# Hamilton-Poisson Realizations for the Lü System

Authors: Camelia Pop; Camelia Petrişor; Dumitru Bălă
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

The Hamilton-Poisson geometry has proved to be an interesting approach for a lot of dynamics arising from different areas like biology (Gümral and Nutku, 1993), economics (Dănăiasă et al., 2008), or engineering (Ginoux and Rossetto, 2006). The Lü system was first proposed by  Lü and  Chen (2002) as a model of a nonlinear electrical circuit, and it was studied from various points of view. We intend to study it
from mechanical geometry point of view and to point out some of its geometrical and dynamical properties.


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## Body

## 1. Introduction

The original Lü system of differential equations onℝ3 has the following form (1.1)ẋ=a(y-x),ẏ=-xz+by,ż=-cz+xy,
where a,b,c∈ℝ.The goal of our paper is to find the relations betweena, b, and c parameters, for which the system (1.1) admits a Hamilton-Poisson realization. The Hamilton-Poisson realization offers us the tools to study the Lü system from mechanical geometry point of view.To do this, one needs first to find the constants of the motion of our system. Due to the numerous parameters of the system and trying to simplify the computation, we will focus on finding only constants of motion being polynomials of degree at most three of the system (1.1).Proposition 1.1.
The following smooth real functionsH are three degree polynomial constants of the motion defined by the system (1.1).  (i)
Ifa∈ℝ*, b=c=0 the system becomes:(1.2)ẋ=a(y-x),ẏ=-xz,ż=xy,(1.3)H(x,y,z)=α(y2+z2)+β,α,β∈R.(ii)
Ifa=0, b,c∈ℝ* the system becomes:(1.4)ẋ=0,ẏ=-xz+by,ż=xy-cz,(1.5)H(x,y,z)=f(x),f∈C1(R).(iii)
Ifa∈ℝ*, b=c∈ℝ the system becomes:(1.6)ẋ=0,ẏ=-xz+by,ż=xy-bz,(1.7)H(x,y,z)=α(xy2-2byz+xz2)+f(x),α∈R,f∈C1(R).(iv)
Ifa=b=c=0 the system becomes:(1.8)ẋ=0,ẏ=-xz,ż=xy,(1.9)H(x,y,z)=α(y2+z2)+β(xy2+xz2)+f(x),α,β∈R,f∈C1(R).Proof.
It is easy to see thatdH=0 for each case mentioned above.


