# Dynamical Analysis of the Lorenz-84 Atmospheric Circulation Model

Authors: Hu Wang; Yongguang Yu; Guoguang Wen
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

The dynamical behaviors of the Lorenz-84 atmospheric circulation model are investigated based on qualitative theory and numerical simulations. The stability and local bifurcation conditions of the Lorenz-84 atmospheric circulation model are obtained. It is also shown that when the bifurcation parameter exceeds a critical value, the Hopf bifurcation occurs in this model. Then, the conditions of the supercritical and subcritical bifurcation are derived through the normal form theory. Finally, the chaotic behavior of the model is also discussed, the bifurcation diagrams and Lyapunov exponents spectrum for the corresponding parameter are obtained, and the parameter interval ranges of limit cycle and chaotic attractor are calculated in further. Especially, a computer-assisted proof of the chaoticity of the model is presented by a topological horseshoe theory.


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## Body

## 1. Introduction

Atmospheric models provided an excellent instrument for complex dynamical behaviors which can be observed in natural science. They involve processes occurring over a wide spectrum of space scales and time scales, from the chemistry of minor constituents in the stratosphere to hurricanes, droughts, or the Quaternary glaciations, and give rise to a variety of intricate behaviors in the form of abrupt transitions, wave propagation, weak chaos, or fully developed turbulence [1–7]. The generally accepted approaches to study the atmospheric and climate dynamics are based on numerical forecasting models, in which all processes deemed to be relevant are included. As for the low-order atmospheric models, they involve a large number equations. Although it is unreasonable to expect solutions to low-dimensional problems to generalize to a million-dimensional spaces, it is unlikely too that problems identified in the simplified models will vanish in operational models [8]. It is equally important to note recent results which indicate the possibility that high-dimensional models may behave in a smooth way with respect to changes in parameter values [9–11]. Thus, low-order models may well have little to do with higher-dimensional operational models.On the other side, the earth’s atmosphere is in constant circulation due to the earth’s atmosphere being heated by the sun and the earth’s rotation. A succession of the heating of the air near the earth’s surface, rising, and cool air coming down sets up a general circulation pattern: air rises near the equator, moves north and south away from the equator at higher altitudes, sinks down near the poles, and flows back along the surface from both poles to the equator [12]. This important type of flow is called Hadley circulation that was first named after Hadley [13]. There is some evidence that the expansion of the Hadley circulation is related to climate change [14]. The majority of earth’s driest and arid regions are located in the areas underneath the descending branches of the Hadley circulation around 30 degrees latitude. Both idealised and more realistic climate model experiments show that the Hadley circulation expands with increased global mean temperature; this can lead to large changes in precipitation in the latitudes at the edge of the cells [15]. Scientists fear that the ongoing presence of global warming might bring changes to the ecosystems in the deep tropics and that the deserts will become drier and expand [16]. Based on the above discussion, the Hadley circulation is very important to the atmospheric science. Furthermore, the stable and unstable atmospheric circulations are closely linked with the dramatic changes and persistent abnormalities of the weather. Therefore, it is very important to research the stable and unstable atmospheric circulation for meteorological phenomena.A very appealing low-order model of atmospheric circulation is introduced by Lorenz in 1984 [17], which is called Lorenz-84 atmospheric circulation model (Lorenz-84 model). The Lorenz-84 model involves just three ordinary differential equations, and it includes some important features of Hadley circulation. So far, this model was known to have a pair of coexisting climates described originally by Lorenz [18]. Due to the importance of Lorenz-84 model, it has received great attention from researchers, and many important results on Lorenz-84 model have also been obtained [19–23]. In 1995, Shil’nikov et al. discussed the bifurcation and predictability of the Lorenz-84 model [19]. Soon afterward, Broer et al. studied the bifurcations and strange attractors in the Lorenz-84 model with seasonal forcing [20]. Van et al. [21] and Roebber [22] investigated the dynamical behaviors of a low-order coupled ocean-atmospheric model. It is well known that the synoptic atmospheric dynamics over the North Atlantic ocean can be dominated by the jet stream, a westerly circulation, and baroclinic waves, which transport heat and momentum northward. Based on this fact, Kuznetsov et al. considered the intensity of the jet stream and discussed the fold-flip bifurcation in the Lorenz-84 model [23]. For some more detailed investigations for the Lorenz-84 model, the interested reader could also see [24–29]. However, the most important results are mainly based on numerical simulations in [19, 20, 22, 28].This paper aims to further investigate the dynamical behaviors of the Lorenz-84 model by theoretical analysis. Some stability conditions, supercritical, and subcritical Hopf bifurcations are obtained by using the qualitative analysis method. Moreover, bifurcation analysis of a nonlinear dynamical system throws useful light on the behavior of the system in different parameter ranges. Generally, equilibrium points play an important role in governing the overall system behavior. It is therefore useful to consider the mathematical expressions of equilibrium points as a function of system parameters. However, it is so difficult to obtain the equilibrium points regarded as the explicit mathematical expressions about the parameters for the Lorenz-84 model. In this paper, one component of equilibrium point is regarded as a parameter, and others are considered as its functions. In this way, it is not necessary to know what kind of the equilibrium point it is, and the stability conditions and the direction of the Hopf bifurcation are still obtained. In addition, there are many important results about the chaotic behaviors of the Lorenz-84 model [17–20, 22, 28], in which the chaotic behaviors of the Lorenz-84 model were studied by considering only one parameter. In this paper, the chaotic behaviors of the Lorenz-84 model are studied by considering every parameter in the model. Furthermore, the topological horseshoe is given in the Lorenz-84 model, which provides a powerful tool in the rigorous study of chaos. Finally, some similar dynamical behaviors of the Lorenz-84 model under different parameters are founded, which will be very useful for discussing the codimension-n (n≥2) bifurcation or other nonlinear phenomena.This paper is arranged as follows. In Section2, the stability and bifurcation of the model are discussed, and the conditions of stability and bifurcation are also given. Especially, the Hopf bifurcation is discussed, and the conditions of the supercritical and subcritical bifurcation are derived. In addition, some numerical simulations are shown to verify our theoretical results and conditions. The chaotic behavior of Lorenz-84 model is researched in Section 3. The bifurcation diagrams and Lyapunov exponent spectrum for every parameter are discussed, and the parameter interval ranges of limit cycle and chaotic attractor of every parameter are calculated. Conclusions in Section 4 close the paper.


