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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 484028, 11 pages
http://dx.doi.org/10.1155/2012/484028
Research Article

A Hamilton-Poisson Model of the Chen-Lee System

Department of Mathematics, “Politehnica” University of Timişoara, Piaţa Victoriei nr. 2, 300006 Timişoara, Romania

Received 14 May 2012; Accepted 23 June 2012

Academic Editor: Chein-Shan Liu

Copyright © 2012 Camelia Pop Arieşanu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We will present some dynamical and geometrical properties of Chen-Lee system from the Poisson geometry point of view.

1. Introduction

Let us consider the Chen-Lee’s system (see [1, 2]) given by the following differential equations system on 3̇𝑥=𝑎𝑥𝑦𝑧,̇𝑦=𝑏𝑦+𝑥𝑧,̇𝑧=𝑐𝑧+𝑥𝑦3,(1.1) where 𝑎,𝑏,𝑐 are real parameters.

In this paper we consider a special case of the Chen-Lee system, realizing this system as a Hamiltonian system and then study it from the mechanical geometry point of view. This means the study of the nonlinear stability, the existence of periodic solutions, and numerical integration. The paper is structured as follows: Section 2 presents the special case of Chen-Lee system for which it admits a Hamilton-Poisson structure; in Section 3 we study the nonlinear stability of the equilibrium states of our dynamics using energy-Casimir method. Periodical orbits are the subject of Section 4. In Section 5 of the paper we give a Lax formulation of the system; Section 6 discusses numerical integration of the system using Poisson and non-Poisson integrators. Numerical simulations using MATHEMATICA 8.0 are presented, too.

For details on Possion geometry and Hamiltonian dynamics, (see [35]).

2. The Poisson Geometry Associated to the Chen-Lee’s System

In this section we will find the parameters values for which Chen-Lee's system admits a Poisson structure. In order to do this, we need to find the system's Hamiltonians. Due to the existence of a numerous parameters, we are looking for polynomial Hamiltonians.

Proposition 2.1. The following smooth real functions 𝐻 are two-degree polynomial constants of the motion defined by the system (1.1). (i) If 𝑎,𝑏=𝑐=0, then the function 𝑦𝐻(𝑥,𝑦,𝑧)=𝛼23𝑧2+𝛽,𝛼,𝛽(2.1) is the Hamiltonian of the system (1.1). (ii) If 𝑏,𝑎=𝑐=0, then the function 𝑥𝐻(𝑥,𝑦,𝑧)=𝛼2+3𝑧2+𝛽,𝛼,𝛽(2.2) is the Hamiltonian of the system (1.1). (iii) If 𝑐,𝑎=𝑏=0, then the function 𝑥𝐻(𝑥,𝑦,𝑧)=𝛼2+𝑦2+𝛽,𝛼,𝛽(2.3) is the Hamiltonian of the system (1.1).

Let us focus now on the first case; if 𝑎,𝑏=𝑐=0, the system (1.1) becomes ̇𝑥=𝑎𝑥𝑦𝑧,̇𝑦=𝑥𝑧,̇𝑧=𝑥𝑦3,(2.4) and we will consider the Hamiltonian given by 1𝐻(𝑥,𝑦,𝑧)=2𝑦23𝑧2.(2.5) In order to find the Poisson structure in this case we will use a method described by Haas and Goedert (see [6] for details). Let us consider the skew-symmetric matrix given by 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.(2.6) In the beginning, let us denote 𝑣1𝑣=𝑎𝑥𝑦𝑧,2𝑣=𝑥𝑧,3=𝑥𝑦3.(2.7) The function 𝑝 is the solution of the following first order ODE: 𝑣1𝜕𝑝𝜕𝑥+𝑣2𝜕𝑝𝜕𝑦+𝑣3𝜕𝑝𝜕𝑧=𝐴𝑝+𝐵,(2.8) where 𝐴=𝜕𝑣1+𝜕𝑥𝜕𝑣2+𝜕𝑦𝜕𝑣3𝜕𝑧𝜕𝑣1/𝜕𝑧(𝜕𝐻/𝜕𝑥)+𝜕𝑣2/𝜕𝑧(𝜕𝐻/𝜕𝑦)+𝜕𝑣3/𝜕𝑧(𝜕𝐻/𝜕𝑧),𝑣𝜕𝐻/𝜕𝑧𝐵=1𝜕𝑣2/𝜕𝑧𝑣2𝜕𝑣1/𝜕𝑧.𝜕𝐻/𝜕𝑧(2.9) the Equation (2.8) becomes (𝑎𝑥𝑦𝑧)𝜕𝑝𝜕𝑥+𝑥𝑧𝜕𝑝+𝜕𝑦𝑥𝑦3𝜕𝑝=𝜕𝑧𝑎+𝑥𝑦3𝑧𝑝𝑎𝑥2.3𝑧(2.10)

If 𝑎=0 then (2.10) has the solution 𝑝(𝑥,𝑦,𝑧)=𝑧.