## 2. Hamilton-Poisson Realizations for the System (1.2)

Let us take for the system (1.2) the Hamiltonian function given by: (2.1)H(x,y,z)=12(y2+z2).
To find the Poisson structure in this case, we will use a method described by Haas and Goedert (see [5] for details). Let us consider the skew-symmetric matrix given by: (2.2)Π:=[0p1(x,y,z)p2(x,y,z)-p1(x,y,z)0p3(x,y,z)-p2(x,y,z)-p3(x,y,z)0].
We have to find the real smooth functions p1,p2,p3:ℝ3→ℝ such that: (2.3)[ẋẏż]=Π⋅∇H,
that is, the following relations hold: (2.4)yp1(x,y,z)+zp2(x,y,z)=a(y-x),zp3(x,y,z)=-xz,-yp3(x,y,z)=xy.
It is easy to see that p3(x,y,z)=-x. Let us denote now p1(x,y,z)=p; from the second equation we obtain (2.5)p2(x,y,z)=ay-xz-yzp.Our goal now is to insertp1,p2,p3 into Jacobi identity and to find the function p(x,y,z). In the beginning, let us denote: (2.6)v1:=a(y-z),v2:=-xz,v3:=xy.
The function p is the solution of the following first order ODE (see [5] for details):(2.7)v1∂p∂x+v2∂p∂y+v3∂p∂z=A⋅p+B,
where (2.8)A=∂v1∂x+∂v2∂y+∂v3∂z-(∂v1/∂z)(∂H/∂x)+(∂v2/∂z)(∂H/∂y)+(∂v3/∂z)(∂H/∂z)∂H/∂z,B=v1(∂v2/∂z)-v2(∂v1/∂z)(∂H/∂z).
Equation (2.7) becomes:(2.9)a(y-x)∂p∂x-xz∂p∂y+xy∂p∂z=(-a+xyz)p-a(y-x)xz.Ifa=0, then (2.9) has the solution p(x,y,z)=xz.Ifa≠0, then finding the solution of (2.9) remains an open problem.Now, one can reach the following result.Proposition 2.1.
Ifa=0, the system (1.2) has the Hamilton-Poisson realization:
(2.10)(R3,Π:=[Πij],H),
where
(2.11)Π=[0xz-xy-xz0-xxyx0],H(x,y,z)=12(y2+z2).Remark 2.2.
There exists only one functionally independent Casimir of our Poisson configuration, given byC:ℝ3→ℝ,
(2.12)C(x,y,z)=2x-y2-z2.Proof.
Indeed, one can easily check that:(2.13)Π⋅∇C=0.
As the rank of Π equals 2, it follows from the general theory of PDEs that C is the only functionally independent Casimir function of the configuration (see, e.g., [6] for details).The phase curves of the dynamics (1.2) are the intersections of the surfaces:(2.14)H=const.C=const,see Figure1.Figure 1
The phase curves of the dynamics (1.2).Remark 2.3.
Ifa=b=c=0, then the system (1.2) becomes:
(2.15)ẋ=0,ẏ=-xz,ż=xy.For the specific casea=b=c=0, we extended the results presented in Proposition 2.1 to the following one.Proposition 2.4 (Alternative Hamilton-Poisson structures).
The system (2.15) may be modeled as an Hamilton-Poisson system in an infinite number of different ways, that is, there exists infinitely many different (in general nonisomorphic) Poisson structures on ℝ3 such that the system (2.15) is induced by an appropriate Hamiltonian.Proof.
The triplets:(2.16)(R3{.,.}αβ,Hγδ),
where
(2.17){f,g}αβ=∇Cαβ⋅(∇f×∇g),∀f,g∈C∞(R3,R),Cαβ=αC+βH,Hγδ=γC+δH,α,β,γ,δ∈R,αδ-βγ=1,H=12(2x-y2-z2),C=12x2,
define Hamilton-Poisson realizations of the dynamics (2.15).