## 2. The Stability and Bifurcation Analysis of the Lorenz-84 Model

In this section, we mainly discuss the stability and local bifurcation of the Lorenz-84 model and obtain the stability conditions and direction of the Hopf bifurcation.


### 2.1. The Lorenz-84 Model

The Lorenz-84 model is a three-dimensional system [17] and is given by
(1)dxdt=-y2-z2-ax+aF,dydt=xy-bxz-y+G,dzdt=bxy+xz-z,
where x represents the strength of the globally averaged westerly current and y and z are the strength of the cosine and sine phases of a chain of superposed waves. The unit of the variable t is equal to the damping time of the waves that is estimated to be five days. F and G represent the thermal forcing terms, and the parameter b represents the advection strength of the waves by the westerly current. Hence, the equilibrium point of model (1) satisfies the following equation:
(2)-y2-z2-ax+aF=0,xy-bxz-y+G=0,bxy+xz-z=0.
That is,
(3)y=1-xG1-2x+1+b2x2,z=bxG1-2x+1+b2x2,aF-x1-2x+1+b2x2=G2.It is well known that when the parameterG=0, the dynamical behaviors of model (1) are simple which have been discussed in [24], while the parameter G≠0, the dynamical behaviors become complicated and display chaotic attractors in model (1). The objective of this paper is to discuss the stability and local bifurcation and to obtain the corresponding stability and bifurcation conditions about model (1) with G≠0. Note from (2) that it is difficult to obtain the explicit mathematical expression of the equilibrium points on the parameters a, b, F, and G. However, we can also view the variable x as a parameter. Then, we can get y=y(x,a,b,F,G), z=z(x,a,b,F,G) and x satisfies the equation a(F-x)(1-2x+(1+b2)x2)=G2, that are shown in (3).


### 2.2. The Stability and Bifurcation Analysis with ParametersG≠0 and b=0

WhenG≠0 and b=0, model (1) is given as
(4)dxdt=-y2-z2-ax+aF,dydt=xy-y+G,dzdt=xz-z.
Then, the equilibrium point of model (4) satisfies the following equation:
(5)-y2-z2-ax+aF=0,xy-y+G=0,xz-z=0.
From the third equation, in (5), we have (x-1)z=0. If x=1, then, from the second equation in (5), G=0, which is contradictory to the hypotheses G≠0 in this section. Hence, it implies z=0 and x≠1. Then, we can obtain an equilibrium point K(x0,G/1-x0,0) of model (1), where x0 satisfies the equation a(F-x)(1-x)2=G2 and x0≠1.Theorem 1.
WhenG≠0 and b=0, there exists at least one equilibrium point K(x0,G/1-x0,0) in model (1)(1)
ifa>0 and the equation a(F-x)1-x2=G2 has solution x0<1, the equilibrium point K is stable,(2)
ifa>0 and the equation a(F-x)1-x2=G2 has solution x0>1, the equilibrium point K is unstable,(3)
ifa<0 and the equation a(F-x)1-x2=G2 has solution 1--2G2/a3<x0<1+a, the equilibrium point K is stable.Proof.
The Jacobian matrix of model (1) at the point K(x0,G/1-x0,0) is given as
(6)J=-a-2y00y0x0-1000x0-1,
and its characteristic equation is given as
(7)fλ=λ+1-x0λ2+(a+1-x0)λhhhhhhhhhhhh+a+1x0+2y02.
Then, eigenvalues λ1, λ2, and λ3 are obtained as follows:
(8)λ1=x0-1,λ2=-(a+1-x0)+a-1+x02-8G2/1-x022,λ3=-(a+1-x0)-a-1+x02-8G2/1-x022.
If a>0 and the equation a(F-x)1-x2=G2 has solution x0<1, then there is λ1<0. Since 8G2/1-x02>0 and a+1-x02-a-1+x02=4(1-x0)a>0, it follows that Re(λ2)<0 and Re(λ3)<0. Therefore, the equilibrium point K is stable. If the solution x0>1, it is easy to verify that the equilibrium point K is unstable. If a<0 and the equation a(F-x)1-x2=G2 has solution 1--2G2/a3<x0<1+a, it follows that Re(λ1)<0, Re(λ2)<0, and Re(λ3)<0. Therefore, the equilibrium point K is stable. The proof is completed.Theorem 2.
IfG≠0 and b=0, then there exists at least one equilibrium point K(x0,G/1-x0,0) in model (1)(1)
whena>0 and 2G2=a4, if x0=a+1 is a solution of the equation a(F-x)1-x2=G2, the equilibrium point K(x0,G/1-x0,0) is unstable and model (1) has a fold bifurcation,(2)
whena<0 and 2G2>a4, if x0=a+1 is a solution of the equation a(F-x)1-x2=G2, the equilibrium point K(x0,G/1-x0,0) is unstable, and model (1) has a Hopf bifurcation.Proof.
It is easy to obtain the result from (8) and is therefore omitted here.


