On an integrable discretisation of the Lotka-Volterra system · all conserved quantities of the integrable system. This problem prompts a concept called integrable discretization[1]

On an integrable discretisation of theLotka-Volterra system

by

Yang He and Yajuan Sun

Report No. ICMSEC-2012-08 September 2012

Research Report

Institute of Computational Mathematics

and Scientific/Engineering Computing

Chinese Academy of Sciences

On an integrable discretisation of the Lotka-Volterra system

Yang He∗ Yajuan Sun †

September 24, 2012

Abstract

In this paper, we study Hirota’s discretization for a three dimensional integrable

Lotka-Volterra system. By using backward error analysis we establish the

corresponding modified equation determined by the numerical discretization and

analyze its properties. As the three dimensional modified system has two first

integrals, it is reduced to a system in one dimension. Numerical results are also

presented for the Lotka-Volterra system by using several numerical methods.

Key Words: Integrable Lotka Volterra system; Hirota’s integrable discretisation;

Backward error analysis; Modified differential equation.

1 Introduction

Since Newton found the exact solution for the Kepler problem, the theory of integrable

system has been an important component in the study of differential equations.

Compared with non-integrable systems, integrable systems have better properties and

more predictable long-term behaviors, thus can be studied in much greater detail by

means of both algebraic and analytic methods. Besides, integrable systems play a

non-negligible role in the description of various physical phenomena, for example in

application fields like fluid physics, quantum physics and biology etc.. Generally, a

system is called integrable if it can be solved by quadrature, i.e. using only evaluation,

inversions and integrals of known functions. However, in different fields of mathematics

and physics, there exist various distinct notions of integrability. In classical mechanics

like Hamiltonian dynamical systems, the most frequently referred notion is complete

∗LSEC, Institute of Computational Mathematics and Scientific/Engineering, Academy ofMathematics

and Systems Science, Chinese Academy of Sciences, Beijing 100190, China.†LSEC, Institute of Computational Mathematics and Scientific/Engineering, Academy ofMathematics

and Systems Science, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100190, China.

1

2 Yang He and Yajuan Sun ICMSEC-RR 2012-08

integrability in the sense of Liouville, which arose in the 19th century and is based on

the notion of first integrals (conserved quantities). In Liouville’s sense, a system (very

often refer to Hamiltonian system) is said to be integrable if it has sufficiently number of

first integrals in involution.

This notion of integrability is also applicable to systems of PDEs and discrete systems

such as lattices. In evolutionary PDEs, the discovery of soliton phenomenon provides a

new view for the study of integrable systems. Then the existence of multi-soliton solutions,

as well as Lax pairs, Backlund transformations or bilinear form in Hirota’s sense can be

used to determine integrability of the given PDEs. More recently, the results are extended

to quantum mechanics and has led to many remarkable results.

Although some important systems can be solved analytically, most systems could not

be treated successfully in this way. Various numerical methods are usually needed to

extract information from models governed by differential equations. For systems with

special structural properties, it is crucial that the numerical results could exhibit the

properties for a long time. Therefore, to preserve as many properties of the original

system as possible is one central idea of constructing methods for numerical integration.

Geometric numerical integrator is a class of numerical methods which are constructed

based on this idea [14]. Specifically, there are symplectic (Poisson) methods, volume-

preserving methods, integral-preserving methods etc. By comparison, geometric numerical

integration usually provides numerical results with superior behavior than that gained

by other methods. For instance, when symplectic methods are applied to an integrable

Hamiltonian system, it is shown by backward error analysis and KAM theory that with

an initial frequency satisfying the diophantine condition, the numerical errors for all the

first integrals of the given system are bounded over exponentially long time and the global

error grows linearly [2]. Specially, for a 2d dimensional completely integrable system there

exist action-angle variables in which the numerical flow with a non-resonant step sizes h

on a Cantor set is linear on a near invariant tori close to the original one [23]. This implies

that when h is restricted on a Cantor set, the modified differential equation is convergent

and integrable with d modified first integrals. Furthermore, it is proved [Shang& Ding11]

that when h is outside the Cantor set all the first integrals are bounded for all time even

though the modified first integrals may not exist.

Since mid 70s there have appeared amounts of theories for integrable difference

systems, and various integrable discretisation methods have been designed to produce

discrete versions of integrable systems admitting soliton solutions. In this paper, we

consider an integrable discretization as a numerical discretization preserving invariant

phase space volume and having enough number of first integrals [25]. One powerful

ICMSEC-RR 2012-08 3

technique to derive integrable discretization is Hirota’s bilinear approach, it is done by

discretizing the bilinear form of the original system, yet mechanics behind it is not fully

understood [3, 4, 5, 6, 7, 8, 9]. One purpose of this paper is to reveal the nature of

integrability when apply integrable discretizations to integrable systems.

In this paper, we study Hirota’s discretization for the three dimensional

Lotka-Volterra system by backward error approach [2]. In backward error theory, a

numerical discretization can be realized to be an exact solution of a modified differential

equation (MDE) with vector field in a formal series of powers of h (time step). We prove

that the MDE of Hirota’ s discretization can carry most properties of the Lotka-Volterra

system.

The outline of the paper is as follows. In Section 2, we give the definitions of integrable

systems and integrable discretisations. In Section 3, we introduce the Lotka-Volterra (LV)

system and analyze its rich geometric properties. We discretize the LV system by Hirota’s

discretisation and establish the corresponding MDE in Section 4. In Section 5 we present

a recurrence formula to derive the coefficients of the MDE corresponding to Hirota’s

discretization and investigate its convergency. Section 6 is devoted to reformulate the

MDE according to its geometric properties. We also show the numerical experiments by

Hirota’s method and other geometric numerical methods in Section 7.

2 Integrable Poisson system and Integrable discretisation

Integrable systems are studied in various research fields ranging from differential geometry

and complex analysis to quantum field theory and fluid dynamics. However, there is no

precise definition of integrability that generally works. The simplest aspect of integrability

is that the solution of integrable systems can be calculated by quadrature. For ordinary

differential equations, this is based on the existence of sufficient number of independent

first integrals (constants of motion). In this section we deal with the integrability of

Poisson systems, it can be deduced from the Arnold-Liouville integrability for Hamiltonian

systems.