Indeed, we have:(2.18){x,Hγδ}αβ=|αx+β-βy-βz100γx+δ-δy-δz|=0=x.;{y,Hγδ}αβ=|αx+β-βy-βz010γx+δ-δy-δz|=-xz=y.;{z,Hγδ}αβ=|αx+β-βy-βz001γx+δ-δy-δz|=xy=z..Let us pass now to study some geometrical and dynamical aspects of the system (2.15).Proposition 2.5 (Lax formulation).
The dynamics (2.15) allows a formulation in terms of Lax pairs.Proof.
Let us take:(2.19)L=[0αx-αβγβ2-γ2z+δαγβ2-γ2x+αβz+γδβ2-γ2-αx+αβγβ2-γ2z-δ0-αβ2β2-γ2y-αγβ2-γ2x-αβz-γδβ2-γ2αβ2β2-γ2y0],B=[0γz-β2-γ2δαβ-β2-γ2z-γδαβ-γz+β2-γ2δαβ0βyβ2-γ2z+γδαβ-βy0],
where α,β,γ,δ∈ℝ, i=-1.
Then, using MATHEMATICA 7.0, we can put the system (2.15) in the equivalent form
(2.20)L.=[L,B]
as desired.Let us continue now with a discussion concerning the nonlinear stability of equilibrium states of our system (2.15) (see [7] for details).It is obvious to see that the equilibrium points of our dynamics are given by:(2.21)e1M=(M,0,0),M∈R,e2M=(0,M,N),M,N∈R.
About their stability,we reached the following result.Proposition 2.6 (A stability result).
The equilibrium statese1M are nonlinearly stable for any  M∈ℝ.Proof.
We shall use energy-Casimir method, see [8] for details. Let
(2.22)Hφ=H+φ(C)=12(y2+z2)+φ(2x-y2-z2)
be the energy-Casimir function, where φ:R→R is a smooth real valued function defined on R.
Now, the first variation ofHφ is given by:(2.23)δHφ=yδy+zδz+φ̇(2x-y2-z2)(2δx-2yδy-2zδz).
This equals zero at the equilibrium of interest if and only if
(2.24)φ̇(2M)=0.
The second variation of Hφ is given by:
(2.25)δ2Hφ=(δy)2+(δz)2+φ̈⋅(2δx-2yδy-2zδz)2-2φ̇((δy)2+(δz)2).
At the equilibrium of interest, the second variation becomes:(2.26)δ2Hφ(M,0,0)=(δy)2+(δz)2+4φ̈⋅(δx)2.
Having chosenφ such that:(2.27)φ̇(2M)=0,φ̈(2M)>0,
we can conclude that the second variation of Hφ at the equilibrium of interest is positive defined and thus eM is nonlinearly stable.As a consequence, we can reach the periodical orbits of the equilibrium pointse1M.Proposition 2.7 (Periodical orbits).
The reduced dynamics to the coadjoint orbit(2.28)2x-y2-z2=2M
has near the equilibrium point e1M at least one periodic solution whose period is close to
(2.29)2π|M|.Proof.
Indeed, we have successively
(i)
the restriction of our dynamics (1.2) to the coadjoint orbit
(2.30)2x-y2-z2=2M
gives rise to a classical Hamiltonian system,(ii)
the matrix of the linear part of the reduced dynamics has purely imaginary roots, more exactly(2.31)λ1=0,λ2,3=±Mi.(iii)
span(∇C(e1M))=V0, where
(2.32)V0=ker⁡(A(e1M)),(iv)
the reduced Hamiltonian has a local minimum at the equilibrium statee1M (see the proof of Proposition 2.4).
Then our assertion follows via the Moser-Weinstein theorem with zero eigenvalue, see [9] for details.Remark 2.8.
The nonlinear stability of the equilibrium statese2M,N remains an open problem, both energy methods (energy-Casimir method and Arnold method) being inconclusive.