### 2.3. Direction and Stability of the Hopf Bifurcation

In this section, the supercritical and subcritical bifurcations of model (1) are considered. According to Theorem 2, model (1) has a Hopf bifurcation when x0=a+1, 2G2>a4, and a<0. Let the eigenvectors corresponding to the eigenvalues λ1 and λ3 be α1+iα2 and α3, respectively, where α1, α2, and α3 are all real vectors. Under the following linear transformation:
(9)x1=x-x0,y1=y-y0,z1=z-z0,
model (1) can be changed into
(10)x˙1=-y12-2y0y1-y02-z12+ax0-ax1+aF,y˙1=x1y1+x1y0+x0y1+x0y0-y1-y0+G,z˙1=x1z1+x0z1-z1.
For (6), (7), and (8), if x0=a+1 is a solution of the equation a(F-x)1-x2=G2,2G2>a4 and a<0, we have
(11)α′0=Reλ′a=12>0,ω′(0)=Im⁡(λ′(a))=4G2-a442G2-a4≠0,ω0=Im⁡λa=2G2-a4a>0,λ3<0.
Therefore, model (1) at the equilibrium point K has a Hopf bifurcation. By calculations, we have
(12)α1=100,α2=aKGaK0,α3=001,T=α2,α1,α3=ak10Gak00001,
where k=2G2-a4/a.For model (10), taking the following transformation x1y1z1=Tx2y2z2, we have
(13)x2~˙=Ax2~+B,
where(14)A=0-k0k0000a,x2~=x2y2z2,B=akx22+x2y2+akGx0y0-y0+G-G2+a4a2k2x22-akx2y2-z22-a2G+2Gakx0y0-ax0+a2Gx0+aF-a2akx2z2+y2z2,.From (13),
(15)x˙2=-ky2+Px2,y2,z2,y˙2=kx2+Qx2,y2,z2,z˙2=az2+Rx2,y2,z2,
where
(16)Px2,y2,z2=akx22+x2y2+akGx0y0-y0+G,Qx2,y2,z2=-G2+a4a2k2x22-akx2y2-z22-a2G+2Gakx0y0-ax0+a2Gx0+aF-a2,Rx2,y2,z2=akx2z2+y2z2.
For model (15), according to the normal form theory in [30, 31], we have
(17)g11=14∂P2∂x22+∂P2∂y22+i∂Q2∂x22+∂Q2∂y22=12ak-G2+a4a2k2i,g02=14∂P2∂x22-∂P2∂y22-2∂Q2∂x2∂y2hhhhh+i∂Q2∂x22-∂Q2∂y22+2∂P2∂x2∂y2=12(1-G2+a4a2k2)i,g20=14∂P2∂x22-∂P2∂y22+2∂Q2∂x2∂y2hhhhh+i∂Q2∂x22-∂Q2∂y22-2∂P2∂x2∂y2=12(-1-G2+a4a2k2)i,G21=18∂P3∂x23+∂P3∂x2∂y22+∂Q3∂x22∂y2+∂Q3∂y23hhhhh+i(∂Q3∂x23+∂Q3∂x2∂y22-∂P3∂x22∂y2-∂P3∂y23)=0.
Since the dimension n=3>2, we obtain the following equations:
(18)h11=14∂R2∂x22+∂R2∂y22=0,h20=14∂R2∂x22-∂R2∂y22-2∂R2∂x2∂y2=0.
Next, we solve the following equations:
(19)λ1ϕ11=-h11,λ1-2kiϕ20=-h20.
Then,
(20)ϕ11=0,ϕ20=0.
Let
(21)G110=12∂P2∂x2∂z2+∂Q2∂y2∂z2+i∂P2∂x2∂z2-∂Q2∂y2∂z2=0,G101=12∂P2∂x2∂z2-∂Q2∂y2∂z2+i∂P2∂x2∂z2+∂Q2∂y2∂z2=0.
Taking m=G2+a4/a2k2, it then follows from (17) and (21) that
(22)g21=G21+(2G110ϕ11+G101ϕ20)=0.
Also, let
(23)M10=i2kg20g11-2g112-13g022+12g21=m+1a8k2-i8k2a2k2+103m2+13m+13.
Then, we have
(24)μ2a,b=-ReM10α′0=-m+1a4k2>0,a<0,β2a,b=2ReM10=m+1a4k2<0,a<0,τ2a,b=-Im⁡M10+μ2a,bω′0b=-18k22a2k2+103m2+13m+13hhhhh-m+1a16k24G2-a442G2-a4<0.Theorem 3.
When2G2>a4,a<0, there is a Hopf bifurcation in model (1) at x0=a+1, and (1)
μ2(a,b)>0; that is to say, the direction of bifurcation is supercritical,(2)
β2(a,b)<0; that is to say, the solutions of bifurcating periodic solutions are orbitally stable,(3)
τ2(a,b)<0; that is to say, the periods of bifurcating periodic solutions increase.Proof.
It is easy to obtain the result from the above derivative process in Section2.3 and is therefore omitted here.