Definition 2.1. The Poisson bracket of two n dimensional smooth functions F and G is

a function {F,G} satisfies

{F,G}(x) = ∇F (x)TJ(x)∇G(x),

where J(x) is a skew-symmetric matrix and satisfies the Jacobi identity [18]

n∑l=1

(∂Jij∂xl

Jlk +∂Jjk∂xl

Jli +∂Jki∂xl

Jlj

)= 0,


J(x) is called the Poisson matrix of the bracket {·, ·}.

By Poisson bracket, the n dimensional Poisson system is defined as

x = {x,H},

which can be rewritten in a matrix form

x = J(x)∇H(x). (1)

H is called the Hamiltonian, the first integral I(x) of the Poisson system (1) is a

nonconstant function that satisfies {I,H} = 0. If the first integral has ∇ITJ(x) = 0 for

all x, it is called the Casimir function of the Poisson system (1) .

In a 2d dimensional space M , if choose the Poisson matrix J(x) as the constant matrix

J(x) = J−1 =

[0 Ed

−Ed 0

], Ed is a d dimensional identity matrix,

we get the canonical Poisson bracket , and the corresponding system (1) is actually a

Hamiltonian system.

Definition 2.2. The 2d dimensional Hamiltonian system

x = J−1∇H(x)

is called completely integrable if there exist d smooth independent first integrals I1(x) =

H(x), I2(x), · · · , Id(x) in involution (i.e. all {Ii, Ij} = 0, i 6= j, where {·, ·} is the canonical

Poisson bracket).

The dynamical behavior of completely integrable Hamiltonian systems is well defined

by Arnold–Liouville theorem (see[2]). Assume that the level set

Ma = {x ∈M ; Ii(x) = ai, i = 1, · · · , d}

is compact and connected, the theorem shows that Ma is an invariant d-dimensional

torus; with d involutive independent first integrals {Ii}di=1, a symplectic transformation

of coordinates can be constructed to the action angle variables, in new coordinates the

system is still a Hamiltonian system whose Hamiltonian depends only on the actions.

Define Td := Rd/2πZd, denote (a, θ) ∈ Rd × Td as the action angle variables, the changed

Hamiltonian system is like

a = 0

θ =∂K(a)

∂a= w(a)

(2)

ICMSEC-RR 2012-08 5

with K(a) = H(x). Therefore, the flow (i.e. the exact solution) of a completely integrable

Hamiltonian system goes linearly on a d dimensional torus. The flow is periodic when

the frequencies ω = (ωi) ∈ Rd are rationally dependent, otherwise it is quasi-periodic,

especially when only ki = 0, i = 1, . . . , d could satisfy the equation∑d

i=1 kiωi = 0.

The results on integrable Hamiltonian systems can be extended to general Poisson

systems [12]. Consider the n dimensional Poisson system (1), assume that the rank of the

Poisson matrix J(x) is a constant 2r, then the system is integrable if it possesses n − rindependent involutive first integrals, among which n− 2r are Casimir functions.

Generally speaking, integrable dynamical systems have regular motions, in contrast

with chaotic motions which may exist for non-integrable systems. But it is usually not the

case once the system is solved numerically, even though the numerical method has inherited

all conserved quantities of the integrable system. This problem prompts a concept called

integrable discretization[1]. Integrable discretisation is often considered as a discrete map

related to a discrete integrable system, but it may not lead to a numerical method. In this

paper, for the integrable Poisson system we consider a kind of integrable discretization

which could also be seen as a numerical method. As follows, we introduce a definition of

integrable discretization for the ODE x = f(x), it is presented by Suris.

Definition 2.3. [25] The integrable discretisation is a map Φ : x 7→ x = Φ(x;h) that

satisfies the following conditions:

(i) Φ(x;h) = x+ hf(x) +O(h2);

(ii) Φ is Poisson with respect to the Poisson bracket {·, ·} or its deformation {·, ·}h =

{·, ·}+O(h), i.e., Φ satisfies {F ◦Φ, G◦Φ}(x) = {F,G}(Φ(x)), or {F ◦Φ, G◦Φ}h(x) =

{F,G}h(Φ(x)).

(iii) Φ possesses enough number of first integrals Ii(x;h)1 which is independent and

satisfies Ii(x;h) = Ii(x) +O(h).

3 The Lotka–Volterra system

The Lotka–Volterra system is an ecological predator-prey model designed by Alfred Lotka

(1925) and Vito Volterra (1926). It describes the dynamics of competing species in a

biological system. Generally the Lotka–Volterra (LV) system is expressed as

xi = xi(bi +

n∑j=1

aijxj), i = 1, . . . , n,

1For map Φ : x 7→ x = Φ(x;h), a h-dependent function I(x;h) satisfying I(x;h) = I(x;h) is called the

first integral of Φ(x, h).


where xi represents the density of the i-th biological species and xi represents the growth

of the density against time; bi are parameters depending on environment and ai,j represent

the interaction between species. Recently, the system is widely applied in many different

areas such as laser physics, plasma physics, and neural networks, etc. In this paper, we

consider the following three dimensional model

xi = xi(xi+1 − xi−1), i = 1, 2, 3, (3)

in the above expression, x4 := x1, x0 := x3. The dynamics (3) has the following two first

integrals

H1 = x1 + x2 + x3, H2 = x1x2x3, (4)

and there are two matrices to make the system (3) a Poisson system

x = B1∇H2 = B2∇H1, (5)

where B1 and B2 are Poisson matrices

B1 =

0 −1 1

1 0 −1

−1 1 0

, B2 =

0 x1x2 −x1x3

−x1x2 0 x2x3

x1x3 −x2x3 0

, (6)

they are skew-symmetric and satisfy the Jacobi identity. With B1 and B2 we define two

Poisson brackets {·, ·}1 and {·, ·}2 satisfying

{f, g}1 := ∇fTB1∇g, {f, g}2 := ∇fTB2∇g, (7)

correspondingly the system (3) can be rewritten as

x = {x,H2}1 = {x,H1}2. (8)

Since ∇HTi Bi = 0 for i = 1, 2, Hi is the Casimir function with respect to {·, ·}i.

As is discussed above, the three dimensional LV system (3) has at least one Poisson

structure of rank 2 and two first integrals, it is integrable. Using one first integral, say

H1 = x1 + x2 + x3 = d, we express x3 by x3 = d − x1 − x2, then the system (3) can be

reduced to a two dimensional system

x1 =x1(2x2 + x1 − d)

x2 =− x2(2x1 + x2 − d).(9)

The system (9) is Hamiltonian with Hamiltonian function H = x1x2(d− x2 − x1), it can

be solved exactly with the solution expressed by an elliptic function of the first kind [16].