If 𝑎0 then finding the solution of (2.10) remains an open problem.

If 𝑎=0 then the system (2.4) becomes ̇𝑥=𝑦𝑧,̇𝑦=𝑥𝑧,̇𝑧=𝑥𝑦3.(2.11) Now, one can reach the following result.

Proposition 2.2. The system (2.11) has the Hamilton-Poisson realization: 3Π,Π=𝑖𝑗,,𝐻(2.12) where 𝑥Π=0𝑧0𝑧030𝑥30,𝐻1(𝑥,𝑦,𝑧)=2𝑦23𝑧2.(2.13)

Proof. Indeed, we have Π𝐻=̇𝑥̇𝑦̇𝑧,(2.14) and the matrix Π is a Poisson matrix, see [7].

It is easy to see that the Poisson structure is degenerate, so we can proceed to find the Casimir functions of our configuration.

Proposition 2.3. The real smooth function 𝐶3, 1𝐶(𝑥,𝑦,𝑧)=2𝑥2+3𝑧2,(2.15) is the only one functionally independent Casimir of the Hamilton-Poisson realization given by Proposition 2.2.

Proof. Indeed, we have (𝐶)𝑡Π=0 and rankΠ=2, as required.

The phase curves of the dynamics (2.11) are the intersections of the surfaces: 𝐶𝐻(𝑥,𝑦,𝑧)=const.,(𝑥,𝑦,𝑧)=const.,(2.16) see the Figure 1.

fig1
Figure 1: The phase curves of the system (2.11).

The next proposition gives other Hamilton-Poisson realizations of the system (2.11).

Proposition 2.4. The system (2.11) may be modeled as a Hamilton-Poisson system in an infinite number of different ways, that is, there exist infinitely more different (in general nonisomorphic) Poisson structures on 3 such that the system (2.11) is induced by an appropriate Hamiltonian.

Proof. The triplets 𝑅3{,}𝛼𝛽,𝐻𝛾𝛿,(2.17) where {𝑓,𝑔}𝛼𝛽=𝐶𝛼𝛽(𝑓×𝑔),𝑓,𝑔𝐶3,𝐶,𝛼𝛽=𝛼𝐶+𝛽𝐻,𝐻𝛾𝛿1=𝛾𝐶+𝛿𝐻,𝛼,𝛽,𝛾,𝛿,𝛼𝛿𝛽𝛾=3,1𝐻=2𝑦23𝑧21,𝐶=2𝑥2+3𝑧2,(2.18) define Hamilton-Poisson realizations of the dynamics (2.11).

3. The Stability Problem

Let us pass now to discuss the stability problem of the system (2.11). It is not difficult to see that the equilibrium states of our dynamics are 𝑒𝑀1=(𝑀,0,0),𝑒𝑀2=(0,𝑀,0),𝑒𝑀3=(0,0,𝑀),𝑀.(3.1) Let 𝐴 be the matrix of the linear part of our system, that is, 𝑦𝐴=0𝑧𝑦𝑧0𝑥3𝑥30.(3.2) Then the characteristic roots of 𝐴(𝑒𝑀1) (resp., 𝐴(𝑒𝑀2), resp. 𝐴(𝑒𝑀3) )are given by, 𝜆1=0,𝜆2,3𝑀=±3,(3.3) (resp., 𝜆1=0,𝜆2,3=(±𝑖𝑀/3), resp., 𝜆1=0,𝜆2,3=±𝑖𝑀), so one gets that the following.

Proposition 3.1. The equilibrium states 𝑒𝑀1,𝑒𝑀2,𝑒𝑀3,𝑀, have the following behavior: (i)𝑒𝑀1 are unstable for any 𝑀;(ii)𝑒𝑀2 are spectrally stable for any 𝑀;(iii)𝑒𝑀3 are spectrally stable for any 𝑀.

Let us begin the nonlinear stability analysis using the energy-Casimir method for the equilibrium state 𝑒𝑀2 and 𝑒𝑀3.

Proposition 3.2. The equilibrium states 𝑒𝑀2,𝑀, are nonlinearly stable for any 𝑀.