### 2.4. The Stability and Bifurcation Analysis with ParametersG≠0 and b≠0

In this section, we consider the stability and bifurcation analysis of model (1) when G≠0 and b≠0. Without loss of generality, we first discuss the stability and bifurcation analysis of model (1) at two special equilibrium points when x=0 and x=1. It is easy to verify that when x=0 and x=1, the equilibrium points of model (1) are K1(0,G,0) and K2(1,0,G/b), respectively.In the following, let us consider the stability and bifurcation analysis of model (1) at the equilibrium point K1.Theorem 4.
For model (1), if a>0, the equilibrium point K1 is stable, and(1)
whena≥1+22G>0 or 0<a≤1-22G, the equilibrium point K1 is stable node point,(2)
when0<1-22G<a<1+22G, the equilibrium point K1 is stable focus point.Proof.
For model (1), the Jacobian matrix at the equilibrium point K1 is
(25)J=-a-2G0G-10bG0-1,
and its characteristic equation is
(26)fλ=λ+1λ2+a+1λ+a+2G2.
We can get λ1,2,3(27)λ1=-1,λ2,3=-a+1±a+12-4a+2G22.
Since a>0, it then follows from (27) that Re(λ1,2,3)<0. Therefore, the equilibrium point K1 is stable. In the same time, note that if Δ=a+12-4(a+2G2)>0, λ1,2,3<0 are real numbers, and it then follows that the equilibrium point K1 is stable node point; if Δ=a+12-4(a+2G2)<0, λ2,3 has conjugate imaginary roots, it then follows that the equilibrium point K1 is stable focus point. The proof is completed.Theorem 5.
For model (1), if a<0 and(1)
whena=-1, F<-1/2 and G2>1/2, then model (1) at the equilibrium point K1 has a Hopf bifurcation,(2)
whena=-1, F=-2 and G2=2, then model (1) at the equilibrium point K1 has a Fold bifurcation,(3)
whena=-1, F=-1/2, then model (1) at the equilibrium point K1 has a flip bifurcation.Proof.
According to (27), we can easily get the conclusion; therefore, we omit it here.Next, let us consider the equilibrium pointK2(1,0,G/b).Theorem 6.
Ifa>0 and F<3/2, then the equilibrium point K2 of model (1) is stable.Proof.
The Jacobian matrix of model (1) at the equilibrium point K2 is
(28)J=-a-2G-2GbG0b-Gbb0,
and its characteristic equation is
(29)fλ=λ3+aλ2+b2+2G2b2λ+ab2-2G2.
Let A=a, B=b2+2G2/b2, and C=ab2-2G2. By the Routh-Hurwitz criterion, when A>0, C>0 and AB-C>0, the eigenvalues of (29) have negative real parts. Therefore, we can get AB-C=2G2(1+a/b2)>0, C>0, and F<3/2. Therefore, the equilibrium point K2 of model (1) is stable. The proof is completed.In the above discussions, we consider the stability and bifurcation analysis of model (1) at two special equilibrium points K1 and K2, respectively. In the following, we will discuss stability and bifurcation analysis of model (1) at the general equilibrium point K(x0,y0,z0).Theorem 7.
SupposeK(x0,y0,z0) is an equilibrium point in model (1), if x0 is a solution of the equation a(F-x)(1-2x+(b2+1)x2)=G2 and satisfies the inequations
(30)x0<1+a2,x02+4+2F+2Fb23(1+b2)x0+1+2F3(1+b2)>0,x03-5a+2b2-ab2+62(1+b2)x02+9a+6+aF+3a2-ab2F2(1+b2)x0-(1+a)(2+2a+aF)2(1+b2)<0,
then the equilibrium point K is stable.Proof.
From (2),
(31)y0=g(x0,b)(1-x0)G,z0=gx0,bx0bG,aF-xfx0,b=G2,
where f(x0,b)=1-2x0+(b2+1)x02 and g(x0,b)=1/f(x0,b). The Jacobian matrix of model (1) at the point K(x0,y0,z0) is given as(32)J(0)=-a-2g(x0,b)(1-x0)G-2g(x0,b)bx0Gg(x0,b)(1-x0-b2x02)Gx0-1-bx0bg(x0,b)Gbx0x0-1.Taking Gg(x0,b)=T, we can get
(33)fλ=λ3+2-2x0+aλ2+2a1-x0+fx0,b1+2T2λ+fx0,ba+2T21-x0-b2x0.
Let A=2-2x0+a, B=2a(1-x0)+f(x0,b)(1+2T2), and C=f(x0,b)(a+2T2(1-x0-b2x0)).
Note that thef(x0,b)>0, g(x0,b)>0, T2>0, and
(34)x0<1+a2,x02+4+2F+2Fb231+b2x0+1+2F31+b2>0,x03-5a+2b2-ab2+621+b2x02+9a+6+aF+3a2-ab2F21+b2x0-1+a2+2a+aF21+b2<0,
and we have A>0, C>0, and AB-C>0. According to the Routh-Hurwitz criterion, the real parts of the roots of (33) are all negative. Therefore, the equilibrium point K is stable. The proof is completed.Theorem 8.
Suppose thatK(x0,y0,z0) is an equilibrium point in model (1). If x0 is a solution of the equation a(F-x)(1-2x+(b2+1)x2)=G2 and satisfies the following inequations and equation:
(35)x0<1+a2,x02+4+2F+2Fb231+b2x0+1+2F31+b2>0,x03-5a+2b2-ab2+621+b2x02+9a+6+aF+3a2-ab2F21+b2x0-1+a2+2a+aF21+b2=0,β1x04-β2x03+β3x02+β4x0+β5≠0,
then model (1) has a Hopf bifurcation, where
(36)β1=4a21-b4G2,β2=4a2G22+2b2+a+ab2+2F-2Fb4,β3=4a2F21-b4G2+8Fa22+2b2+a+ab2G2β3=+3b2+1,β4=4ab2-14a-8-4b2-4a2F2+2b2+a+ab2G2,β5=12a+2aF+2ab2F+4a2+3.Proof.
From (33), we can obtain A=2-2x0+a, B=2a(1-x0)+f(x0,b)(1+2T2), and C=f(x0,b)(a+2T2(1-x0-b2x0)).
Since(37)x0<1+a2,x02+4+2F+2Fb231+b2x0+1+2F31+b2>0,x03-5a+2b2-ab2+621+b2x02+9a+6+aF+3a2-ab2F21+b2x0-1+a2+2a+aF21+b2=0,
it follows that A(x0)>0, B(x0)=C(x0)/A(x0)>0, and C(x0)>0. Similarly, since β1x04-β2x03+β3x02+β4x0+β5≠0, it follows that C′(x0)≠A′(x0)B(x0)-A(x0)B′(x0), where
(38)β1=4a21-b4G2,β2=4a2G22+2b2+a+ab2+2F-2Fb4,β3=4a2F21-b4G2+8Fa22+2b2+a+ab2G2β3=+3b2+1,β4=4ab2-14a-8-4b2β3=-4a2F2+2b2+a+ab2G2,β5=12a+2aF+2ab2F+4a2+3.
Hence, we have proven that A(x0)>0, B(x0)=C(x0)/A(x0)>0, C(x0)>0, and C′(x0)≠A′(x0)B(x0)-A(x0)B′(x0). Therefore, according to the Hopf bifurcation theorem [30], model (1) has a Hopf bifurcation. The proof is completed.