ICMSEC-RR 2012-08 7

Geometrically, the solution of the system is bounded in the intersection of two surfaces

H1 = const and H2 = const, shown as the red curve in Fig.1. Notice that H1 = const

defines a plane, the orbits can be shown in a plane as in Fig.2. It’s easy to check that the

LV system has four equilibrium points (0, 0, d), (0, d, 0), (d, 0, 0) and (d, d, d). In [19] C.Pop

and A.Aron prove that only the equilibrium states (d, d, d) is elliptic and nonlinearly stable,

near the point there exists periodic solution, while the other three equilibrium states are

hyperbolic and nonstable. Actually only when the start point satisfies xi > 0, i = 1, 2, 3

will the orbit be closed, and we can see from Fig.2 that H2 = 0 defines the separatrix

between bounded and unbounded orbits.

Fig 1: Exact orbit of three dimensional LV system with initial point (0.1, 0.4, 0.4).

Besides the Poisson structures, the LV system (3) has more intriguing properties. It

follows from (3) that

x = ∇H1 ×∇H2, (10)

where × is the cross product. ODEs in form of (10) is in fact a three dimensional kind of

Nambu system. Nambu system was firstly introduced by Nambu in [17] as a generalization

of Hamiltonian systems and has multiple Hamiltonians. In three dimensional case, Nambu

system is in the form

x =∂(F,G)

∂(y, z), y =

∂(F,G)

∂(z, x), z =

∂(F,G)

∂(x, y), (11)

where F and G are two smooth functions of (x, y, z), and ∂(F,G)∂(·,·) is the Jacobian. If denote


−0.14

−0.14

−0.1

4

−0.14

−0.1

4

−0.14

−0.14

−0.03

−0.03

−0.03

−0.0

3

−0.03−0

.03

−0.03

−0.03

0

0

0

0

0 0

0

0

0

0

0.01

0.010.

01

0.01

0.01

0.01

0.01

0.01

0.02

0.02

0.02

0.02

0.02

0.03

0.03

0.03

0.03

0.07

0.07

0.070.14 0.14

Fig 2: Exact orbit projected ont the plane defined by H1 = 1, the legend numbers are

values of H2 = x1x2x3.

X = (x, y, z), (11) in vector formulation is

X = ∇F ×∇G. (12)

It is easy to check that (12) is a Poisson system, because it can be written in the form

X = ∇F∇G = −∇G∇F, (13)

where the skew-symmetric matrix ∇F =

0 ∂F

∂z −∂F∂y

−∂F∂z 0 ∂F

∂x

∂F∂y −∂F

∂x 0

derived from the three

dimensional vector ∇F satisfies Jacobi identity. Therefore, F and G are two first integrals

of the Nambu system (12), they are still called Hamiltonians. Furthermore, taking the

divergence of equations (12) gives

∇ · (∇F ×∇G) ≡ 0,

ICMSEC-RR 2012-08 9

which implies that the Nambu system (12) is source-free and the exact flow of the given

system preserves volume in phase space [2].

Above all, the three dimensional LV system (3) has the following properties:

• It is a Nambu system. So it possesses two poisson structures, two independent first

integrals, and is source free.

• It is ρ-reversible [2] with respect to ρ = diag(−1,−1,−1), i.e. ρf(x) = −f(ρx) for all

x with f := ∇H1×∇H2, so the exact flow ϕt of the system satisfies ρ◦ϕt = ϕ−1t ◦ρ.

• It is integrable.

4 Numerical discretisation and modified equations

In this section, we study Hirota’s integrable discretization for the LV system (3) and the

corresponding modified equations. Based on bilinear approach, Hirota’s discretisation was

presented in [9] following three steps: Firstly, transform the given system into a system in

bilinear form by the transformation of dependent variables; Secondly, discretize the bilinear

equation under the constraint of gauge invariance, and find multi-soliton solutions of the

discrete bilinear system to determine the integrability; Thirdly, transform the discrete

bilinear equation into a discrete nonlinear system by an associated transformation.

Applying Hirota’s bilinear approach to the LV system (3) provides the following

discrete map

Φ : x 7→ x, xi − xi = h(xixi+1 − xixi−1), i = 1, 2, 3, (14)

where h is the difference step size. If set x := x(t), x := x(t+h), it can be proved that the

difference system derived from (14) has multi-soliton solutions [9], and the discretisation

is consequently integrable in soliton theories’ sense.

By some simple calculations the discrete map can be expressed explicitly

xi = Ψh(xi) =xi(h2xi+1xi−1 + hxi+1 + 1

)h2xi−1xi + hxi−1 + 1

or xi = −1

h− xi+1, i = 1, 2, 3. (15)

To be considered as a numerical method we choose the first form x = Ψh(x). We have the

following proposition on the properties of Ψh.

Proposition 4.1. The numerical method Ψh (15) defined by the discrete system (14) has

the following properties:

(i) It is of order 1 and explicit.


(ii) It preserves volume in phase space.

(iii) It preserves the Poisson bracket {·, ·}2, i.e., {f ◦ Ψh, g ◦ Ψh}2(x) = {f, g}2(Ψh(x))

holds for two arbitrary smooth functions f and g.

(iv) It preserves two first integrals[16]

I1(x) := H1(x) + hG(x) = x1 + x2 + x3 + h(x2x3 + x1x2 + x1x3)

and

I2(x) := H2(x) = x1x2x3,

i.e., Ii(x, h) = Ii(x, h), i = 1, 2.

Proof. (i) Write (15) into Taylor series

xi = Ψh(xi) = xi + hxi(xi+1 − xi−1) +O(h2),

which implies that the numerical method Ψh is of order 1.

(ii) By calculation it is easy to know that det(∂x∂x) = det(∂Φh(x)∂x ) ≡ 1, so the numerical

method Ψh can preserve volume in phase space.

(iii) Denote B2 =

0 x1x2 −x1x3

−x1x2 0 x2x3

x1x3 −x2x3 0

, we have

(∂Ψh(x)

∂x

)B2(x)

(∂Ψh(x)

∂x

)T= B2(Ψh(x)),

i.e., {f ◦Ψh, g ◦Ψh}2(x) = {f, g}2(x).

(iv) By calculation it follows from (15) that Ii(x) = Ii(x) for i = 1, 2.

