Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 842325 | https://doi.org/10.1155/2011/842325

Camelia Pop, Camelia Petrişor, Dumitru Bălă, "Hamilton-Poisson Realizations for the Lü System", Mathematical Problems in Engineering, vol. 2011, Article ID 842325, 13 pages, 2011. https://doi.org/10.1155/2011/842325

Hamilton-Poisson Realizations for the Lü System

Academic Editor: Yuri Vladimirovich Mikhlin
Received06 Dec 2010
Accepted05 Feb 2011
Published31 Mar 2011

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.

1. Introduction

The original Lü system of differential equations on 3 has the following form ̇𝑥=𝑎(𝑦𝑥),̇𝑦=𝑥𝑧+𝑏𝑦,̇𝑧=𝑐𝑧+𝑥𝑦,(1.1) where 𝑎,𝑏,𝑐.

The goal of our paper is to find the relations between 𝑎, 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 functions 𝐻 are three degree polynomial constants of the motion defined by the system (1.1). (i) If 𝑎, 𝑏=𝑐=0 the system becomes:𝑦̇𝑥=𝑎(𝑦𝑥),̇𝑦=𝑥𝑧,̇𝑧=𝑥𝑦,(1.2)𝐻(𝑥,𝑦,𝑧)=𝛼2+𝑧2+𝛽,𝛼,𝛽.(1.3)(ii) If 𝑎=0, 𝑏,𝑐 the system becomes:̇𝑥=0,̇𝑦=𝑥𝑧+𝑏𝑦,̇𝑧=𝑥𝑦𝑐𝑧,(1.4)𝐻(𝑥,𝑦,𝑧)=𝑓(𝑥),𝑓𝐶1().(1.5)(iii)If 𝑎, 𝑏=𝑐 the system becomes:̇𝑥=0,̇𝑦=𝑥𝑧+𝑏𝑦,̇𝑧=𝑥𝑦𝑏𝑧,(1.6)𝐻(𝑥,𝑦,𝑧)=𝛼𝑥𝑦22𝑏𝑦𝑧+𝑥𝑧2+𝑓(𝑥),𝛼,𝑓𝐶1().(1.7)(iv)If 𝑎=𝑏=𝑐=0 the system becomes:𝑦̇𝑥=0,̇𝑦=𝑥𝑧,̇𝑧=𝑥𝑦,(1.8)𝐻(𝑥,𝑦,𝑧)=𝛼2+𝑧2+𝛽𝑥𝑦2+𝑥𝑧2+𝑓(𝑥),𝛼,𝛽,𝑓𝐶1().(1.9)

Proof. It is easy to see that 𝑑𝐻=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: 1𝐻(𝑥,𝑦,𝑧)=2𝑦2+𝑧2.(2.1) 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: 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.(2.2) We have to find the real smooth functions 𝑝1,𝑝2,𝑝33 such that: ̇𝑥̇𝑦̇𝑧=Π𝐻,(2.3) that is, the following relations hold: 𝑦𝑝1(𝑥,𝑦,𝑧)+𝑧𝑝2(𝑥,𝑦,𝑧)=𝑎(𝑦𝑥),𝑧𝑝3(𝑥,𝑦,𝑧)=𝑥𝑧,𝑦𝑝3(𝑥,𝑦,𝑧)=𝑥𝑦.(2.4) 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 𝑝2(𝑥,𝑦,𝑧)=𝑎𝑦𝑥𝑧𝑦z𝑝.(2.5)

Our goal now is to insert 𝑝1,𝑝2,𝑝3 into Jacobi identity and to find the function 𝑝(𝑥,𝑦,𝑧). In the beginning, let us denote: 𝑣1𝑣=𝑎(𝑦𝑧),2𝑣=𝑥𝑧,3=𝑥𝑦.(2.6) The function 𝑝 is the solution of the following first order ODE (see [5] for details):𝑣1𝜕𝑝𝜕𝑥+𝑣2𝜕𝑝𝜕𝑦+𝑣3𝜕𝑝𝜕𝑧=𝐴𝑝+𝐵,(2.7) 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)(2.8) Equation (2.7) becomes:𝑎(𝑦𝑥)𝜕𝑝𝜕𝑥𝑥𝑧𝜕𝑝𝜕𝑦+𝑥𝑦𝜕𝑝=𝜕𝑧𝑎+𝑥𝑦𝑧𝑥𝑝𝑎(𝑦𝑥)𝑧.(2.9)

If 𝑎=0, then (2.9) has the solution 𝑝(𝑥,𝑦,𝑧)=𝑥𝑧.

If 𝑎0, then finding the solution of (2.9) remains an open problem.

Now, one can reach the following result.