### 2.5. Simulation

In this section, some numerical examples and simulations are presented to illustrate the effectiveness of our theoretical results. Here, we mainly discuss and verify the conditions of Theorems1, 2, and 3.Firstly, we verify the effectiveness of Theorem1 with a>0. Taking a=0.25, F=8, then the curves of a(F-x)1-x2=G2 and straight line of x=1 are shown in Figure 1(a), in which x-axis represents the parameter G and y-axis represents x values of the equilibrium point K(x,G/1-x,0). Note that when a=0.25, F=8, and G=3, there are three equilibrium points A=(-1,1.5,0), B=(7,-0.5,0), and C=(4,-1,0), which are shown in Figure 1(a). From Figure 1(b), the converging trajectories for points B and C are Heteroclinic orbits, and they will converge to the equilibrium point A. Hence, the equilibrium posits B, C are unstable. Take an arbitrary point D (see Figure 1) under the conditions a=0.25, F=8, and G=3. It is shown from Figure 1(b) that the trajectory starting from D converges to A. Therefore, we have verified that the point A is stable.(a) The curve ofa(F-x)1-x2=G2. (b) Phase plane of system (1) with the parameter G=3.
(a)
(b)In addition, we considera<0 in Theorem 1. Choose a=-0.25, F=-6, and then the curves of a(F-x)1-x2=G2 and straight line of x=1 are shown in Figure 2(a). When a=-0.25, F=-6, and G=3, we can obtain three equilibrium points A(-2,1,0), B(3,-1.5,0), and C(-5,0.5,0), which are shown in Figure 2(a). From Figure 2(b), the converging trajectories for points B and C are heteroclinic orbits, and they will converge to the equilibrium point A. Hence, the equilibrium posits B, C are unstable, and the equilibrium posits A is stable.(a) The curve ofa(F-x)1-x2=G2. (b) Phase plane of of system (1) with the parameter G=3.
(a)
(b)Next, we verify the effectiveness of Theorems2 and 3. Take the parameters G=2, F=-4, and a=-1 which satisfy the Hopf bifurcation conditions a3(F-a-1)=G2 and 2G2>a4. Figure 3(a) shows the phase plane of model (1) with G=1.9, F=-4, and a=-1. It can be seen from Figure 3(a) that there is a stable limit cycle. Figure 3(b) shows the phase plane of model (1) with G=2.1, F=-4, and a=-1. It can be seen from Figure 3(b) that there is a stable equilibrium point. Therefore, we have verified that model (1) exists a Hopf bifurcation. In addition, we can calculate μ2(a,b)=1/7, according to Theorem 5, and Hopf bifurcation is supercritical. Also, we can calculate β2(a,b)=-94/7<0, according to Theorem 5, and the periodic solution is stable.(a) Phase plane of system (1) with parameter G=1.9. (b) Phase plane of system (1) with parameters G=2.1.
(a)
(b)


## 2.1. The Lorenz-84 Model

The Lorenz-84 model is a three-dimensional system [17] and is given by
(1)dxdt=-y2-z2-ax+aF,dydt=xy-bxz-y+G,dzdt=bxy+xz-z,
where x represents the strength of the globally averaged westerly current and y and z are the strength of the cosine and sine phases of a chain of superposed waves. The unit of the variable t is equal to the damping time of the waves that is estimated to be five days. F and G represent the thermal forcing terms, and the parameter b represents the advection strength of the waves by the westerly current. Hence, the equilibrium point of model (1) satisfies the following equation:
(2)-y2-z2-ax+aF=0,xy-bxz-y+G=0,bxy+xz-z=0.
That is,
(3)y=1-xG1-2x+1+b2x2,z=bxG1-2x+1+b2x2,aF-x1-2x+1+b2x2=G2.It is well known that when the parameterG=0, the dynamical behaviors of model (1) are simple which have been discussed in [24], while the parameter G≠0, the dynamical behaviors become complicated and display chaotic attractors in model (1). The objective of this paper is to discuss the stability and local bifurcation and to obtain the corresponding stability and bifurcation conditions about model (1) with G≠0. Note from (2) that it is difficult to obtain the explicit mathematical expression of the equilibrium points on the parameters a, b, F, and G. However, we can also view the variable x as a parameter. Then, we can get y=y(x,a,b,F,G), z=z(x,a,b,F,G) and x satisfies the equation a(F-x)(1-2x+(1+b2)x2)=G2, that are shown in (3).