From the above Proposition, we know that the map (15) is integrable by definition

(2.3). It is shown later in section 6 that H1 is bounded and the global error of numerical

solution grows linearly.

To further study behaviors of the numerical solution we employ the tool of modified

equations. Consider a numerical method ϕh applied to an ordinary differential equation

x = f(x), (16)

h is the step size, given an initial value x, the numerical solution ϕh(x) gives an

approximation to the solution of (16). By the idea of backward error analysis, the

numerical solution ϕh(x) is related to the exact solution of the following equation

˙x = f(x) := f(x) + hf2(x) + h2f3(x) + · · · (17)

ICMSEC-RR 2012-08 11

by x(nh) = xn = ϕh(xn−1), the vector field f is a perturbation of original vector field f in

a series of powers h. Equation (17) is called the modified differential equation (MDE) of

ϕh corresponding to ODE (16). To obtain the expression of the modified equation (17),

we assume that the numerical method ϕh can be expanded in a Taylor series

ϕh(x) = x+ hd1(x) + h2d2(x) + h3d3(x) + · · · , (18)

put x := x(0), the solution of MDE (17) satisfies

x(h) = x+ hf(x) +h2

2f ′f(x) +

h3

3!(f ′′f2 + f ′2f)(x) + · · · , (19)

the coefficients of modified equation (17) can be computed iteratively by requiring x(h) =

ϕh(x).

Lemma 4.1. [2] If the numerical method has an expansion of the form (18), the functions

fj of the MDE (17) satisfy

fj = dj −j∑i=2

1

i!

∑k1+k2+···+ki=j

Dk1Dk2 · · ·Dki−1fki , (20)

where Dig := g′fi is the Lie derivative, and km ≥ 1 for all m.

In particular, if equation (16) is a system with polynomial vector field, and each

coefficient dj of the expansion (18) is a polynomial, the corresponding MDE (17) of ϕh is

also a system with polynomial vector field, i.e, fi, i ≥ 2 are polynomials.

Denote

HPn = {p(x) | each component of p(x) is a homogeneous polynomial of degreen},

we have the following Corollary.

Corollary 4.1. Assume that f ∈ HPn, the numerical method ϕh can be expanded in the

form of (18) and dj ∈ HP j(n−1)+1, then each component in the coefficients fj of MDE

(17) is a homogeneous polynomial of degree j(n− 1) + 1, i.e. fj ∈ Hj(n−1)+1.

Proof. Define f1 := f . From the assumption of this Corollary we have f1 ∈ HPn. Suppose

that for all j < r, fj ∈ HP j(n−1)+1, so for any s,m < r,

Dsfm = f ′mfs ∈ HP (m+s)(n−1)+1.

Furthermore, when r ≥ 1, k1 + · · ·+ ki = r we have

Dk1Dk2 · · ·Dki−1fki ∈ HP

(k1+k2+···+ki)(n−1)+1 = HP r(n−1)+1.


As dr ∈ HP r(n−1)+1, it follows from (20) that

fr ∈ HP r(n−1)+1.

Consider the LV system (3) and Hirota’s numerical method (14), we derive that

d1 = f,

d2 =[x1x3(x3 − x1), x1x2(x1 − x2), x2x3(x2 − x3)

]T,

dj = −

x3 0 0

0 x1 0

0 0 x2

dj−1 −

x1x3 0 0

0 x1x2 0

0 0 x2x3

dj−2, j > 2,

so we have

f1 := f ∈ HP 2,

d1 ∈ HP 2,

d2 ∈ HP 3

dj ∈ HP j+1, j > 2.

Thus, by Corollary (4.1) we know

fj ∈ HP j+1, j ≥ 1.

For example, when j = 2 we have

f2 =

−1/2x1

2x3 + 1/2x32x1 − 1/2x2

2x1 + 1/2x12x2

−1/2x22x1 + 1/2x1

2x2 − 1/2x32x2 + 1/2x2

2x3

−1/2x32x2 + 1/2x2

2x3 + 1/2x32x1 − 1/2x1

2x3

,each component of f2 is a homogeneous polynomial of degree 3.

As a consequence, the MDE of Hirota’s method for the LV system (3) is a polynomial

system with homogeneous polynomials {fj}∞j=1. It is possible to simplify the computation

from using formula (20) to applying relations between coefficients of fj ’s. In use of Matlab

we calculate the coefficients of fj ’s for j < 35. Denote

Cj := max | coefficients of fj |,

it is figured out that C2 = 1/2, C3 = 1/3, C4 = 1/4, Fig. 3 shows the numerical result of

Cj for j from 1 to 35. From Fig. 3, for j = 2, . . . , 35 we observe that Cj < Cj−1 < 1 and

|fj | ≤ |x1 + x2 + x3|j+1. We have the following Conjecture.


0 5 10 15 20 25 30 350.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

maximal absolute value of fj’s coefficients

j

C j

Fig 3: Maximal absolute value of coefficients of fj for j ≤ 35.

Conjecture 4.1. Apply Hirota’s method to the LV system (3), we have that Cj < 1 for

all integer j > 2. Thus, |fj | ≤ |x1 + x2 + x3|j+1 and

|f | ≤∞∑j=1

hj−1|x1 + x2 + x3|j+1 =|x1 + x2 + x3|2

(1− h|x1 + x2 + x3|), for |h(x1 + x2 + x3)| < 1.

Therefore the corresponding modified equation is convergent if |h(x1 + x2 + x3)| < 1.

5 Qualitative analysis of modified differential equation

Proposition 4.1 tells us that when is applied to the LV system (3), Hirota’s bilinear method

preserves the Poisson structure {·, ·}2 and two first integrals

I1 = H1 + hG, I2 = H2,

where H1 = x1 + x2 + x3 and H2 = x1x2x3 are first integrals of the LV system (3) and

G = x2x3 + x1x2 + x1x3.

From Corollary 4.1, we already know that coefficients of the MDE corresponding to

Hirota’s method (14) can be computed recursively. In this section, we analyze qualitative

properties of the MDE.

Proposition 5.1. When Hirota’s method (14) is applied to the LV system (3), the

corresponding modified equations satisfies the following properties:


(i) The MDE has two independent first integrals I1(x) and I2(x). Furthermore, there

exist two skew-symmetric matrices

S(x, h) :=∞∑i=1

hi−1Si(x) and R(x, h) :=∞∑i=1

hi−1Ri(x)

such that the MDE can be written as

x = f(x) = S(x, h)∇I1 = R(x, h)∇I2.

