Hamilton-Poisson Realizations for the Lü System

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 ℝ³ has the following form 𝑦𝑧𝑥̇̇̇===−−𝑎𝑥𝑐(𝑦𝑧𝑧−++𝑥𝑥𝑏)𝑦𝑦,,,(1.1) where 𝑎,𝑏,𝑐∈ℝ.

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∈ℝ^∗, 𝑏=𝑐=0 the system becomes:𝐻(𝑥,𝑦,𝑧𝑥𝑧𝑦)̇̇̇====𝑎𝑥−𝛼𝑦(𝑥𝑦,𝑦𝑧₂−,𝑥),(1.2)+𝑧²+𝛽,𝛼,𝛽∈ℝ.(1.3)(ii) If 𝑎=0, 𝑏,𝑐∈ℝ^∗ the system becomes:𝐻(𝑥,𝑦,𝑧𝑥𝑧𝑦)̇̇̇====𝑥−𝑓0,𝑦𝑥(𝑥𝑧−)+𝑐,𝑧𝑏,𝑦𝑓,∈𝐶₁(1.4)(ℝ).(1.5)(iii)If 𝑎∈ℝ^∗, 𝑏=𝑐∈ℝ the system becomes:𝐻(𝑥,𝑦,𝑧𝑥𝑧𝑦)̇̇̇====−𝛼0𝑥,𝑦𝑥𝑥𝑧−𝑦+𝑏₂𝑧𝑏,𝑦,(1.6)−2𝑏𝑦𝑧+𝑥𝑧²+𝑓(𝑥),𝛼∈ℝ,𝑓∈𝐶¹(ℝ).(1.7)(iv)If 𝑎=𝑏=𝑐=0 the system becomes:𝐻(𝑥,𝑦,𝑧𝑥𝑧𝑦)̇̇̇====𝛼0𝑥−,𝑦𝑥,𝑦𝑧₂,(1.8)+𝑧²+𝛽𝑥𝑦²+𝑥𝑧²+𝑓(𝑥),𝛼,𝛽∈ℝ,𝑓∈𝐶¹(ℝ).(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: 𝐻(𝑥,𝑦,𝑧)=12𝑦²+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−𝑝₁(𝑥,𝑦,𝑧)𝑝₂(𝑥,𝑦,𝑧)−𝑝₂(𝑥,𝑦,𝑧)0𝑝₃(𝑥,𝑦,𝑧)(𝑥,𝑦,𝑧)−𝑝₃(𝑥,𝑦,𝑧)0⎤⎥⎥⎦.(2.2) We have to find the real smooth functions 𝑝₁,𝑝₂,𝑝₃∶ℝ³→ℝ such that: ⎡⎢⎢⎣𝑥𝑧𝑦̇⎤⎥⎥⎦̇̇=Π⋅∇𝐻,(2.3) that is, the following relations hold: 𝑦𝑝₁(𝑥,𝑦,𝑧)+𝑧𝑧𝑝𝑝₂₃(𝑥,𝑦,𝑧)=𝑎(𝑦−𝑥),−𝑦𝑝₃(𝑥,𝑦,𝑧)=−𝑥𝑧,(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)=𝑎𝑦−𝑥𝑧−𝑦z𝑝.(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(𝑦−𝑧),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=𝐴⋅𝑝+𝐵,(2.7) where 𝐴=𝜕𝑣¹𝜕𝑥+𝜕𝜕𝑣𝑦²+𝜕𝜕𝑣𝑧³−𝜕𝑣¹/𝜕𝑧(𝜕𝐻/𝜕𝑥)+𝜕𝑣₂/𝜕𝑧(𝜕𝐻/𝜕𝑦)+𝜕𝑣₃/𝜕𝑧(𝜕𝐻/𝜕𝑧)𝐵=𝑣₁𝜕𝐻/𝜕𝑧,𝜕𝑣₂/𝜕𝑧−𝑣₂𝜕𝑣₁/𝜕𝑧(𝜕𝐻/𝜕𝑧).(2.8) Equation (2.7) becomes:𝑎(𝑦−𝑥)𝜕𝑝𝜕𝑥−𝑥𝑧𝜕𝜕𝑦𝑝+𝑥𝑦𝜕𝜕𝑧𝑝=−𝑎+𝑥𝑧𝑦𝑝−𝑎(𝑦−𝑥)𝑥𝑧.(2.9)

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

If 𝑎≠0, 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: ℝ³,Π∶=Π^𝑖𝑗,𝐻,(2.10) where 𝐻(𝑥,𝑦,𝑧Π)==⎡⎢⎢⎣1−𝑥0𝑥𝑦𝑧𝑥𝑥0𝑧−−0𝑥𝑥𝑦⎤⎥⎥⎦,2𝑦²+𝑧².(2.11)

Remark 2.2. There exists only one functionally independent Casimir of our Poisson configuration, given by 𝐶∶ℝ³→ℝ, 𝐶(𝑥,𝑦,𝑧)=2𝑥−𝑦²−𝑧².(2.12)

Proof. Indeed, one can easily check that: Π⋅∇𝐶=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:𝐻𝐶==ccoonnsstt.,(2.14)

see Figure 1.

