a)( k , k , k , k ) = (0 , , − , −
1) and (b) ( k , k , k , k ) = ( − , − , − , x , x , x plane for the integrablesystems: (a) ( k , k , k , k ) = (0 , , − , −
1) and (b) ( k , k , k , k ) = ( − , − , − , -6 -5 -4 -3 -2 -1 t -6 -5 -4 -3 -2 -1 t Figure 7: The largest Lyapunov exponent λ for the system with ( k , k , k , k ) =( − , − , − ,
2) at (b) E = 10 and (c) E = 72.a periodic orbit, as is illustrated in Fig.6(a). However, a similar behavior appears in othercases, not described as integrable by Theorem 5.2. Such an example is given in Fig.5(b)and corresponds to ( k , k , k , k ) = ( − , − , − , x and x tend asymptotically to zero as e − t , while x and x asymptotically converge to theperiodic orbit shown in Fig.6(b). Furthermore, we carefully examine the largest Lyapunovexponent λ in Fig.7 for constantly increasing energies and we find that λ ∝ /t , even when E = 72, which strongly indicates that the system is integrable in this case too.Similarly to the case ( k , k , k , k ) = ( − , − , − ,
2) we find other cases which displayintegrable behavior, manifested by asymptotically vanishing Lyapunov exponents. Few ofthe cases that we checked are listed in the following table k k k k n = 4 system (7) emerges when a i > k i < a i < k i > Conclusions
We presented a new class of Hamiltonian parametric Lotka–Volterra systems with non-zerolinear terms and we proved that, for particular choices of parameters, Liouville integrabilityand superintegrability is established. Different choices of parameters when n = 4, notdescribed by the theory, were studied numerically, showing that both chaotic and newintegrable cases appear. Concerning these new cases with integrable behavior, we aim tostudy them in detail in order to detect additional integrals and complete our investigationby including all the odd dimensional cases too.In the present work we restricted our analysis to the even-dimensional case; however, asimilar approach can be considered for odd dimensions. Finally, we believe that a similarapproach can be considered for integrable Lotka–Volterra systems with different communitymatrices, or integrable deformations of them such as the systems presented in [6, 7, 8], byinserting parametric linear terms in the corresponding vector fields. Acknowledgements
HC is supported by the State Scholarship Foundation (IKY) operational Program: ‘Edu-cation and Lifelong Learning–Supporting Postdoctoral Researchers’ 2014-2020, and is co–financed by the European Union and Greek national funds; she is also grateful to SMSAS,Kent for hosting her as a visitor. ANWH is supported by Fellowship EP/M004333/1 fromthe Engineering & Physical Sciences Research Council, UK, and is grateful to the Schoolof Mathematics & Statistics, UNSW for hosting him as a Visiting Professorial Fellow withfunding from the Distinguished Researcher Visitor scheme; he also thanks Prof. WolfgangSchief for additional financial support in 2019. TEK would like to thank Prof. ReinoutQuispel, Dr Peter Van Der Kamp and Dr Charalambos Evripidou for their hospitality atLa Trobe University, and for their useful comments on this topic.
A Comments and examples on the odd dimensionalcases
As it is stated in Section 3, in the odd dimensional cases the described Hamiltonian for-malism, i.e. the log-canonical Poisson structure (6) along with the Hamiltonian H ( x ) = P ni =1 ( a i x i + k i log x i ) , is not sufficient to include all the cases of vector fields (1) for arbi-trary r i , since matrix (5) is not invertible. Therefore, in this setting we can only restrictto the cases with r = P k , that is systems of the form (7). For n = 3, the integrabilityof (7) follows directly from its Hamiltonian formalism and the existence of the Casimirfunction x x x . More interesting integrable cases emerge for odd n >
3, by considering thecorresponding integrals of the k = 0 case as they appear in [14] and the correspondingpermutation symmetry of the system. We will illustrate this in the following example for n = 5. 16et us consider the system˙ x i = x i X j =1 P ij ( a j x j + k j ) ! , i = 1 , . . . , , (16)with parameters a = ( a , . . . , a ) , k = ( k , . . . , k ) ∈ R . According to [14], for k = thissystem admits the first integral F = x x ( a x + a x + a x ) . We compute its Poisson bracket with the Hamiltonian H = P ni =1 ( a i x i + k i log x i ) of (16)to get { F, H } = x x ( a ( k + k ) x + a ( k − k ) x − a ( k + k )) . Hence, F is a first integral of (16) if and only if a ( k + k ) = a ( k − k ) = a ( k + k ) = 0 . (17)If the parameters a , k satisfy (17), then the integral F in addition to the Casimir function C = x x x x x ensures the complete integrability of the system. Furthermore, the invarianceof (16) under the transformation x i x − i , a i
7→ − a − i , k i
7→ − k − i , implies that˜ F = x x ( a x + a x + a x ) . is a first integral of (16) if and only if a ( k + k ) = a ( k − k ) = a ( k + k ) = 0 . (18)So we conclude that system (16) is integrable if the parameters a , k satisfy (17) or (18).For example, in the case of a = 0, system (16) is integrable if k = − k = k or k = − k = k , while the case of k = − k = k = − k = k which leads to superintegrability isequivalent to the k = 0 case. References [1] ´A. Ballesteros, A. Blasco and F. Musso, Integrable deformations of Lotka–Volterrasystems,
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