On Some Properties and Symmetries of the 5-Dimensional Lorenz System

The importance of the 5-dimensional Lorenz system [1] in the study of geophysical fluid dynamics is well known. This system describes coupled Rossby waves and gravity waves. It wasmainly investigated from the existence of a slowmanifold point of view [2–5]. Among other studies regarding 5-dimensional Lorenz system we mention Hamiltonian structure [6], chaotic behaviour [7–9], and analytic integrability [10]. According to [10], the 5-dimensional Lorenz system has at most three functionally independent global analytic first integrals. We mention that two first integrals are known [1]. It raises the following question: how can the third first integral be determined, provided that it exists? A possible answer is given by the connection between symmetries and the existence of conservative laws [11]. Ourmain purpose is to try to determine the third first integral using symmetries.This attempt was successful in the case of 5-dimensionalMaxwellBloch equations with the rotating wave approximation [12]. “Intuitively speaking, a symmetry is a transformation of an object leaving this object invariant” [13]. In our case, a transformation means a vector field and an object means a differential equation. Recently, this field is widely investigated. We refer to some new progress [14–17]. In our paper, the constants of motion of the 5-dimensional Lorenz system are used to study some stability problems and the existence of periodic orbits. “The stability of an orbit of a dynamical system characterizes whether nearby (i.e., perturbed) orbits will remain in a neighborhood of that orbit or be repelled away from it” [18]. Also, with the aid of these constants of motion, a symplectic realization and a Lagrangian formulation are given. In the last part of our work some symmetries are pointed out.


Introduction
The importance of the 5-dimensional Lorenz system [1] in the study of geophysical fluid dynamics is well known. This system describes coupled Rossby waves and gravity waves. It was mainly investigated from the existence of a slow manifold point of view [2][3][4][5]. Among other studies regarding 5-dimensional Lorenz system we mention Hamiltonian structure [6], chaotic behaviour [7][8][9], and analytic integrability [10].
According to [10], the 5-dimensional Lorenz system has at most three functionally independent global analytic first integrals. We mention that two first integrals are known [1]. It raises the following question: how can the third first integral be determined, provided that it exists? A possible answer is given by the connection between symmetries and the existence of conservative laws [11]. Our main purpose is to try to determine the third first integral using symmetries. This attempt was successful in the case of 5-dimensional Maxwell-Bloch equations with the rotating wave approximation [12]. "Intuitively speaking, a symmetry is a transformation of an object leaving this object invariant" [13]. In our case, a transformation means a vector field and an object means a differential equation. Recently, this field is widely investigated. We refer to some new progress [14][15][16][17].
In our paper, the constants of motion of the 5-dimensional Lorenz system are used to study some stability problems and the existence of periodic orbits. "The stability of an orbit of a dynamical system characterizes whether nearby (i.e., perturbed) orbits will remain in a neighborhood of that orbit or be repelled away from it" [18]. Also, with the aid of these constants of motion, a symplectic realization and a Lagrangian formulation are given. In the last part of our work some symmetries are pointed out.