Figure 1 

The phase curves of the dynamics (1.2).

Remark 2.3. If 𝑎=𝑏=𝑐=0, then the system (1.2) becomes: 𝑥𝑧𝑦̇̇̇===𝑥−0,𝑦𝑥.𝑧,(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 ℝ³ such that the system (2.15) is induced by an appropriate Hamiltonian.

Proof. The triplets: 𝑅³{.,.}_𝛼𝛽,𝐻_𝛾𝛿,(2.16) where {𝑓,𝑔𝐶}_𝛼𝛽=∇𝐶_𝛼𝛽⋅(∇𝑓×∇𝑔),∀𝑓,𝑔∈𝐶^∞ℝ³,ℝ_𝛼𝛽,𝐻==𝛼1𝐶+𝛽𝐻,𝐻_𝛾𝛿=𝛾𝐶+𝛿𝐻,𝛼,𝛽,𝛾,𝛿∈ℝ,𝛼𝛿−𝛽𝛾=1,22𝑥−𝑦²−𝑧²,𝐶=12𝑥²,(2.17) define Hamilton-Poisson realizations of the dynamics (2.15).
Indeed, we have: 𝑥𝑦,,𝐻𝐻_{𝛾𝛿𝛾𝛿}_𝛼𝛽=||||||𝛼𝛾𝑥𝑥1++𝛿𝛽−−0𝛽𝛿𝑦𝑦−−0𝛽𝛿𝑧𝑧||||||=0=𝑥̇;𝑧,𝐻_𝛾𝛿_𝛼𝛽=||||||𝛼𝛾𝑥𝑥0++𝛿𝛽−−1𝛽𝛿𝑦𝑦−−0𝛽𝛿𝑧𝑧||||||=−𝑥𝑧=𝑦;̇_𝛼𝛽=||||||𝛼𝛾𝑥𝑥0++𝛿𝛽−−0𝛽𝛿𝑦𝑦−−1𝛽𝛿𝑧𝑧||||||=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=⎡⎢⎢⎢⎢⎢⎢⎢⎣0𝛼𝑥−𝛼𝛽𝛾√𝛽²−𝛾²𝑧+𝛿𝛼𝛾√𝛽²−𝛾²𝑥+𝛼𝛽𝑧+𝛾𝛿√𝛽²−𝛾²−𝛼𝑥+𝛼𝛽𝛾√𝛽²−𝛾²𝑧−𝛿0−𝛼𝛽²√𝛽²−𝛾²−𝛼𝛾𝑦√𝛽²−𝛾²𝑥−𝛼𝛽𝑧−𝛾𝛿√𝛽²−𝛾²𝛼𝛽²√𝛽²−𝛾²𝐵=⎡⎢⎢⎢⎢⎢⎢⎣0𝛾𝑧−√𝑦0⎤⎥⎥⎥⎥⎥⎥⎥⎦,𝛽²−𝛾²𝛿𝛼𝛽−√𝛽²−𝛾²𝑧−𝛾𝛿−𝛾𝑧+√𝛼𝛽𝛽²−𝛾²𝛿√𝛼𝛽0𝛽𝑦𝛽²−𝛾²𝑧+𝛾𝛿𝛼𝛽−𝛽𝑦0⎤⎥⎥⎥⎥⎥⎥⎦,(2.19) where 𝛼,𝛽,𝛾,𝛿∈ℝ, 𝑖=√−1.
Then, using MATHEMATICA 7.0, we can put the system (2.15) in the equivalent form 𝐿̇=[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: 𝑒𝑒^𝑀1=(𝑀,0,0),𝑀∈ℝ,_𝑀2=(0,𝑀,𝑁),𝑀,𝑁∈ℝ.(2.21) About their stability,we reached the following result.

Proposition 2.6 (A stability result). The equilibrium states 𝑒^𝑀1 are nonlinearly stable for any 𝑀∈ℝ.