Proposition 2.1. If 𝑎=0, the system (1.2) has the Hamilton-Poisson realization: 3Π,Π=𝑖𝑗,,𝐻(2.10) where ,1Π=0𝑥𝑧𝑥𝑦𝑥𝑧0𝑥𝑥𝑦𝑥0𝐻(𝑥,𝑦,𝑧)=2𝑦2+𝑧2.(2.11)

Remark 2.2. There exists only one functionally independent Casimir of our Poisson configuration, given by 𝐶3, 𝐶(𝑥,𝑦,𝑧)=2𝑥𝑦2𝑧2.(2.12)

Proof. Indeed, one can easily check that: Π𝐶=0.(2.13) 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:𝐻=const.𝐶=const,(2.14)

see Figure 1.

Remark 2.3. If 𝑎=𝑏=𝑐=0, then the system (1.2) becomes: ̇𝑥=0,̇𝑦=𝑥𝑧,̇𝑧=𝑥𝑦.(2.15)

For the specific case 𝑎=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 3 such that the system (2.15) is induced by an appropriate Hamiltonian.

Proof. The triplets: 𝑅3{.,.}𝛼𝛽,𝐻𝛾𝛿,(2.16) where {𝑓,𝑔}𝛼𝛽=𝐶𝛼𝛽(𝑓×𝑔),𝑓,𝑔𝐶3,𝐶,𝛼𝛽=𝛼𝐶+𝛽𝐻,𝐻𝛾𝛿1=𝛾𝐶+𝛿𝐻,𝛼,𝛽,𝛾,𝛿,𝛼𝛿𝛽𝛾=1,𝐻=22𝑥𝑦2𝑧21,𝐶=2𝑥2,(2.17) define Hamilton-Poisson realizations of the dynamics (2.15).
Indeed, we have: 𝑥,𝐻𝛾𝛿𝛼𝛽=||||||||||||𝛼𝑥+𝛽𝛽𝑦𝛽𝑧100𝛾𝑥+𝛿𝛿𝑦𝛿𝑧=0=̇𝑥;𝑦,𝐻𝛾𝛿𝛼𝛽=||||||||||||𝛼𝑥+𝛽𝛽𝑦𝛽𝑧010𝛾𝑥+𝛿𝛿𝑦𝛿𝑧=𝑥𝑧=̇𝑦;𝑧,𝐻𝛾𝛿𝛼𝛽=||||||||||||𝛼𝑥+𝛽𝛽𝑦𝛽𝑧001𝛾𝑥+𝛿𝛿𝑦𝛿𝑧=𝑥𝑦=̇𝑧.(2.18)

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: 𝐿=0𝛼𝑥𝛼𝛽𝛾𝛽2𝛾2𝑧+𝛿𝛼𝛾𝛽2𝛾2𝑥+𝛼𝛽𝑧+𝛾𝛿𝛽2𝛾2𝛼𝑥+𝛼𝛽𝛾𝛽2𝛾2𝑧𝛿0𝛼𝛽2𝛽2𝛾2𝑦𝛼𝛾𝛽2𝛾2𝑥𝛼𝛽𝑧𝛾𝛿𝛽2𝛾2𝛼𝛽2𝛽2𝛾2,𝑦0𝐵=0𝛾𝑧𝛽2𝛾2𝛿𝛼𝛽𝛽2𝛾2𝑧𝛾𝛿𝛼𝛽𝛾𝑧+𝛽2𝛾2𝛿𝛼𝛽0𝛽𝑦𝛽2𝛾2𝑧+𝛾𝛿,𝛼𝛽𝛽𝑦0(2.19) where 𝛼,𝛽,𝛾,𝛿, 𝑖=1.
Then, using MATHEMATICA 7.0, we can put the system (2.15) in the equivalent form ̇[]𝐿=𝐿,𝐵(2.20) 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: 𝑒𝑀1𝑒=(𝑀,0,0),𝑀,𝑀2=(0,𝑀,𝑁),𝑀,𝑁.(2.21) About their stability,we reached the following result.

Proposition 2.6 (A stability result). The equilibrium states 𝑒𝑀1 are nonlinearly stable for any 𝑀.