(ii) The MDE is a Poisson system corresponding to the Poisson bracket {·, ·}2, i.e., three

exists a function Fj(x) such that

fj = B2∇Fj , j ≥ 1.

(iii) The MDE is a Nambu system, i.e. f = ∇H ×∇H2 with H = H1 + hF2 + · · · .

(iv) Two first integrals I1 and I2 are in involution with respect to Poisson bracket {·, ·}2,

i.e. {I1, I2}2 = 0. I1 and I2 are independent on the manifold exclude the equilibrium

points and points satisfying xi = xi+1 = −1/h .

(v) The MDE is source-free, i.e. ∇ · f ≡ 0.

Proof. (i) It is known that H1 is a first integral of (3), i.e. ∇HT1 f = 0. Therefore, there

exists a skew symmetric matrix S1(x) such that f = S1∇H1 [21]. Let S0(x) := 0,

then f1 can be expressed as f1 := f = S1∇H1 +S0∇G. Assume that for j = 2, . . . , r

there exist skew-symmetric matrices Sj such that

fj = Sj∇H1 + Sj−1∇G. (21)

Consider the following ordinary differential equation

x = f [r](x) :=

r∑j=1

hj−1Sj(x)

∇I1(x) =

r∑j=1

hj−1fj(x) + hrSr(x)∇G(x), (22)

whose flow ϕr,t(x(0)), compared to the flow Ψt(x(0)) of the modified equation (17),

satisfies

Φh(x(0)) = ϕr,h(x(0)) + hr+1(fr+1 − Sr∇G)(x(0)) +O(hr+2).


It is obvious that ∇IT1 f [r] = 0, which implies that I1 is a first integral of (22), i.e.,

I1(ϕr,h(x)) = I1(x). Therefore, we have

I1(x(0)) = I1(Φh(x(0)))

= I1(ϕr,h(x(0))) + hr+1∇IT1 (fr+1 − Sr∇G)(x(0)) + · · ·

= I1(x(0)) + hr+1∇HT1 (fr+1 − Sr∇G)(x(0)) +O(hr+2).

This implies that∇HT1 (x)(fr+1(x)−Sr(x)∇G(x)) = 0, there exists a skew-symmetric

matrix Sr+1(x) such that fr+1(x)− Sr(x)∇G(x) = Sr+1(x)∇H1(x). Then

fr+1(x) = Sr+1(x)∇H1(x) + Sr(x)∇G(x).

By the induction hypothesis, the vector field of MDE has the following form

f(x) =

∞∑i=1

hi−1fi(x) =

∞∑i=1

hi−1Si(x)∇H1(x) +

∞∑i=0

hiSi(x)∇G(x) = S(x, h)∇I1(x).

Similarly, we can prove that there exists a skew-symmetric matrix Rj(x) such that

f(x) =

∞∑i=1

hi−1Ri(x)∇I2(x) = R(x, h)∇I2(x).

So I1 and I2 are two first integrals of the MDE.

(ii) From Proposition 4.1, it is known that Hirota’s method Ψh is a Poisson integrator2.

It is pointed out in [2] that the corresponding modified equation is a Poisson system

with respect to {·, ·}2. More precisely, there exist smooth functions Fj(x) such that

the following equality holds

fj = B2∇Fj .

(iii) Define H = H1 + hF2 + h2F3 + · · · , from (ii) it is known that the modified vector

field is of the form

f = B2∇H = ∇H ×∇H2.

This implies that the MDE is a Nambu system.

(iv) It has been pointed out that H2 is a Casimir function respect to {·, ·}2, so {I1, H2}2 ≡0. I1 and I2 are dependent if and only if ∂I1/∂xi

∂I1/∂xi+1= ∂I2/∂xi

∂I2/∂xi+1, i = 1, 2, 3, solving this

system of algebraic equations we get x1 = x2 = x3 or xi = xi+1 = 0 or xi = xi+1 =

−1/h.

2If a method preserves a Poisson structure {·, ·} and the corresponding Casimir function, it is a Poisson

integrator.


(v) By (iii), it is known that the LV system is a three dimensional Nambu system which

is source-free.

Remark 5.1. Proposition 5.1 tells us that the MDE (17) has two closed first integrals

and is formally a Nambu system. This is not enough to determine the integrability of the

MDE, because the vector field is a series and might be divergent. However, we can still

find the transformation with which the MDE is reduced.

Lemma 5.1. A three dimensional ordinary differential equation x = f(x) possesses two

independent first integrals I1 and I2 iff there exists a scalar function m(x) such that

f(x) = m(x)∇I1(x)×∇I2(x).

Moreover, if f(x) is source free i.e. ∇ · f(x) = 0, then m(x) can be expressed as the

function of I1, I2.

Proof. If I1 and I2 are two independent first integrals, then ∇ITi f = 0, i = 1, 2. In three

dimensional space, this implies that the vector ∇I1×∇I2 is parallel to the vector f , there

exists a scalar function m(x) such that f = m∇I1 ×∇I2.

Denote (∇m,∇I1,∇I2) as a 3×3 matrix whose column vectors are ∇m,∇I1, and ∇I2.

If f is source free, then

0 = ∇ · f = det(∇m,∇I1,∇I2). (23)

According to Theorem 2.16 in [18], we know that m(x), I1(x) and I2(x) are functionally

dependent3. Therefore, there exists a smooth function F which is not identically zero such

that

F(m(x), I1(x), I2(x)) = 0. (24)

It is known from (23) that there exist two functions α(x) and β(x) such that

∇m(x) = α(x)∇I1(x) + β(x)∇I2(x), (25)

given the fact that ∇I1(x) and ∇I2(x) are linearly independent. Differentiating (24) w.r.t

x and substituting (25) read

0 =dFdx

(x) =

(∂F∂m

α+∂F∂I1

)∇I1(x) +

(∂F∂m

β +∂F∂I2

)∇I2(x). (26)

3Let I1(x), . . . , Ik(x) be smooth, real valued functions, they are called functionally dependent if there

exists a smooth function F(z1, · · · , zk) (F is not identically zero) such that F(I1(x), . . . , Ik(x)) = 0 [18].


Notice that ∇I1(x) and ∇I2(x) are linearly independent, this gives(∂F∂m

α+∂F∂I1

)(x) =

(∂F∂m

β +∂F∂I2

)(x) ≡ 0

which reads ∂F∂m 6= 0 otherwise F ≡ 0. By the implicit function theorem, m(x) is

determined as a function of I1(x) and I2(x) from (24).