Proof. We shall use energy-Casimir method, see [8] for details. Let 𝐻_𝜑=𝐻+𝜑(𝐶)=12𝑦²+𝑧²+𝜑2𝑥−𝑦²−𝑧²(2.22) be the energy-Casimir function, where 𝜑∶𝑅→𝑅 is a smooth real valued function defined on 𝑅.
Now, the first variation of 𝐻_𝜑 is given by:𝛿𝐻_𝜑=𝑦𝛿𝑦+𝑧𝛿𝑧+𝜑̇2𝑥−𝑦²−𝑧²(2𝛿𝑥−2𝑦𝛿𝑦−2𝑧𝛿𝑧).(2.23) This equals zero at the equilibrium of interest if and only if 𝜑̇(2𝑀)=0.(2.24) The second variation of 𝐻_𝜑 is given by: 𝛿²𝐻_𝜑=(𝛿𝑦)²+(𝛿𝑧)²+𝜑̈⋅(2𝛿𝑥−2𝑦𝛿𝑦−2𝑧𝛿𝑧)²−2𝜑̇(𝛿𝑦)²+(𝛿𝑧)².(2.25)
At the equilibrium of interest, the second variation becomes:𝛿²𝐻_𝜑(𝑀,0,0)=(𝛿𝑦)²+(𝛿𝑧)²+4𝜑̈⋅(𝛿𝑥)².(2.26)
Having chosen 𝜑 such that:𝜑𝜑̇̈((22𝑀𝑀))=>00,,(2.27) we can conclude that the second variation of 𝐻_𝜑 at the equilibrium of interest is positive defined and thus 𝑒^𝑀 is nonlinearly stable.

As a consequence, we can reach the periodical orbits of the equilibrium points 𝑒^𝑀1.

Proposition 2.7 (Periodical orbits). The reduced dynamics to the coadjoint orbit 2𝑥−𝑦²−𝑧²=2𝑀(2.28) has near the equilibrium point 𝑒^𝑀1 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 2𝑥−𝑦²−𝑧²=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,𝜆_2,3=±𝑀𝑖.(2.31)(iii)span(∇𝐶(𝑒^𝑀1))=𝑉₀, where 𝑉₀=ker𝐴𝑒^𝑀1,(2.32)(iv)the reduced Hamiltonian has a local minimum at the equilibrium state 𝑒^𝑀1 (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 𝑒^2𝑀,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 ℝ³,Π∶=Π^𝑖𝑗,𝐻,(3.1) where Π∶=⎡⎢⎢⎢⎢⎣−−𝑥𝑏𝑧𝑧0++𝑥𝑏𝑦𝑦−1𝑥𝑧−0𝑏𝑦12𝑏𝑧𝑦²−+𝑥𝑧𝑦²2𝑦²𝐻(𝑥,𝑦,𝑧)=𝑥.+𝑧²0⎤⎥⎥⎥⎥⎦,(3.2)

Using a method described in [6], we have found the Casimir of the configuration given by. 𝐶(𝑥,𝑦,𝑧)=12𝑦²+𝑧²𝑥−𝑏𝑦𝑧,𝑏∈ℝ^∗,(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 ℝ³ such that the system (1.4) is induced by an appropriate Hamiltonian.

