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]:(1)x˙1=-x2x3+bx2x5,x˙2=x1x3-bx1x5,x˙3=-x1x2,x˙4=-x5,x˙5=x4+bx1x2,where b∈R.

Recall that, for system (1), the functions H,C∈C∞(R5,R),(2)Hx1,x2,x3,x4,x5=12x12+2x22+x32+x42+x52,(3)Cx1,x2,x3,x4,x5=12x12+x22are constants of motion. The functions H and C 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(4)π=00-x20bx200x10-bx1x2-x10000000-1-bx2bx1010, system (1) has the Hamiltonian form [8]:(5)x˙=x,H, where the Hamiltonian H is given by (2). Hence (R5,π,XH) is a Hamilton-Poisson realization of dynamics (1), where(6)XH=-x2x3+bx2x5,x1x3-bx1x5,-x1x2,-x5,x4+bx1x2. It is easy to see that the function C 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:(7)E1=M,0,0,0,0∣M∈R,E2=0,M,0,0,0∣M∈R∗,E3=0,0,M,0,0∣M∈R∗. Let eM1=(M,0,0,0,0)∈E1. Considering the function L∈C∞(R5,R),(8)Lx1,x2,x3,x4,x5=x22+x32+x42+x52+x12+x22-M22,we have(9)LeM1=0,Lx1,x2,x3,x4,x5>0,∀x1,x2,x3,x4,x5∈V∖eM1,L˙x1,x2,x3,x4,x5=2x2x˙2+2x3x˙3+2x4x˙4+2x5x˙5+4x12+x22-M2x1x˙1+x2x˙2=0,∀x1,x2,x3,x4,x5∈V,for some neighbourhood V of eM1. By [19, 20], we deduce that all the equilibrium states from the family E1 are nonlinearly stable.

The characteristic polynomial associated with the linear part of system (1) at the equilibrium eM2=(0,M,0,0,0), M≠0, is given by(10)peM2λ=-λλ4-M2+b2M2-1λ2-M2. We notice that a root of pEM2 is strictly positive, whence eM2 is an unstable equilibrium state. Therefore, all the equilibrium states from the family E2 are unstable.

Let eM3∈E3. The roots of the characteristic polynomial associated with the linear part of system (1) at eM3 are(11)λ1=0,λ2,3=±i,λ4,5=±iM. Hence all the equilibrium states from the family E3 are spectrally stable.

Now, we study the existence of periodic orbits of system (1) around the equilibrium states from the family E1∖{(0,0,0,0,0)}.

Since the eigenvalues of the linear part of system (1) at the equilibrium eM1=(M,0,0,0,0), M≠0, are(12)λ1=0,λ2,3=±i-y1,λ4,5=±i-y2,where y1 and y2 are the roots of the equation(13)y2+M2+b2M2+1y+M2=0,we apply Theorem 2.1 from [21]. The eigenspace corresponding to the eigenvalue λ1=0 has one dimension. Taking the constant of motion I:R5→R,(14)Ix1,x2,x3,x4,x5=x22+x32+x42+x52,it follows that(15)dIeM1=0,d2IeM1W×W>0,where(16)W=kerdCeM1=SpanR0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1. Therefore, for each sufficiently small ɛ∈R+∗, any integral hypersurface(17)ΣɛeM1:x22+x32+x42+x52=ɛ2contains at least one periodic orbit of system (1) whose period is close to 2π/-y1 and at least one periodic orbit of system (1) whose period is close to 2π/-y2.

In the case of the equilibrium states from E3, we cannot apply the above method. On the other hand the dynamics of system (1) are carried out at the intersection of the hypersurfaces(18)Hx1,x2,x3,x4,x5=H0,0,M,0,0,Cx1,x2,x3,x4,x5=C0,0,M,0,0;that is,(19)x32+x42+x52=M2,x12+x22=0.Then the solution of system (1) is(20)x1=0,x2=0,x3=±M2-k12-k22,x4=k1cost+k2sint,x5=k1sint-k2cost,where k1,k2∈R, k12+k22≤M2. We remark that (20) represents periodic orbits around equilibrium state (0,0,M,0,0), M∈R∗ (see Figure 1).