Stability and Periodic Orbits
We consider the 5-dimensional Lorenz system [1]: where ∈ R.
Recall that, for system (1), the functions , ∈ C ∞ (R 5 , R), system (1) has the Hamiltonian form [8]: where the Hamiltonian is given by (2). Hence (R 5 , , ) is a Hamilton-Poisson realization of dynamics (1), where It is easy to see that the function is a Casimir for the above Poisson bracket. In the following we study the stability of system (1). The equilibrium states of system (1) are given as the union of the following families:    Let 1 = ( , 0, 0, 0, 0) ∈ E 1 . Considering the function ∈ C ∞ (R 5 , R), we have = 2 22 + 2 33 + 2 44 + 2 55 for some neighbourhood of 1 .
By [19,20], we deduce that all the equilibrium states from the family E 1 are nonlinearly stable. The characteristic polynomial associated with the linear part of system (1) at the equilibrium 2 = (0, , 0, 0, 0), ̸ = 0, is given by We notice that a root of 2 is strictly positive, whence 2 is an unstable equilibrium state. Therefore, all the equilibrium states from the family E 2 are unstable. Let 3 ∈ E 3 . The roots of the characteristic polynomial associated with the linear part of system (1) at 3 are Hence all the equilibrium states from the family E 3 are spectrally stable. Now, we study the existence of periodic orbits of system (1) around the equilibrium states from the family Since the eigenvalues of the linear part of system (1) at the equilibrium 1 = ( , 0, 0, 0, 0), ̸ = 0, are where 1 and 2 are the roots of the equation we apply Theorem 2.1 from [21]. The eigenspace corresponding to the eigenvalue 1 = 0 has one dimension. Taking the constant of motion : R 5 → R, it follows that Therefore, for each sufficiently small ∈ R * + , any integral hypersurface contains at least one periodic orbit of system (1) whose period is close to 2 /√− 1 and at least one periodic orbit of system (1) whose period is close to 2 /√− 2 .
In the case of the equilibrium states from E 3 , we cannot apply the above method. On the other hand the dynamics of system (1) are carried out at the intersection of the hypersurfaces that is, Then the solution of system (1) is We remark that (20) represents periodic orbits around equilibrium state (0, 0, , 0, 0), ∈ R * (see Figure 1).

Symplectic Realization and Symmetries
First result shows that system (1) can be regarded as a Hamiltonian mechanical system.

Theorem 1. The Hamilton-Poisson mechanical system
) has a full symplectic realization ( * R 3 ≃ R 6 , ,̃), where and the corresponding Hamiltonian vector field is as follows: Proof. Using the Hamiltoniañone obtains the corresponding Hamilton's equations: We consider the mapping : R 6 → R 5 , Using the standard symplectic bracket one gets Hence the canonical structure {⋅, ⋅} is mapped onto the Poisson structure . Taking into account relations (23), we havė Therefore the Hamiltonian vector field̃is mapped onto the Hamiltonian vector field . Moreover is a surjective submersion and ∘ =̃, which finishes the proof.
For details about Lagrangian and Hamiltonian formalism see, for example, [22,23].
In the sequel we study the Lie-point symmetries for Euler-Lagrange equations (28).
We recall that a vector field is a Lie-point symmetry for Euler-Lagrange equations if the action of its second prolongation on these equations vanishes.
In the case ̸ = 0, it follows that , ∈ R.
Remark 4. Let = = = 0. Denoting k 1 = / and k 2 = / 3 , it follows that k 1 , k 2 are variational symmetries. Moreover, (i) for = 0 and ̸ = 0, we have k = k 1 that represents the time translation symmetry which generates the conservation of energỹ; 6 Mathematical Problems in Engineering (ii) for = 0 and ̸ = 0, we have k = k 2 that represents a translation in the cyclic 3 direction which is related to the conservation of 3 .
We notice that the vector field u leads to the vector field Also, we can consider the vector field The last result furnishes some symmetries of system (1) in the case = 0. (45) is a Lie-point symmetry of system (1) in the case = 0. Also, if = = 0, then X is a symmetry of system (1) in the case = 0. Moreover, the vector field Y given by (46) has the same properties.

Proposition 5. The vector field X given by
Proof. It is easy to see that the action of the first prolongation of X on (1) in the case = 0 vanishes. Therefore X is a Liepoint symmetry.

Conclusions
In this paper the 5-dimensional Lorenz system is considered. This is a system of five differential equations which couples the Rossby waves and gravity waves. In Section 2 some stability problems and the existence of periodic orbits are studied. The equilibrium states of considered system are given as the union of three families of points. For one of these families, all the equilibria are spectrally stable, but it remains an open problem to establish if these equilibria are nonlinearly stable. In the third part of the paper a symplectic realization and the corresponding Lagrangian formulation are given. In the last part of our work, some symmetries of the 5-dimensional Lorenz system are studied. Knowing the connection between symmetries and conservative laws, we tried to determine a third first integral of the considered system, provided that it exists.