Proof. The triples: 𝑅³{.,.}_𝛼𝛽,𝐻_𝛾𝛿,(3.4) where {𝑓,𝑔𝐶}_𝛼𝛽=∇𝐶_𝛼𝛽⋅(∇𝑓×∇𝑔),∀𝑓,𝑔∈𝐶^∞ℝ³,ℝ_𝛼𝛽,𝐻==𝑥𝛼,𝐶+𝐶𝛽=𝐻1,𝐻_𝛾𝛿=𝛾𝐶+𝛿𝐻,𝛼,𝛽,𝛾,𝛿∈ℝ,𝛼𝛿−𝛽𝛾=1,2𝑥𝑦²+𝑧²−𝑏𝑦𝑧,𝑏∈ℝ,(3.5) define Hamilton-Poisson realizations of the dynamics (1.4).
Indeed, we have: 𝑥,𝐻_𝛾𝛿_𝛼𝛽=||||||||𝛼+𝛽2𝑦²𝛾+𝛿1+𝑧²𝛽(𝑥𝑦0−𝑏𝑧)𝛽(𝑥𝑧0−𝑏𝑦)2𝑦²𝑦,𝐻_𝛾𝛿+𝑧²𝛿(𝑥𝑦−𝑏𝑧)𝛿(𝑥𝑧−𝑏𝑦)||||||||=0=𝑥̇;_𝛼𝛽=||||||||𝛼+𝛽2𝑦²𝛾+𝛿0+𝑧²𝛽(𝑥𝑦1−𝑏𝑧)𝛽(𝑥𝑧0−𝑏𝑦)2𝑦²𝑧,𝐻_𝛾𝛿+𝑧²𝛿(𝑥𝑦−𝑏𝑧)𝛿(𝑥𝑧−𝑏𝑦)||||||||=𝑥𝑧−𝑏𝑦=𝑦;̇_𝛼𝛽=||||||||𝛼+𝛽2𝑦²𝛾+𝛿0+𝑧²𝛽(𝑥𝑦0−𝑏𝑧)𝛽(𝑥𝑧1−𝑏𝑦)2𝑦²+𝑧²𝛿(𝑥𝑦−𝑏𝑧)𝛿(𝑥𝑧−𝑏𝑦)||||||||=𝑏𝑧−𝑥𝑦=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=⎡⎢⎢⎣−−0𝑣𝑢−0𝑢𝑤𝑤𝑣0⎤⎥⎥⎦,(3.7) where 𝑢=𝛼+𝛼𝛽²+𝛾²−𝛿²𝑖√𝛽²+𝛾²2𝑏𝛿𝛽²+𝛾²𝑥+𝛼𝛽𝛾𝛿𝛽²+𝛾²−𝛿²𝑖√𝛽²+𝛾²2𝑏𝛿𝛽²+𝛾²²𝑦−𝛼𝛽𝛾𝛽²+𝛾²−𝛿²𝑣=−𝛼𝑖√2𝑏𝛿𝛽²+𝛾²𝑧,𝛽²+𝛾²𝛽+𝛼𝛾²𝑖√𝛽²+𝛾²𝛽𝛽²+𝛾²+𝛼𝛽𝛽²+𝛾²−𝛿²2𝑏𝛿𝛽²+𝛾²𝑥+𝛼𝛾𝛽²+𝛾²−𝛿²+𝛼𝛾𝛽₂2𝑏𝛽²+𝛾²𝑦+𝛾²−𝛿²𝑖√𝛽²+𝛾²𝑤=𝛼𝛾₂2𝑏𝛿𝛽²+𝛾²𝑧,𝛽²+𝛾²−𝛿²𝑖√𝛽²+𝛾²2𝑏𝛽²+𝛾²²𝑦−𝛼𝛾²𝛽²+𝛾²−𝛿²𝐵=⎡⎢⎢⎢⎢⎢⎢⎣−0𝜔𝜔0𝛾𝑦−2𝑏𝛿𝜑𝛽𝛿²+𝛾²𝑧,𝑖√𝛽²+𝛾²−𝜑−𝛾𝑦−𝛿𝑧𝑖√𝛽²+𝛾²𝑧0⎤⎥⎥⎥⎥⎥⎥⎦,(3.8) where 𝜔=−𝑏𝑖√𝛽²+𝛾²𝛽²+𝛾²𝛽²+𝛾²+𝛿²+𝛽𝛾𝛽²+𝛾²−𝛿²𝛽²+𝛾²𝑦+𝑖𝛿√𝛽²+𝛾²𝑧𝜑=−𝑏𝛽𝛽₂𝛾𝛽²+𝛾²𝛽²+𝛾²−𝛿²,+𝛾²+𝛿²𝛾𝛽²+𝛾²−𝛿²−𝑖𝛽²+𝛾²𝑦+𝛿𝑧,𝑖=√−1(3.9) and 𝛼,𝛽,𝛾,𝛿∈ℝ.
Then, using MATHEMATICA 7.0, we can put the system (1.4) in the equivalent form𝐿̇=[𝐿,𝐵](3.10) as desired.

The equilibrium points of the dynamics (1.4) are given by𝑒𝑒^𝑀1=(𝑀,0,0),𝑀∈ℝ,_𝑀2𝑒=(−𝑏,𝑀,−𝑀),𝑀,𝑏∈ℝ,_𝑀3=(𝑏,𝑀,𝑀),𝑀,𝑏∈ℝ.(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 𝑒^𝑀1 are nonlinearly stable.

As a consequence, we can find the periodical orbits of the equilibrium points 𝑒^𝑀1.

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 2𝜋√𝑀²−𝑏².(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,𝜆_2,3=±𝑖√𝑀²−𝑏².(3.14)(iii)span(∇𝐶(𝑒^𝑀1))=𝑉₀, where 𝑉₀=ker𝐴𝑒^𝑀1,(3.15)(iv)if 𝑀>𝑏,𝑏>0, then the reduced Hamiltonian has a local minimum at the equilibrium state 𝑒^𝑀1 (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.

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.