From Lemma 5.1, it reads that the three dimensional system x = f(x) which is source

free and has two independent first integrals Ii(x), i = 1, 2 can be written in the form

x = m(I1(x), I2(x))∇I1(x)×∇I2(x), (27)

where m(I1, I2) is a scalar function of I1 and I2. The formulation is crucial to derive the

reduced expression for a three dimensional source free system with two independent first

integrals. Based on (27) the reducing process can be accomplished by a volume preserving

coordinate transformation. Theoretically, any volume preserving transformation can be

related to a generating function [20, 22, 24]. In particular, the relation is shown in the

following Lemma presented by Shang.

Lemma 5.2. [22] Consider x ∈ Rn, let α =

(Aα Bα

Cα Dα

)be a 2n dimensional invertible

matrix and S(w) = (S1(w), · · · , Sn(w)) be a differentiable mapping satisfying the relation

det

(∂S

∂w(w)Cα −Aα

)= det

(Bα −

∂S

∂w(w)Dα

)6= 0,

then the relation

Aαx+Bαx = S(Cαx+Dαx)

gives a volume-preserving mapping g : Rn → Rn, x = g(x). S is called the generating

function of g.

When n = 3, if choose Aα = Dα =

0 0 0

0 0 0

0 0 1

, Bα = Cα =

1 0 0

0 1 0

0 0 0

, the given

volume preserving transformation, say x 7→ x, can be generated implicitly by

x1 = S1(x1, x2, x3)

x2 = S2(x1, x2, x3)

x3 = S3(x1, x2, x3)

(28)

with S satisfying

det

(∂S3

∂x3

)= det

(∂(S1, S2)

(x1, x2)

).


Theorem 5.1. Consider the three dimensional system x = f(x) which is source free and

possesses two independent first integrals I1(x), I2(x). There exists a local

volume-preserving transformation g : (x1, x2, x3) 7→ (x1, x2, x3) such that in new variables

(x1, x2, x3) this system becomes

˙x1 = 0,

˙x2 = 0,

˙x3 = m(x1, x2).

(29)

Proof. Define

x1 = I1(x1, x2, x3), x2 = I2(x1, x2, x3), (30)

then˙x1 = I1 = 0, ˙x2 = I2 = 0. (31)

Since I1 and I2 are independent, the rank of Jacobi matrix ∂(I1,I2)∂(x1,x2,x3) is 2, without loss of

generality, we suppose that ∂(I1,I2)∂(x1,x2) is invertible. By implicit function theorem, x1 and x2

can be locally determined from (30) by

xi = Si(x1, x2, x3), i = 1, 2.

Solving the following differential equation

det

(∂S3

∂x3

)= det

(∂(S1, S2)

∂(x1, x2)

)= det

(∂(I1, I2)

∂(x1, x2)

)−1

,

we can get function S3(x1, x2, x3). So S3 together with S1 and S2 generates a volume-

preserving mapping x 7→ x by (28). By Lemma 5.2, we know x = f(x) can be written in

the form (27). Therefore, in new coordinate variables we have

˙x3 =

(∂x3

∂x

)Tx

=

(∂x3

∂x

)Tm(I1, I2)∇I1 ×∇I2

= m(I1, I2) det

(∂(x1, x2, x3)

∂(x1, x2, x3)

).

Notice that g : x 7→ x is volume-preserving, it follows from the above equality that

˙x3 = m(x1, x2). (32)

Equation (32) together with (31) gives (29).


As to the MDE of Hirota’s method being applied to the LV system (3), we know from

Proposition 5.1 that the MDE is source-free and has two independent first integrals I1 =

H1 +hG and I2 = H2. According to Theorem 5.1, there exists a coordinate transformation

to formally reduce the form of the MDE .

Theorem 5.2. When Hirota’s method is applied to the LV system (3), the corresponding

modified equation x = f(x) has the following properties:

(i) There exists a scalar function A(x, h) = A(I1(x), I2(x), h) which is the function of

I1, I2 and h, such that

x = f(x) = A(I1(x), I2(x), h)∇I1(x)×∇I2(x). (33)

(ii) There exists a volume-preserving transformation

g : (x1, x2, x3) 7→ (x1, x2, x3)

such that in new variables the MDE becomes˙x1 =0,

˙x2 =0,

˙x3 =A(x1, x2, h).

(34)

Expanding A(x, h) as A(x, h) =∑∞

i=1 hi−1Ai(x) and substituting it into (33) gives

x = f(x) =∞∑i=1

hi−1Ai(x)∇I1(x)×∇I2(x) (35)

Notice that I1 = H1 + hG, I2 = H2 and f =∞∑i=1

hj−1fj , it follows from (35) that

fj = Aj∇H1 ×∇H2 +Aj−1∇G×∇H2, (36)

where f1 := f = ∇H1 ×∇H2, A0 := 0. Left-multiplying (∇G)T on both sides of (36), we

obtain

(∇G)T fj = Aj(∇G)T f. (37)

Consequently, when (∇G)T f 6= 0 (that means that G is not a Casimir function of the LV

system)

Aj(x) =(∇G(x))T fj(x)

(∇G(x))T f(x), j = 1, 2, . . . (38)


Corollary 5.1. Under the assumption of Theorem 5.1, the MDE can be written in form

of (33) with A(x, h) =∑∞

i=1 hi−1Ai(x), where Ai(x) can be calculated by (38).

Specially,

A1(x) = 1,

A2(x) = −1

2x1 −

1

2x2 −

1

2x3,

A3(x) =1

3x1

2 +1

3x2

2 +1

3x3

2 +1

6x1x2 +

1

6x1x3 +

1

6x2x3.

Remark 5.2. The MDE (17) can be transformed to a linear equation

˙x3 = A(I1(x(0)), I2(x(0)), h).

Formally, we can say that this system is integrable. As a consequence, the numerical flow

is on the one dimensional manifold defined by I1(x) = I1(x(0)) and I2(x) = I2(x(0)).

There is another way to observe the dynamics of the MDE, that is reducing the order

with help of its first integral H2. It is known that the MDE has the form of x = B2∇H.