## 2.2. The Stability and Bifurcation Analysis with ParametersG≠0 and b=0

WhenG≠0 and b=0, model (1) is given as
(4)dxdt=-y2-z2-ax+aF,dydt=xy-y+G,dzdt=xz-z.
Then, the equilibrium point of model (4) satisfies the following equation:
(5)-y2-z2-ax+aF=0,xy-y+G=0,xz-z=0.
From the third equation, in (5), we have (x-1)z=0. If x=1, then, from the second equation in (5), G=0, which is contradictory to the hypotheses G≠0 in this section. Hence, it implies z=0 and x≠1. Then, we can obtain an equilibrium point K(x0,G/1-x0,0) of model (1), where x0 satisfies the equation a(F-x)(1-x)2=G2 and x0≠1.Theorem 1.
WhenG≠0 and b=0, there exists at least one equilibrium point K(x0,G/1-x0,0) in model (1)(1)
ifa>0 and the equation a(F-x)1-x2=G2 has solution x0<1, the equilibrium point K is stable,(2)
ifa>0 and the equation a(F-x)1-x2=G2 has solution x0>1, the equilibrium point K is unstable,(3)
ifa<0 and the equation a(F-x)1-x2=G2 has solution 1--2G2/a3<x0<1+a, the equilibrium point K is stable.Proof.
The Jacobian matrix of model (1) at the point K(x0,G/1-x0,0) is given as
(6)J=-a-2y00y0x0-1000x0-1,
and its characteristic equation is given as
(7)fλ=λ+1-x0λ2+(a+1-x0)λhhhhhhhhhhhh+a+1x0+2y02.
Then, eigenvalues λ1, λ2, and λ3 are obtained as follows:
(8)λ1=x0-1,λ2=-(a+1-x0)+a-1+x02-8G2/1-x022,λ3=-(a+1-x0)-a-1+x02-8G2/1-x022.
If a>0 and the equation a(F-x)1-x2=G2 has solution x0<1, then there is λ1<0. Since 8G2/1-x02>0 and a+1-x02-a-1+x02=4(1-x0)a>0, it follows that Re(λ2)<0 and Re(λ3)<0. Therefore, the equilibrium point K is stable. If the solution x0>1, it is easy to verify that the equilibrium point K is unstable. If a<0 and the equation a(F-x)1-x2=G2 has solution 1--2G2/a3<x0<1+a, it follows that Re(λ1)<0, Re(λ2)<0, and Re(λ3)<0. Therefore, the equilibrium point K is stable. The proof is completed.Theorem 2.
IfG≠0 and b=0, then there exists at least one equilibrium point K(x0,G/1-x0,0) in model (1)(1)
whena>0 and 2G2=a4, if x0=a+1 is a solution of the equation a(F-x)1-x2=G2, the equilibrium point K(x0,G/1-x0,0) is unstable and model (1) has a fold bifurcation,(2)
whena<0 and 2G2>a4, if x0=a+1 is a solution of the equation a(F-x)1-x2=G2, the equilibrium point K(x0,G/1-x0,0) is unstable, and model (1) has a Hopf bifurcation.Proof.
It is easy to obtain the result from (8) and is therefore omitted here.


## 2.3. Direction and Stability of the Hopf Bifurcation

In this section, the supercritical and subcritical bifurcations of model (1) are considered. According to Theorem 2, model (1) has a Hopf bifurcation when x0=a+1, 2G2>a4, and a<0. Let the eigenvectors corresponding to the eigenvalues λ1 and λ3 be α1+iα2 and α3, respectively, where α1, α2, and α3 are all real vectors. Under the following linear transformation:
(9)x1=x-x0,y1=y-y0,z1=z-z0,
model (1) can be changed into
(10)x˙1=-y12-2y0y1-y02-z12+ax0-ax1+aF,y˙1=x1y1+x1y0+x0y1+x0y0-y1-y0+G,z˙1=x1z1+x0z1-z1.
For (6), (7), and (8), if x0=a+1 is a solution of the equation a(F-x)1-x2=G2,2G2>a4 and a<0, we have
(11)α′0=Reλ′a=12>0,ω′(0)=Im⁡(λ′(a))=4G2-a442G2-a4≠0,ω0=Im⁡λa=2G2-a4a>0,λ3<0.
Therefore, model (1) at the equilibrium point K has a Hopf bifurcation. By calculations, we have
(12)α1=100,α2=aKGaK0,α3=001,T=α2,α1,α3=ak10Gak00001,
where k=2G2-a4/a.For model (10), taking the following transformation x1y1z1=Tx2y2z2, we have
(13)x2~˙=Ax2~+B,
where(14)A=0-k0k0000a,x2~=x2y2z2,B=akx22+x2y2+akGx0y0-y0+G-G2+a4a2k2x22-akx2y2-z22-a2G+2Gakx0y0-ax0+a2Gx0+aF-a2akx2z2+y2z2,.From (13),
(15)x˙2=-ky2+Px2,y2,z2,y˙2=kx2+Qx2,y2,z2,z˙2=az2+Rx2,y2,z2,
where
(16)Px2,y2,z2=akx22+x2y2+akGx0y0-y0+G,Qx2,y2,z2=-G2+a4a2k2x22-akx2y2-z22-a2G+2Gakx0y0-ax0+a2Gx0+aF-a2,Rx2,y2,z2=akx2z2+y2z2.
For model (15), according to the normal form theory in [30, 31], we have
(17)g11=14∂P2∂x22+∂P2∂y22+i∂Q2∂x22+∂Q2∂y22=12ak-G2+a4a2k2i,g02=14∂P2∂x22-∂P2∂y22-2∂Q2∂x2∂y2hhhhh+i∂Q2∂x22-∂Q2∂y22+2∂P2∂x2∂y2=12(1-G2+a4a2k2)i,g20=14∂P2∂x22-∂P2∂y22+2∂Q2∂x2∂y2hhhhh+i∂Q2∂x22-∂Q2∂y22-2∂P2∂x2∂y2=12(-1-G2+a4a2k2)i,G21=18∂P3∂x23+∂P3∂x2∂y22+∂Q3∂x22∂y2+∂Q3∂y23hhhhh+i(∂Q3∂x23+∂Q3∂x2∂y22-∂P3∂x22∂y2-∂P3∂y23)=0.
Since the dimension n=3>2, we obtain the following equations:
(18)h11=14∂R2∂x22+∂R2∂y22=0,h20=14∂R2∂x22-∂R2∂y22-2∂R2∂x2∂y2=0.
Next, we solve the following equations:
(19)λ1ϕ11=-h11,λ1-2kiϕ20=-h20.
Then,
(20)ϕ11=0,ϕ20=0.
Let
(21)G110=12∂P2∂x2∂z2+∂Q2∂y2∂z2+i∂P2∂x2∂z2-∂Q2∂y2∂z2=0,G101=12∂P2∂x2∂z2-∂Q2∂y2∂z2+i∂P2∂x2∂z2+∂Q2∂y2∂z2=0.
Taking m=G2+a4/a2k2, it then follows from (17) and (21) that
(22)g21=G21+(2G110ϕ11+G101ϕ20)=0.
Also, let
(23)M10=i2kg20g11-2g112-13g022+12g21=m+1a8k2-i8k2a2k2+103m2+13m+13.
Then, we have
(24)μ2a,b=-ReM10α′0=-m+1a4k2>0,a<0,β2a,b=2ReM10=m+1a4k2<0,a<0,τ2a,b=-Im⁡M10+μ2a,bω′0b=-18k22a2k2+103m2+13m+13hhhhh-m+1a16k24G2-a442G2-a4<0.Theorem 3.
When2G2>a4,a<0, there is a Hopf bifurcation in model (1) at x0=a+1, and (1)
μ2(a,b)>0; that is to say, the direction of bifurcation is supercritical,(2)
β2(a,b)<0; that is to say, the solutions of bifurcating periodic solutions are orbitally stable,(3)
τ2(a,b)<0; that is to say, the periods of bifurcating periodic solutions increase.Proof.
It is easy to obtain the result from the above derivative process in Section2.3 and is therefore omitted here.


