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,
4 Yang He and Yajuan Sun ICMSEC-RR 2012-08
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).
6 Yang He and Yajuan Sun ICMSEC-RR 2012-08
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
8 Yang He and Yajuan Sun ICMSEC-RR 2012-08
−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.
10 Yang He and Yajuan Sun ICMSEC-RR 2012-08
(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.
12 Yang He and Yajuan Sun ICMSEC-RR 2012-08
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.
ICMSEC-RR 2012-08 13
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:
14 Yang He and Yajuan Sun ICMSEC-RR 2012-08
(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).
ICMSEC-RR 2012-08 15
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.
16 Yang He and Yajuan Sun ICMSEC-RR 2012-08
(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].
ICMSEC-RR 2012-08 17
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)
).
18 Yang He and Yajuan Sun ICMSEC-RR 2012-08
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).
ICMSEC-RR 2012-08 19
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)
20 Yang He and Yajuan Sun ICMSEC-RR 2012-08
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.
ICMSEC-RR 2012-08 21
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
22 Yang He and Yajuan Sun ICMSEC-RR 2012-08
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
ICMSEC-RR 2012-08 23
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.
24 Yang He and Yajuan Sun ICMSEC-RR 2012-08
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.
28 Yang He and Yajuan Sun ICMSEC-RR 2012-08
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.
ICMSEC-RR 2012-08 29
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
30 Yang He and Yajuan Sun ICMSEC-RR 2012-08
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.
ICMSEC-RR 2012-08 31
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.