Put x1x2 6= 0, x3 = H2/x1x2, then the MDE becomes a two dimensional Poisson system

x = x1x2J−1∇K(x, h) (39)

with x = (x1, x2) and K(x1, x2, h) = H(x1, x2, x3 = H2/x2x1, h). It is easy to know that

(39) can be transformed into

y = J−1∇L(y, h)

by φ2 : (y1, y2) 7→ (x1 = ln y1, x2 = ln y2) and L(y, h) = K(x, h). This is a one degree

Hamiltonian system, so there exists a symplectic transformation

φ : (a, θ) 7→ (y1, y2),

under which this system has the form of

a = 0, θ = w(a, h).

However, the explicit expressions for a, θ is not easy to get.


6 Numerical experiments

We have analyzed Hirota’s discretisation (15) for the LV system, it is integrable and

preserves many geometric structures, because of which it belongs to the family of

geometric integrators. In this section we do some numerical experiments to verify our

results. Because the LV system (3) has multiple geometric structures, we examine six

other geometric integrators: the Average Vector Field (AVF) method, Kahan’s method,

the midpoint rule, 1st and 2nd order splitting method and the symplectic Euler method.

As a comparison the explicit Euler method is tested.

Recall that the LV system (3) is in the form

xi = f(x)i = xi(xi+1 − xi−1), i = 1, 2, 3,

it has two first integrals H1 = x1 + x2 + x3 and H2 = x1x2x3. To preserve them a kind of

discrete gradient method, the AVF method [15], is used. The general form for the AVF

method applied to x = f(x) is

x− xh

=

∫ 1

0f((1− c)x+ cx)dc.

As the vector field of the LV system is polynomial, calculating the integral exactly by

quadrature gives

xi = xi +h

6((2xi + xi)(xi+1 − xi−1) + (xi + 2xi)(xi+1 − xi−1)), (40)

it preserves H1 and H2 exactly.

The LV system is ρ-reversible with ρ = diag(−1,−1,−1), i.e. the exact flow ϕt of

the system satisfies ρ ◦ ϕt = ϕ−1t ◦ ρ. For quadratic systems, Kahan derived a kind of

reflexive nonstandard method [10, 11]. When it is applied to the LV system, the derived

discretization

xi = xi +h

2((xixi+1 + xixi+1)− (xixi−1 + xixi−1)) (41)

is ρ-reversible, i.e., ρ ◦ Φ = Φ−1 ◦ ρ with Φ as the discrete map.

Besides, as the LV system has two Poisson structures (7), three Poisson integrators are

used. The first one is the midpoint rule, the derived discretisation

xi = xi + hxi + xi

2(xi+1 + xi+1

2− xi−1 + xi−1

2) (42)

preserves the constant Poisson structure with B1 in (6) and the linear first integral H1.

The others are constructed by splitting [13]. Based on the from (3), split the LV system


as

x = B2∇H1 = B2(x)∇

(3∑i=1

xi

)=

3∑i=1

B2(x)∇xi

:= X(x) = X1(x) +X2(x) +X3(x),

where X and Xi = (B2(x)∇xi)j ∂∂xj

, i = 1, 2, 3 are vector fields. The exact solution to the

system is x(t) = exp(tX)(x(0)), so a first order approximation is chosen as

Q1(h, x) = exp(hX1) exp(hX2) exp(hX3)(x), (43)

and a second order approximation is

S2(h, x) = exp(hX1/2) exp(hX2/2) exp(hX3) exp(hX2/2) exp(hX1/2)(x). (44)

The two splitting methods preserve the Poisson structure with B2 in (6) and the first

integral H2.

Furthermore, it has been shown that the LV system can be reduced to the two

dimensional Hamiltonian system (9). Therefore the symplectic Euler method is applied

to preserve the symplectic structure. With d = x(0)1 + x(0)2 + x(0)3 the discrete form is{x1 = x1 + hx1(2x2 + x1 − d),

x2 = x2 − hx2(2x1 + x2 − d).(45)

As a comparison we use the explicit Euler method which reads

xi = xi + hxi(xi+1 − xi−1). (46)

It is known that the midpoint rule (42), S2(h, x) (44) and the AVF method (40) are

2nd order symmetric methods; Kahan’s method (41) is a 2nd order reversible symmetric

method; the others are 1st order methods. Besides, the midpoint rule (42) and the AVF

method (40) are implicit; the symplectic method is semi-implicit and the the others are

explicit.

With initial values x(0) = (0.3, 0.3, 0.4) and step size h = 0.1, we compute by these

eight methods over 3000 steps, and compare the errors of numerical solutions and first

integrals. In Figs. 4 we present the numerical solutions in three dimensional space.

This shows that except the explicit Euler method (46) which gives a numerical solution

that spirals outwards, all the methods guarantee closed orbits. In addition, Figs. 5

compares numerical solutions and the exact solution projected on the (x1, x2) plane. For

all the 2nd order methods (40,41,42,44) together with Hirota’s method numerical solutions


approximate the exact orbit well, while numerical results gained by Q1 (43) and the

symplectic Euler method (45) deflect a little. This verifies that all the geometric integrators

are able to guarantee closed orbits, Hirota’s method in particular approximates the orbit

quite well as an explicit 1st order method.

Our next experiment studies the conservation of invariants H1 = x1 + x2 + x3 and

H2 = x1x2x3 . The relative errors calculated by errH(t) = H(t)−H(0)H(0) are shown in Figs.

6. It can be found that the explicit Euler method can only preserve the linear first integral

H1, while the others can preserve at least one of the first integrals to round error and all

the relative numerical errors are bounded over long time.

It seems that the rate of a relative error is related to the order of the method, so in

Fig. 7 we plot numerical errors of the two first integrals with time step h ranging from

10−5 to 103. In I and II we find that every relative error grows with increasing h by a rate

approximately equal to the order. When h becomes larger, explicit methods including

splitting methods Q1, S2 and the symplectic Euler method can not work because of the

restriction of stability. However, symmetric methods including Kahan’s method and the

midpoint rule could still bound the errors, Hirota’s method as an explicit method could

not only be stable but also lower the error bound for large h. It implies that it might

preserve modified first integrals in closed form. This is consistent with the theoretical

result. From III we can see that the exactly preserved first integrals could be preserved

up to round error for large h, but when it turns to the AVF method we can see from IV

that it is no longer stable for h > 10.

In Fig.8, we plot global errors of numerical solutions for the eight methods. The main

observation is that the global errors shows linear growth for those geometric integrators

compared to a quadratic growth for the explicit Euler method. Compare the upper two

figures with the lower two figures we find that global errors depend on the order of the

method, among methods with the same order, splitting methods show half times smaller

error growth than the others.