Proof. We shall use energy-Casimir method, see [8] for details. Let 𝐻𝜑1=𝐻+𝜑(𝐶)=2𝑦2+𝑧2+𝜑2𝑥𝑦2𝑧2(2.22) be the energy-Casimir function, where 𝜑𝑅𝑅 is a smooth real valued function defined on 𝑅.
Now, the first variation of 𝐻𝜑 is given by:𝛿𝐻𝜑=𝑦𝛿𝑦+𝑧𝛿𝑧+̇𝜑2𝑥𝑦2𝑧2(2𝛿𝑥2𝑦𝛿𝑦2𝑧𝛿𝑧).(2.23) This equals zero at the equilibrium of interest if and only if ̇𝜑(2𝑀)=0.(2.24) The second variation of 𝐻𝜑 is given by: 𝛿2𝐻𝜑=(𝛿𝑦)2+(𝛿𝑧)2+̈𝜑(2𝛿𝑥2𝑦𝛿𝑦2𝑧𝛿𝑧)22̇𝜑(𝛿𝑦)2+(𝛿𝑧)2.(2.25)
At the equilibrium of interest, the second variation becomes:𝛿2𝐻𝜑(𝑀,0,0)=(𝛿𝑦)2+(𝛿𝑧)2+4̈𝜑(𝛿𝑥)2.(2.26)
Having chosen 𝜑 such that:̇𝜑(2𝑀)=0,̈𝜑(2𝑀)>0,(2.27) 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 points 𝑒𝑀1.

Proposition 2.7 (Periodical orbits). The reduced dynamics to the coadjoint orbit 2𝑥𝑦2𝑧2=2𝑀(2.28) has near the equilibrium point 𝑒𝑀1 at least one periodic solution whose period is close to 2𝜋||𝑀||.(2.29)

Proof. Indeed, we have successively
(i)the restriction of our dynamics (1.2) to the coadjoint orbit 2𝑥𝑦2𝑧2=2𝑀(2.30) gives rise to a classical Hamiltonian system,(ii)the matrix of the linear part of the reduced dynamics has purely imaginary roots, more exactly𝜆1=0,𝜆2,3=±𝑀𝑖.(2.31)(iii)span(𝐶(𝑒𝑀1))=𝑉0, where 𝑉0𝐴𝑒=ker𝑀1,(2.32)(iv)the reduced Hamiltonian has a local minimum at the equilibrium state 𝑒𝑀1 (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 states 𝑒2𝑀,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. If 𝑎=0 and 𝑏=𝑐, the system (1.4) has the Hamilton-Poisson realization 3Π,Π=𝑖𝑗,𝐻,(3.1) where 1Π=0𝑥𝑧𝑏𝑦𝑏𝑧𝑥𝑦𝑥𝑧+𝑏𝑦02𝑦2+𝑧21𝑏𝑧+𝑥𝑦2𝑦2+𝑧20,𝐻(𝑥,𝑦,𝑧)=𝑥.(3.2)

Using a method described in [6], we have found the Casimir of the configuration given by. 1𝐶(𝑥,𝑦,𝑧)=2𝑦2+𝑧2𝑥𝑏𝑦𝑧,𝑏,(3.3) (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 3 such that the system (1.4) is induced by an appropriate Hamiltonian.

Proof. The triples: 𝑅3{.,.}𝛼𝛽,𝐻𝛾𝛿,(3.4) where {𝑓,𝑔}𝛼𝛽=𝐶𝛼𝛽(𝑓×𝑔),𝑓,𝑔𝐶3,𝐶,𝛼𝛽=𝛼𝐶+𝛽𝐻,𝐻𝛾𝛿1=𝛾𝐶+𝛿𝐻,𝛼,𝛽,𝛾,𝛿,𝛼𝛿𝛽𝛾=1,𝐻=𝑥,𝐶=2𝑥𝑦2+𝑧2𝑏𝑦𝑧,𝑏,(3.5) define Hamilton-Poisson realizations of the dynamics (1.4).
Indeed, we have: 𝑥,𝐻𝛾𝛿𝛼𝛽=||||||||𝛽𝛼+2𝑦2+𝑧2𝛿𝛽(𝑥𝑦𝑏𝑧)𝛽(𝑥𝑧𝑏𝑦)100𝛾+2𝑦2+𝑧2||||||||𝛿(𝑥𝑦𝑏𝑧)𝛿(𝑥𝑧𝑏𝑦)=0=̇𝑥;𝑦,𝐻𝛾𝛿𝛼𝛽=||||||||𝛽𝛼+2𝑦2+𝑧2𝛽𝛿(𝑥𝑦𝑏𝑧)𝛽(𝑥𝑧𝑏𝑦)010𝛾+2𝑦2+𝑧2||||||||𝛿(𝑥𝑦𝑏𝑧)𝛿(𝑥𝑧𝑏𝑦)=𝑥𝑧𝑏𝑦=̇𝑦;𝑧,𝐻𝛾𝛿𝛼𝛽=||||||||𝛽𝛼+2𝑦2+𝑧2𝛿𝛽(𝑥𝑦𝑏𝑧)𝛽(𝑥𝑧𝑏𝑦)001𝛾+2𝑦2+𝑧2||||||||𝛿(𝑥𝑦𝑏𝑧)𝛿(𝑥𝑧𝑏𝑦)=𝑏𝑧𝑥𝑦=̇𝑧.(3.6)