## 2.4. The Stability and Bifurcation Analysis with ParametersG≠0 and b≠0

In this section, we consider the stability and bifurcation analysis of model (1) when G≠0 and b≠0. Without loss of generality, we first discuss the stability and bifurcation analysis of model (1) at two special equilibrium points when x=0 and x=1. It is easy to verify that when x=0 and x=1, the equilibrium points of model (1) are K1(0,G,0) and K2(1,0,G/b), respectively.In the following, let us consider the stability and bifurcation analysis of model (1) at the equilibrium point K1.Theorem 4.
For model (1), if a>0, the equilibrium point K1 is stable, and(1)
whena≥1+22G>0 or 0<a≤1-22G, the equilibrium point K1 is stable node point,(2)
when0<1-22G<a<1+22G, the equilibrium point K1 is stable focus point.Proof.
For model (1), the Jacobian matrix at the equilibrium point K1 is
(25)J=-a-2G0G-10bG0-1,
and its characteristic equation is
(26)fλ=λ+1λ2+a+1λ+a+2G2.
We can get λ1,2,3(27)λ1=-1,λ2,3=-a+1±a+12-4a+2G22.
Since a>0, it then follows from (27) that Re(λ1,2,3)<0. Therefore, the equilibrium point K1 is stable. In the same time, note that if Δ=a+12-4(a+2G2)>0, λ1,2,3<0 are real numbers, and it then follows that the equilibrium point K1 is stable node point; if Δ=a+12-4(a+2G2)<0, λ2,3 has conjugate imaginary roots, it then follows that the equilibrium point K1 is stable focus point. The proof is completed.Theorem 5.
For model (1), if a<0 and(1)
whena=-1, F<-1/2 and G2>1/2, then model (1) at the equilibrium point K1 has a Hopf bifurcation,(2)
whena=-1, F=-2 and G2=2, then model (1) at the equilibrium point K1 has a Fold bifurcation,(3)
whena=-1, F=-1/2, then model (1) at the equilibrium point K1 has a flip bifurcation.Proof.
According to (27), we can easily get the conclusion; therefore, we omit it here.Next, let us consider the equilibrium pointK2(1,0,G/b).Theorem 6.
Ifa>0 and F<3/2, then the equilibrium point K2 of model (1) is stable.Proof.
The Jacobian matrix of model (1) at the equilibrium point K2 is
(28)J=-a-2G-2GbG0b-Gbb0,
and its characteristic equation is
(29)fλ=λ3+aλ2+b2+2G2b2λ+ab2-2G2.
Let A=a, B=b2+2G2/b2, and C=ab2-2G2. By the Routh-Hurwitz criterion, when A>0, C>0 and AB-C>0, the eigenvalues of (29) have negative real parts. Therefore, we can get AB-C=2G2(1+a/b2)>0, C>0, and F<3/2. Therefore, the equilibrium point K2 of model (1) is stable. The proof is completed.In the above discussions, we consider the stability and bifurcation analysis of model (1) at two special equilibrium points K1 and K2, respectively. In the following, we will discuss stability and bifurcation analysis of model (1) at the general equilibrium point K(x0,y0,z0).Theorem 7.
SupposeK(x0,y0,z0) is an equilibrium point in model (1), if x0 is a solution of the equation a(F-x)(1-2x+(b2+1)x2)=G2 and satisfies the inequations
(30)x0<1+a2,x02+4+2F+2Fb23(1+b2)x0+1+2F3(1+b2)>0,x03-5a+2b2-ab2+62(1+b2)x02+9a+6+aF+3a2-ab2F2(1+b2)x0-(1+a)(2+2a+aF)2(1+b2)<0,
then the equilibrium point K is stable.Proof.
From (2),
(31)y0=g(x0,b)(1-x0)G,z0=gx0,bx0bG,aF-xfx0,b=G2,
where f(x0,b)=1-2x0+(b2+1)x02 and g(x0,b)=1/f(x0,b). The Jacobian matrix of model (1) at the point K(x0,y0,z0) is given as(32)J(0)=-a-2g(x0,b)(1-x0)G-2g(x0,b)bx0Gg(x0,b)(1-x0-b2x02)Gx0-1-bx0bg(x0,b)Gbx0x0-1.Taking Gg(x0,b)=T, we can get
(33)fλ=λ3+2-2x0+aλ2+2a1-x0+fx0,b1+2T2λ+fx0,ba+2T21-x0-b2x0.
Let A=2-2x0+a, B=2a(1-x0)+f(x0,b)(1+2T2), and C=f(x0,b)(a+2T2(1-x0-b2x0)).
Note that thef(x0,b)>0, g(x0,b)>0, T2>0, and