Proof. To study the nonlinear stability of the equilibrium state 𝑒𝑀 we are using energy-Casimir method ([8]). To do that, let 𝐻𝜑𝐶(3,) be defined by 𝐻𝜑=𝐶+𝜑(𝐻),(3.4) where 𝜑 is a smooth real valued function defined on .
Now, the first variation of 𝐻𝜑 at the equilibrium of interest equals zero if and only if 𝑀̇𝜑22=0.(3.5) Using (3.5), the second variation of 𝐻𝜑 at the equilibrium of interest is given by 𝛿2𝐻𝜑𝑒𝑀2=(𝛿𝑥)2+3(𝛿𝑧)2+𝑀2..𝜑𝑀22(𝛿𝑦)2.(3.6)
If we choose now the function 𝜑 such that 𝑀̇𝜑22=0,..𝜑𝑀22>0(3.7) we can conclude that the second variation of 𝐻𝜑 at the equilibrium of interest is positively defined for any 𝑀 and thus 𝑒𝑀2 are nonlinear stable for any 𝑀.

Proposition 3.3. The equilibrium states 𝑒𝑀3,𝑀, are nonlinearly stable for any 𝑀.

Proof. To study the nonlinear stability of the equilibrium state 𝑒𝑀 we are using energy-Casimir method ([8]). To do that, let 𝐻𝜑𝐶(3,) be defined by 𝐻𝜑=𝐶+𝜑(𝐻),(3.8) where 𝜑 is a smooth real valued function defined on .
Now, the first variation of 𝐻𝜑 at the equilibrium of interest equals zero if and only if ̇𝜑3𝑀22=1.(3.9) Using (3.9), the second variation of 𝐻𝜑 at the equilibrium of interest is given by 𝛿2𝐻𝜑𝑒𝑀2=(𝛿𝑥)2+(𝛿𝑦)2+9𝑀2..𝜑3𝑀22(𝛿𝑧)2.(3.10)
If we choose now the function 𝜑 such that ̇𝜑3𝑀22=1,..𝜑3𝑀22>0(3.11) we can conclude that the second variation of 𝐻𝜑 at the equilibrium of interest is positively defined for any 𝑀 and thus 𝑒𝑀3 are nonlinear stable for any 𝑀.

4. Periodical Orbits

As we have proved in the previous section, the equilibrium states 𝑒𝑀2 are nonlinear stable so we can try to find the periodic orbits around them. As we know, the dynamics described by a Hamilton-Poisson system take place on the symplectic leaves of the Poisson configuration. In order to do this we consider the system restricted to a regular coadjoint orbit of (𝑅3) that contains 𝑒𝑀2; we will obtain the existence of periodic solutions for the restricted system; these are, also, the periodic solutions for the unrestricted system.

Proposition 4.1. Near to 𝑒𝑀2 the reduced dynamics has for each sufficiently small value of the reduced energy at least 1-periodic solution whose period is close to (2𝜋3/𝑀).

Proof. Indeed, we have successively the following. (i) The reduction of the system (2.11) to the regular coadjoint orbit: Ω𝑒𝑀2=(𝑥,𝑦,𝑧)𝑅,𝑦23𝑧2=𝑀2(4.1) gives rise to a classical Hamiltonian system. (ii) The matrix of the linear part of the reduced dynamics has purely imaginary roots. More exactly, 𝜆2,3𝑀=±𝑖3.(4.2)(iii) One  has span(𝐶(𝑒𝑀2))=𝑉0, where 𝑉0𝐴𝑒=ker𝑀2.(4.3)(iv) The equilibrium state 𝑒𝑀2 are nonlinear stable for any 𝑀.Then our assertion follows via the Moser-Weinstein theorem with zero eigenvalue, see [9] for details.

Similar arguments lead us to the following result.

Proposition 4.2. Near to 𝑒𝑀3 the reduced dynamics has for each sufficiently small value of the reduced energy at least 1-periodic solution whose period is close to 2𝜋/𝑀.

5. Lax Formulation of the Dynamics (2.11)

Proposition 5.1. The dynamics (2.11) allows a formulation in terms of Lax pairs.

Proof. Let us take the following. 𝑖𝐿=02𝑖𝑥32𝑦3𝑧3𝑥𝑦+𝑖𝑖3𝑧2𝑖𝑥+32𝑦+3𝑧02𝑥2𝑖𝑧3𝑥+𝑦𝑖0,𝑖3𝑧2𝑥+2𝑖𝑧𝐵=0𝑖𝑥+31𝑦3𝑖𝑥+𝑦𝑖𝑥31𝑦0𝑥3.𝑥𝑦𝑥0(5.1)
Then, using MATHEMATICA 8.0, we can put the system (2.11) in the equivalent form: ̇[],𝐿=𝐿,𝐵(5.2) as desired.

