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

Let us consider the Chen-Lee’s system (see [

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

For details on Possion geometry and Hamiltonian dynamics, (see [

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.

The following smooth real functions

If

If

If

Let us focus now on the first case; if

If

If

If

The system (

Indeed, we have

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

The real smooth function

Indeed, we have

The phase curves of the dynamics (

The phase curves of the system (

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

The system (

The triplets

Let us pass now to discuss the stability problem of the system (

The equilibrium states

Let us begin the nonlinear stability analysis using the energy-Casimir method for the equilibrium state

The equilibrium states

To study the nonlinear stability of the equilibrium state

Now, the first variation of

If we choose now the function

The equilibrium states

To study the nonlinear stability of the equilibrium state

Now, the first variation of

If we choose now the function

As we have proved in the previous section, the equilibrium states

Near to

Indeed, we have successively the following.

The reduction of the system (

The matrix of the linear part of the reduced dynamics has purely imaginary roots. More exactly,

One has span

The equilibrium state

Similar arguments lead us to the following result.

Near to

The dynamics (

Let us take the following.

Then, using MATHEMATICA 8.0, we can put the system (

We will discuss now the numerical integration of the dynamics (

Their corresponding integral curves are, respectively, given by

and

Then the Lie-Trotter integrator is given by

The following proposition sketches the Lie-Trotter integrator properties.

The Lie-Trotter integrator (

It preserves the Poisson structure

It preserves the Casimir

It does not preserve the Hamiltonian

Its restriction to the coadjoint orbit

We will discuss now the numerical integration of the dynamics (

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 (

The Kahan integrator (

As we can see from Figure

Runge-Kutta 4 steps, Lie-Trotter, and Kahan integrator, respectively (

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 [

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.