From the numerical results, we notice that when time step is small, geometric

integrators can simulate the closed solution orbit, and provide numerical solutions with

linear error growth. However, when h becomes larger, only the numerical solutions

computed by integrable discretisations: Hirota’s method and Kahan’s method can still

be bounded and exhibit good approximation to the original solution orbit.


7 Conclusion

In this paper, we studied Hirota’s discretisation constructed by the bilinear approach. As

a discrete map, Hirota’ discretisation is applied to the three dimensional Lotka Volterra

system and gives a numerical method which is integrable in the sense of preserving two first

integrals and a Poisson structure. With the help of backward error analysis, we studied

dynamics of the discrete system. It is proved that the modified differential equation

determined by the discretisation is a polynomial, source free Nambu system. By the

generating function theory, we construct a volume-preserving transformation under which

the modified system was reduced. We also gave the numerical experiments with the explicit

Euler method and seven geometric integrators including the integrable discretizations:

Hirota’s method and Kahan method.

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0.20.3

0.4

0.20.3

0.4

0.3

0.4

x1

Hirota

x2

x 3

0.20.3

0.4

0.20.3

0.4

0.3

0.4

x1

AVF

x2

x 3

0.20.3

0.4

0.20.3

0.4

0.3

0.4

x1

Kahan

x2

x 3

0.20.3

0.4

0.20.3

0.4

0.3

0.4

x1

midpoint

x2

x 3

0.20.3

0.4

0.20.3

0.4

0.3

0.4

x1

splitting Q1

x2

x 3

0.20.3

0.4

0.20.3

0.4

0.3

0.4

x1

splitting S2

x2

x 3

0.20.3

0.4

0.20.3

0.4

0.3

0.4

x1

symplectic Euler

x2

x 3

00.5

1

00.5

10

0.5

1

x1

explicit Euler

x2

x 3

Fig 4: The numerical solution. The initial point is (0.3, 0.3, 0.4) and the step size is

h = 0.1.


x1

x 2

splitting Q1

0.25 0.3 0.35 0.4 0.450.25

0.3

0.35

0.4

0.45

originalnumerical

x1

x 2

splitting S2

0.25 0.3 0.35 0.4 0.450.25

0.3

0.35

0.4

0.45

x1

x 2

Kahan

0.25 0.3 0.35 0.4 0.450.25

0.3

0.35

0.4

0.45

x1

x 2

Hirota

0.25 0.3 0.35 0.4 0.450.25

0.3

0.35

0.4

0.45

x1

x 2

AVF

0.25 0.3 0.35 0.4 0.450.25

0.3

0.35

0.4

0.45

x1

x 2

midpoint

0.25 0.3 0.35 0.4 0.450.25

0.3

0.35

0.4

0.45

x1

x 2

symplectic Euler

0.25 0.3 0.35 0.4 0.450.25

0.3

0.35

0.4

0.45

x1

x 2

explicit Euler

0 0.5 10

0.2

0.4

0.6

0.8

1

Fig 5: The numerical solution projected on (x1, x2) plane. The initial point is (0.3, 0.3, 0.4)

and the step size is h = 0.1. The dashed line expresses the numerical results and the solid

line expresses the exact solution which is the circle x1x2(x(0)1 +x(0)2 +x(0)3−x1−x2) =

x(0)1x(0)2x(0)3.


0 100 200 300−20

−15

−10

−5

0

t/10

erro

r of

firs

t int

egra

ls

splitting Q1

error of H1=x

1+x

2+x

3

error of H2=x

1x

2x

3

0 100 200 300−20

−15

−10

−5

0splitting S2

t/10

erro

r of

firs

t int

egra

ls

0 100 200 300−20

−15

−10

−5Kahan

t/10

erro

r of

firs

t int

egra

ls

0 100 200 300−20

−15

−10

−5

0Hirota

t/10

erro

r of

firs

t int

egra

ls

0 100 200 300−16

−15.5

−15

−14.5

−14AVF

t/10

erro

r of

firs

t int

egra

ls

0 100 200 300−20

−15

−10

−5midpoint

t/10

erro

r of

firs

t int

egra

ls

0 100 200 300−20

−15

−10

−5

0symplectic Euler

t/10

erro

r of

firs

t int

egra

ls

0 100 200 300−20

−15

−10

−5

0explicit Euler

t/10

erro

r of

firs

t int

egra

ls

Fig 6: The relative errors of the two first integrals H1 = x1 + x2 + x3 and H2 = x1x2x3

with initial value (0.3, 0.3, 0.4) and step size h = 0.1. The relative error is calculate

by errH(t) = H(t)−H(0)H(0) . The dashed line represents the error of H1 and the solid line

represents the error of H2


10−4

10−2

100

102

104

10−20

10−15

10−10

10−5

100

h

err

or

of firs

t in

teg

rals

I

Hirota to H1

Kahan to H2

midpoint to H2

explicit Euler to H2

10−4

10−2

100

102

104

10−20

10−15

10−10

10−5

100

h

err

or

of firs

t in

teg

rals

II

splitting Q1 to H

1

splitting S2 to H

1

symplectic Euler to H2

−5 −4 −3 −2 −1 0 1 2 30

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

−15

log10(h)

err

or

of firs

t in

teg

rals

III

−5 −4 −3 −2 −1 0 1 2 3

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

log10(h)

err

or

of firs

t in

teg

rals

IV

AVF to H1

AVF to H2

Hirota to H2

Kahan to H1

midpoint to H1

splitting Q1to H

2

splitting S2 to H

2

explicit Euler to H1

Fig 7: The numerical errors of two first integrals w.r.t h.


0 500 1000 1500 2000 2500 3000 35000

1

2

3x 10

−3

0 500 1000 1500 2000 2500 3000 35000

1

2

3x 10

−3

AVFKahan

midpointsplitting S2

0 500 1000 1500 2000 2500 3000 35000

0.020.040.060.08

0.10.120.14

Hirotasplitting Q

1

0 500 1000 1500 2000 2500 3000 35000

0.10.20.30.40.50.60.70.8

symplectic Eulerexplicit Euler

Fig 8: The numerical error growth of x1 along with t.

Documents

On an integrable discretisation of the Lotka-Volterra system · all conserved quantities of the integrable system. This problem prompts a concept called integrable discretization[1]