## 3. Hamilton-Poisson Realizations of the System (1.4)

As we have proved in [10], the system (1.4) admits a Hamilton-Poisson realization only in the special case b=c; more exactly, we have reached the following result.Proposition 3.1.
Ifa=0 and b=c, the system (1.4) has the Hamilton-Poisson realization
(3.1)(R3,Π:=[Πij],H),
where
(3.2)Π:=[0xz-bybz-xy-xz+by012(y2+z2)-bz+xy-12(y2+z2)0],H(x,y,z)=x.Using a method described in [6], we have found the Casimir of the configuration given by.
(3.3)C(x,y,z)=12(y2+z2)x-byz,b∈R*,
(see [10]).Now we can broaden this result to the following one.Proposition 3.2 (Alternative Hamilton-Poisson structures).
The system (1.4) may be realized as a Hamilton-Poisson system in an infinite number of different ways, that is, there exists infinitely many different (in general nonisomorphic) Poisson structures on ℝ3 such that the system (1.4) is induced by an appropriate Hamiltonian.Proof.
The triples:(3.4)(R3{.,.}αβ,Hγδ),
where
(3.5){f,g}αβ=∇Cαβ⋅(∇f×∇g),∀f,g∈C∞(R3,R),Cαβ=αC+βH,Hγδ=γC+δH,α,β,γ,δ∈R,αδ-βγ=1,H=x,C=12x(y2+z2)-byz,b∈R,
define Hamilton-Poisson realizations of the dynamics (1.4).
Indeed, we have:(3.6){x,Hγδ}αβ=|α+β2(y2+z2)β(xy-bz)β(xz-by)100γ+δ2(y2+z2)δ(xy-bz)δ(xz-by)|=0=x.;{y,Hγδ}αβ=|α+β2(y2+z2)β(xy-bz)β(xz-by)010γ+δ2(y2+z2)δ(xy-bz)δ(xz-by)|=xz-by=y.;{z,Hγδ}αβ=|α+β2(y2+z2)β(xy-bz)β(xz-by)001γ+δ2(y2+z2)δ(xy-bz)δ(xz-by)|=bz-xy=z..Let us pass to discuss some dynamical and geometrical properties of the system (1.4).Proposition 3.3 (Lax formulation).
The dynamics (1.4) allows a formulation in terms of Lax pairs.Proof.
Let us take(3.7)L=[0uv-u0w-v-w0],
where
(3.8)u=α+α(β2+γ2-δ2)iβ2+γ22bδ(β2+γ2)x+αβγδ(β2+γ2-δ2)iβ2+γ22bδ(β2+γ2)2y-αβγ(β2+γ2-δ2)2bδ(β2+γ2)z,v=-αiβ2+γ2β+αγ2iβ2+γ2β(β2+γ2)+αβ(β2+γ2-δ2)2bδ(β2+γ2)x+αγ(β2+γ2-δ2)2b(β2+γ2)y+αγ(β2+γ2-δ2)iβ2+γ22bδ(β2+γ2)z,w=αγ2(β2+γ2-δ2)iβ2+γ22b(β2+γ2)2y-αγ2(β2+γ2-δ2)2bδ(β2+γ2)z,B=[0ωφ-ω0γ(y-δiβ2+γ2z)-φ-γ(y-δiβ2+γ2z)0],
where
(3.9)ω=-biβ2+γ2(β2+γ2)(β2+γ2+δ2)+βγ(β2+γ2-δ2)((β2+γ2)y+iδβ2+γ2z)γ(β2+γ2)(β2+γ2-δ2),φ=-bβ(β2+γ2+δ2)γ(β2+γ2-δ2)-iβ2+γ2y+δz,i=-1
and α,β,γ,δ∈ℝ.
Then, using MATHEMATICA 7.0, we can put the system (1.4) in the equivalent form(3.10)L.=[L,B]
as desired.The equilibrium points of the dynamics (1.4) are given by(3.11)e1M=(M,0,0),M∈R,e2M=(-b,M,-M),M,b∈R,e3M=(b,M,M),M,b∈R.About their stability, we have proven in [10] the following result,Proposition 3.4 (Stability problem).
IfM>b or M<-b,b>0, then the equilibrium states e1M are nonlinearly stable.As a consequence, we can find the periodical orbits of the equilibrium pointse1M.Proposition 3.5 (Periodical orbits).
IfM>b, b>0, the reduced dynamics to the coadjoint orbit x=M has near the equilibrium point at least one periodic solution whose period is close to
(3.12)2πM2-b2.Proof.
Indeed, we have successively(i)
the restriction of our dynamics (1.4) to the coadjoint orbit(3.13)x=M
gives rise to a classical Hamiltonian system,(ii)
the matrix of the linear part of the reduced dynamics has purely imaginary roots, more exactly(3.14)λ1=0,λ2,3=±iM2-b2.(iii)
span(∇C(e1M))=V0, where(3.15)V0=ker⁡(A(e1M)),(iv)
ifM>b,b>0, then the reduced Hamiltonian has a local minimum at the equilibrium state e1M (see the proof of Proposition 3.4 [10]).
Then our assertion follows via the Moser-Weinstein theorem with zero eigenvalue, see [9] for details.


## 4. Conclusion

The paper presents Hamilton-Poisson realizations of a dynamical system arising from electrical engineering; due to its chaotic behavior, finding the solution of the system could be very difficult. A Hamilton-Poisson realization offers us the possibility to find this solution as the intersection of two surfaces, the surfaces equation being given by the Hamiltonian and the Casimir of our configuration. The first paragraph of the paper presents the only four cases for which the Lü system admits as Hamiltonian a three degree polynomial function. Finding another kind of function as a Hamiltonian of the Lü system remains an open problem. The first case,a∈ℝ, b,c=0 is the subject of the second paragraph. For this specific case, we have proved that a Hamilton-Poisson realization exists if and only if a=0. Lax formulation, stability problems, and the existence of the periodical orbits are discussed, too. The third part of the paper analyses the case a=0, b,c∈ℝ. We have proved that Hamilton-Poisson realization exists only if b=c. The last two cases, a=0, b=c and a=b=c=0, can be found as the first studied cases. We can conclude that the Lü system admits Hamilton-Poisson realization with a three degree polynomial function as the Hamiltonian only if a=0, b=c∈ℝ, or a=b=c=0.



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