6. Numerical Integration of the Dynamics (2.11)

We will discuss now the numerical integration of the dynamics (2.11) via the Lie-Trotter integrator [7]. For the beginning, let us observe that the Hamiltonian vector field 𝑋𝐻 splits in 𝑋𝐻=𝑋𝐻1+𝑋𝐻2, where 𝐻1=12𝑦2,𝐻21=2𝑧2.(6.1)

Their corresponding integral curves are, respectively, given by 𝑥1(𝑡)𝑥2(𝑡)𝑥3(𝑡)=𝐴𝑖𝑥1(0)𝑥2(0)𝑥3𝐴(0),𝑖=1,2,where1=cos𝑎𝑡303sin𝑎𝑡310103sin𝑎𝑡30cos𝑎𝑡3,𝐴2=,100𝑏𝑡10001(6.2)

and 𝑎=𝑥(0),𝑏=𝑧(0).

Then the Lie-Trotter integrator is given by 𝑥1𝑛+1𝑥2𝑛+1𝑥3𝑛+1𝑡=𝐴1𝐴2𝑥𝑛1𝑥𝑛2𝑥𝑛3𝑡,(6.3) that is, 𝑥𝑛+1=cos𝑎𝑡3𝑥𝑛3sin𝑎𝑡3𝑧𝑛,𝑦𝑛+1=𝑏𝑡𝑥𝑛+𝑦𝑛,𝑧𝑛+1=13sin𝑎𝑡3𝑥𝑛+cos𝑎𝑡3𝑧𝑛,(6.4)

The following proposition sketches the Lie-Trotter integrator properties.

Proposition 6.1. The Lie-Trotter integrator (6.4) has the following properties.(i)It preserves the Poisson structure Π. (ii)It preserves the Casimir C of our Poisson configuration (3,Π). (iii)It does not preserve the Hamiltonian H of our system (2.11). (iv)Its restriction to the coadjoint orbit (𝒪k,𝜔k), where 𝒪𝑘=(𝑥,𝑦,𝑧)𝑅,𝑦23𝑧2=𝑘(6.5) and 𝜔𝑘 is the Kirilov-Kostant-Souriau symplectic structure on 𝒪𝑘, gives rise to a symplectic integrator.

We will discuss now the numerical integration of the dynamics (2.11) via the Kahan integrator and also via Runge-Kutta 4th steps integrator and we will point out some properties of Kahan integrator. The Kahan integrator [10] of the system (2.11) is given by 𝑥𝑛+1𝑥𝑛=2𝑦𝑛+1𝑧𝑛+𝑧𝑛+1𝑦𝑛,𝑦𝑛+1𝑦𝑛=2𝑥𝑛+1𝑧𝑛+𝑧𝑛+1𝑥𝑛,𝑧𝑛+1𝑧𝑛=6𝑥𝑛+1𝑦𝑛+𝑦𝑛+1𝑥𝑛.(6.6)

Using MATHEMATICA 8.0, we can prove the following proposition which shows the incompatibility of the Kahan integrator with the Poisson structure of the system (2.11).

Proposition 6.2. The Kahan integrator (6.6) does not preserve the Poisson structure and does not preserve the Hamiltonian and the Casimir of our configuration.

Remark 6.3. As we can see from Figure 2 the three integrators give us almost the same results. However, Lie-Trotter and Kahan integrators have the advantage of being easily implemented.

fig2
Figure 2: Runge-Kutta 4 steps, Lie-Trotter, and Kahan integrator, respectively (𝑥(0)=𝑦(0)=𝑧(0)=1).

7. Conclusion

The Chen-Lee system is a system arisen from engineering field. Its chaotic behavior makes it good to applied in secure communications, complete synchronization, or optimization of nonlinear system performance. The geometric overview gives it a different perspective and points out new properties. It is easy to see that, like other chaotic systems studied before—the Rikitake system [11], the Lü system [12], the Lorenz system [13]—finding the corresponding Poisson structure implies the study of particular values for its parameters. Unlike the other studied systems, the Chen-Lee one needs to vanish all its parameters to admit a Hamilton-Poisson realization. The connexion between the existence of a Hamilton-Poisson realization and the number of the parameters which should be vanished of a chaotic system remains an open problem.

Acknowledgments

This paper was supported by the project "Development and support of multidisciplinary postdoctoral programmes in major technical areas of national strategy for research-development-innovation" 4D-POSTDOC, Contract no. POSDRU/89/1.5/S/52603, project cofunded by the European Social Fund through Sectorial Operational Programme Human Resources Development 2007–2013.

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