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 𝐿=0𝑢𝑣𝑢0𝑤𝑣𝑤0,(3.7) where 𝛼𝛽𝑢=𝛼+2+𝛾2𝛿2𝑖𝛽2+𝛾2𝛽2𝑏𝛿2+𝛾2𝛽𝑥+𝛼𝛽𝛾𝛿2+𝛾2𝛿2𝑖𝛽2+𝛾2𝛽2𝑏𝛿2+𝛾22𝛽𝑦𝛼𝛽𝛾2+𝛾2𝛿2𝛽2𝑏𝛿2+𝛾2𝑧,𝑣=𝛼𝑖𝛽2+𝛾2𝛽+𝛼𝛾2𝑖𝛽2+𝛾2𝛽𝛽2+𝛾2+𝛽𝛼𝛽2+𝛾2𝛿2𝛽2𝑏𝛿2+𝛾2𝛽𝑥+𝛼𝛾2+𝛾2𝛿2𝛽2𝑏2+𝛾2𝑦+𝛽𝛼𝛾2+𝛾2𝛿2𝑖𝛽2+𝛾2𝛽2𝑏𝛿2+𝛾2𝑧,𝑤=𝛼𝛾2𝛽2+𝛾2𝛿2𝑖𝛽2+𝛾2𝛽2𝑏2+𝛾22𝑦𝛼𝛾2𝛽2+𝛾2𝛿2𝛽2𝑏𝛿2+𝛾2𝛿𝑧,𝐵=0𝜔𝜑𝜔0𝛾𝑦𝑖𝛽2+𝛾2𝑧𝛿𝜑𝛾𝑦𝑖𝛽2+𝛾2𝑧0,(3.8) where 𝜔=𝑏𝑖𝛽2+𝛾2𝛽2+𝛾2𝛽2+𝛾2+𝛿2𝛽+𝛽𝛾2+𝛾2𝛿2𝛽2+𝛾2𝑦+𝑖𝛿𝛽2+𝛾2𝑧𝛾𝛽2+𝛾2𝛽2+𝛾2𝛿2,𝛽𝜑=𝑏𝛽2+𝛾2+𝛿2𝛾𝛽2+𝛾2𝛿2𝑖𝛽2+𝛾2𝑦+𝛿𝑧,𝑖=1(3.9) and 𝛼,𝛽,𝛾,𝛿.
Then, using MATHEMATICA 7.0, we can put the system (1.4) in the equivalent forṁ[]𝐿=𝐿,𝐵(3.10) as desired.

The equilibrium points of the dynamics (1.4) are given by𝑒𝑀1𝑒=(𝑀,0,0),𝑀,𝑀2=𝑒(𝑏,𝑀,𝑀),𝑀,𝑏,𝑀3=(𝑏,𝑀,𝑀),𝑀,𝑏.(3.11)

About their stability, we have proven in [10] the following result,

Proposition 3.4 (Stability problem). If 𝑀>𝑏 or 𝑀<𝑏,𝑏>0, then the equilibrium states 𝑒𝑀1 are nonlinearly stable.

As a consequence, we can find the periodical orbits of the equilibrium points 𝑒𝑀1.

Proposition 3.5 (Periodical orbits). If 𝑀>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 2𝜋𝑀2𝑏2.(3.12)

Proof. Indeed, we have successively(i)the restriction of our dynamics (1.4) to the coadjoint orbit 𝑥=𝑀(3.13) gives rise to a classical Hamiltonian system,(ii)the matrix of the linear part of the reduced dynamics has purely imaginary roots, more exactly 𝜆1=0,𝜆2,3=±𝑖𝑀2𝑏2.(3.14)(iii)span(𝐶(𝑒𝑀1))=𝑉0, where 𝑉0𝐴𝑒=ker𝑀1,(3.15)(iv)if 𝑀>𝑏,𝑏>0, then the reduced Hamiltonian has a local minimum at the equilibrium state 𝑒𝑀1 (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, 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. 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, or 𝑎=𝑏=𝑐=0.

Acknowledgments

The work of C. Pop was supported by the project “Development and support for multidisciplinary postdoctoral programs in primordial technical areas of the national strategy for research development innovation” 4D-POSTDOC, contract no. POSDRU/89/1.5/S/52603, project cofunded from the European Social Fund through the Sectorial Operational Program Human Resources 2007–2013.

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Copyright © 2011 Camelia Pop et al. 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.


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