Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 438694, 7 pages

http://dx.doi.org/10.1155/2015/438694

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

Department of Mathematics, Politehnica University of Timişoara, Piaţa Victoriei No. 2, 300006 Timişoara, Romania

Received 11 August 2015; Revised 3 October 2015; Accepted 5 October 2015

Academic Editor: Yan-Wu Wang

Copyright © 2015 Cristian Lăzureanu and Tudor Bînzar. 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

The 5-dimensional Lorenz system for the gravity-wave activity is considered. Some stability problems and the existence of periodic orbits are studied. Also, a symplectic realization and some symmetries are given.

#### 1. 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 [2–5]. 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]. 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 [14–17].

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.

#### 2. Stability and Periodic Orbits

We consider the 5-dimensional Lorenz system [1]:where .

Recall that, for system (1), the functions ,are constants of motion. The functions and are linearly related to analogs of the energy and, respectively, enstrophy of the nine-component “primitive equations” model introduced by Lorenz [1, 8].

Considering the matrix formulation of the Poisson bracket , given in coordinates by system (1) has the Hamiltonian form [8]: where the Hamiltonian is given by (2). Hence 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 . Considering the function ,we have By [19, 20], we deduce that all the equilibrium states from the family are nonlinearly stable.

The characteristic polynomial associated with the linear part of system (1) at the equilibrium , , is given by We notice that a root of is strictly positive, whence is an unstable equilibrium state. Therefore, all the equilibrium states from the family are unstable.

Let . The roots of the characteristic polynomial associated with the linear part of system (1) at are Hence all the equilibrium states from the family 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 , , arewhere and are the roots of the equationwe apply Theorem from [21]. The eigenspace corresponding to the eigenvalue has one dimension. Taking the constant of motion ,it follows thatwhere Therefore, for each sufficiently small , any integral hypersurfacecontains at least one periodic orbit of system (1) whose period is close to and at least one periodic orbit of system (1) whose period is close to .

In the case of the equilibrium states from , we cannot apply the above method. On the other hand the dynamics of system (1) are carried out at the intersection of the hypersurfacesthat is,Then the solution of system (1) iswhere , . We remark that (20) represents periodic orbits around equilibrium state , (see Figure 1).