Periodic orbits around equilibrium state (0,0,M,0,0) (3D view in the x3-x4-x5 space).

3. 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 (R5,π,XH) has a full symplectic realization (T∗R3≃R6,ω,XH~), where (21)ω=dp1∧dq1+dp2∧dq2+dp3∧dq3,H~=12p12+p22+p32+12q2-bp12+12p32sin2q1,and the corresponding Hamiltonian vector field is as follows:(22)XH~=1+b2p1-bq2∂∂q1+p2∂∂q2+p3+p3sin2q1∂∂q3-p32sinq1cosq1∂∂p1+bp1-q2∂∂p2.

Proof.

Using the Hamiltonian H~ one obtains the corresponding Hamilton’s equations:(23)q˙1=1+b2p1-bq2,q˙2=p2,q˙3=p3+p3sin2q1,p˙1=-p32sinq1cosq1,p˙2=bp1-q2,p˙3=0.We consider the mapping φ:R6→R5,(24)φq1,q2,q3,p1,p2,p3=p3cosq1,p3sinq1,p1,p2,q2-bp1=x1,x2,x3,x4,x5. Using the standard symplectic bracket(25)f,gω=∑i=13∂f∂qi·∂g∂pi-∂f∂pi·∂g∂qi,one gets(26)x1,x2ω=0,x1,x3ω=-x2,x1,x4ω=0,x1,x5ω=bx2,x2,x3ω=x1,x2,x4ω=0,x2,x5ω=-bx1,x3,x4ω=0,x3,x5ω=0,x4,x5ω=-1.Hence the canonical structure {·,·}ω is mapped onto the Poisson structure π.

Taking into account relations (23), we have(27)x˙1=-p3sinq1·q˙1=-x2x3+bx2x5,x˙2=p3cosq1·q˙1=x1x3-bx1x5,x˙3=p˙1=-x1x2,x˙4=p˙2=-x5,x˙5=q˙2-bp˙1=x4+bx1x2.Therefore the Hamiltonian vector field XH~ is mapped onto the Hamiltonian vector field XH. Moreover φ is a surjective submersion and H∘φ=H~, which finishes the proof.

Denoting C~≔C∘φ, it follows that C~=p3.

The next result states that system (23) can be written in Lagrangian formalism.

Theorem 2.

System (23) has the form(28)q¨1+1+b2sinq1cosq11+sin2q12q˙32+bq˙2=0,q¨2-b1+b2q˙1+11+b2q2=0,q¨3-2sinq1cosq11+sin2q1q˙1q˙3=0,on the tangent bundle TR3. Also, system (28) represents the Euler-Lagrange equations generated by the Lagrangian(29)L=121+b2q˙12+12q˙22+121+sin2q1q˙32+b1+b2q˙1q2-121+b2q22.

Proof.

By Hamilton’s equations (23) we obtain(30)p1=11+b2q˙1+b1+b2q2p2=q˙2p3=11+sin2q1q˙3,whence(31)p˙1=11+b2q¨1+b1+b2q˙2,p˙2=q¨2,p˙3=-2sinq1cosq11+sin2q12q˙1q˙3+11+sin2q1q¨3.Substituting p1, p2, p3, p˙1, p˙2, p˙3 into (23), one gets (28). For the Lagrangian L given by (29), the Euler-Lagrange equations,(32)ddt∂L∂q˙i-∂L∂qi=0,i∈1,2,3,have the form (28). The relation between the Hamiltonian H~ and the Lagrangian L,(33)H~=∑i=13piq˙i-L,where(34)pi=∂L∂q˙i,i∈1,2,3,follows by relations (23).

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(35)v=ξq1,q2,q3,t∂∂t+η1q1,q2,q3,t∂∂q1+η2q1,q2,q3,t∂∂q2+η3q1,q2,q3,t∂∂q3is a Lie-point symmetry for Euler-Lagrange equations if the action of its second prolongation on these equations vanishes. For more details about symmetries see, for example, [24–26].

Applying the second prolongation of v,(36)pr2v=v+∑i=13η˙i-ξ˙q˙i∂∂q˙i+∑i=13η¨i-ξ¨q˙i-2ξ˙q¨i∂∂q¨i,on (28) one obtains(37)η¨1-ξ¨q˙1-2q¨1ξ˙+1+b21-5sin2q1+2sin4q11+sin2q13q˙32η1+bη˙2-ξ˙q˙2+21+b2sinq1cosq11+sin2q12η˙3-ξ˙q˙3q˙3=0,η¨2-ξ¨q˙2-2q¨2ξ˙+11+b2η2-b1+b2η˙1-ξ˙q˙1=0,η¨3-ξ¨q˙3-2q¨3ξ˙+23sin2q1-11+sin2q12η1q˙1q˙3-2sinq1cosq11+sin2q1η˙1-ξ˙q˙1q˙3-2sinq1cosq11+sin2q1η˙3-ξ˙q˙3q˙1=0. The resulting equations obtained by expanding ξ˙, ξ¨, η˙1, η¨1, η˙2, η¨2, η˙3, η¨3 and replacing q¨1, q¨2, and q¨3 must be satisfied identically in t, q1, q2, q3, q˙1, q˙2, q˙3, which are all independent variables.

In the case b≠0, it follows that(38)ξq1=ξq2=ξq3=0,η1,t=η1,q2=η1,q3=0,η1,q1q1=0,η2,q1=η2,q3=0,η2,q2q2=0,η3,t=η3,q1=η3,q2=0,η3,q3q3=0,η2+1+b2η2,tt-q2η2,q2+2q2ξt=0,η1,q1+ξt-η2,q2=0,2η2,tq2-ξtt=0,bη2,t=0,bη2,q2+ξt-η1,q1=0,1-5sin2q1+2sin4q11+sin2q1η1+sinq1cosq12η3,q3-η1,q1=0,6sin2q1-21+sin2q1η1-2sinq1cosq1η1,q1=0. The last relation implies η1=0. It results in(39)ξ=α,η1=0,η2=0,η3=β,α,β∈R.

In the case b=0, it follows that(40)ξq1=ξq2=ξq3=0,η1=0,η2,q1=η2,q3=0,η2,q2q2=0,η2,tq2=0,η3,t=η3,q1=η3,q2=η3,q3=0,η2+η2,tt-q2η2,q2+2q2ξt=0. It results in(41)ξ=at+b,η2=cq2+ft,a,b,c∈R. Then a=0 and f′′(t)+f(t)=0. Therefore(42)ξ=α,η1=0,η2=γq2+δcost+θsint,η3=β,α,β,γ,δ,θ∈R.

We can conclude the following result.

Theorem 3.

The symmetries of (28) are given by(43)v=α∂∂t+β∂∂q3,where α,β∈R, in the case b∈R∗, respectively, and(44)u=α∂∂t+γq2+δcost+θsint∂∂q2+β∂∂q3,where α,β,γ,δ,θ∈R, in the case b=0.

Remark 4.

Let γ=δ=θ=0. Denoting v1=∂/∂t and v2=∂/∂q3, it follows that v1, v2 are variational symmetries. Moreover,

(i) for β=0 and α≠0, we have v=αv1 that represents the time translation symmetry which generates the conservation of energy H~;

(ii) for α=0 and β≠0, we have v=βv2 that represents a translation in the cyclic q3 direction which is related to the conservation of p3.

We notice that the vector field u leads to the vector field(45)X=α∂∂t+γx4-δsint+θcost∂∂x4+γx5+δcost+θsint∂∂x5.Also, we can consider the vector field(46)Y=α∂∂t+γx5-δsint+θcost∂∂x4+-γx4+δcost+θsint∂∂x5.

The last result furnishes some symmetries of system (1) in the case b=0.

Proposition 5.

The vector field X given by (45) is a Lie-point symmetry of system (1) in the case b=0. Also, if δ=θ=0, then X is a symmetry of system (1) in the case b=0. Moreover, the vector field Y given by (46) has the same properties.

Proof.

It is easy to see that the action of the first prolongation of X on (1) in the case b=0 vanishes. Therefore X is a Lie-point symmetry.

Considering δ=θ=0, it immediately follows that(47)∂X∂t+X,Xf=0,where(48)Xf=-x2x3,x1x3,-x1x3,-x5,x4,whence X is a symmetry of system (1) in the case b=0.

4. 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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