(34)x0<1+a2,x02+4+2F+2Fb231+b2x0+1+2F31+b2>0,x03-5a+2b2-ab2+621+b2x02+9a+6+aF+3a2-ab2F21+b2x0-1+a2+2a+aF21+b2<0,
and we have A>0, C>0, and AB-C>0. According to the Routh-Hurwitz criterion, the real parts of the roots of (33) are all negative. Therefore, the equilibrium point K is stable. The proof is completed.Theorem 8.
Suppose thatK(x0,y0,z0) is an equilibrium point in model (1). If x0 is a solution of the equation a(F-x)(1-2x+(b2+1)x2)=G2 and satisfies the following inequations and equation:
(35)x0<1+a2,x02+4+2F+2Fb231+b2x0+1+2F31+b2>0,x03-5a+2b2-ab2+621+b2x02+9a+6+aF+3a2-ab2F21+b2x0-1+a2+2a+aF21+b2=0,β1x04-β2x03+β3x02+β4x0+β5≠0,
then model (1) has a Hopf bifurcation, where
(36)β1=4a21-b4G2,β2=4a2G22+2b2+a+ab2+2F-2Fb4,β3=4a2F21-b4G2+8Fa22+2b2+a+ab2G2β3=+3b2+1,β4=4ab2-14a-8-4b2-4a2F2+2b2+a+ab2G2,β5=12a+2aF+2ab2F+4a2+3.Proof.
From (33), we can obtain A=2-2x0+a, B=2a(1-x0)+f(x0,b)(1+2T2), and C=f(x0,b)(a+2T2(1-x0-b2x0)).
Since(37)x0<1+a2,x02+4+2F+2Fb231+b2x0+1+2F31+b2>0,x03-5a+2b2-ab2+621+b2x02+9a+6+aF+3a2-ab2F21+b2x0-1+a2+2a+aF21+b2=0,
it follows that A(x0)>0, B(x0)=C(x0)/A(x0)>0, and C(x0)>0. Similarly, since β1x04-β2x03+β3x02+β4x0+β5≠0, it follows that C′(x0)≠A′(x0)B(x0)-A(x0)B′(x0), where
(38)β1=4a21-b4G2,β2=4a2G22+2b2+a+ab2+2F-2Fb4,β3=4a2F21-b4G2+8Fa22+2b2+a+ab2G2β3=+3b2+1,β4=4ab2-14a-8-4b2β3=-4a2F2+2b2+a+ab2G2,β5=12a+2aF+2ab2F+4a2+3.
Hence, we have proven that A(x0)>0, B(x0)=C(x0)/A(x0)>0, C(x0)>0, and C′(x0)≠A′(x0)B(x0)-A(x0)B′(x0). Therefore, according to the Hopf bifurcation theorem [30], model (1) has a Hopf bifurcation. The proof is completed.


## 2.5. Simulation

In this section, some numerical examples and simulations are presented to illustrate the effectiveness of our theoretical results. Here, we mainly discuss and verify the conditions of Theorems1, 2, and 3.Firstly, we verify the effectiveness of Theorem1 with a>0. Taking a=0.25, F=8, then the curves of a(F-x)1-x2=G2 and straight line of x=1 are shown in Figure 1(a), in which x-axis represents the parameter G and y-axis represents x values of the equilibrium point K(x,G/1-x,0). Note that when a=0.25, F=8, and G=3, there are three equilibrium points A=(-1,1.5,0), B=(7,-0.5,0), and C=(4,-1,0), which are shown in Figure 1(a). From Figure 1(b), the converging trajectories for points B and C are Heteroclinic orbits, and they will converge to the equilibrium point A. Hence, the equilibrium posits B, C are unstable. Take an arbitrary point D (see Figure 1) under the conditions a=0.25, F=8, and G=3. It is shown from Figure 1(b) that the trajectory starting from D converges to A. Therefore, we have verified that the point A is stable.(a) The curve ofa(F-x)1-x2=G2. (b) Phase plane of system (1) with the parameter G=3.
(a)
(b)In addition, we considera<0 in Theorem 1. Choose a=-0.25, F=-6, and then the curves of a(F-x)1-x2=G2 and straight line of x=1 are shown in Figure 2(a). When a=-0.25, F=-6, and G=3, we can obtain three equilibrium points A(-2,1,0), B(3,-1.5,0), and C(-5,0.5,0), which are shown in Figure 2(a). From Figure 2(b), the converging trajectories for points B and C are heteroclinic orbits, and they will converge to the equilibrium point A. Hence, the equilibrium posits B, C are unstable, and the equilibrium posits A is stable.(a) The curve ofa(F-x)1-x2=G2. (b) Phase plane of of system (1) with the parameter G=3.
(a)
(b)Next, we verify the effectiveness of Theorems2 and 3. Take the parameters G=2, F=-4, and a=-1 which satisfy the Hopf bifurcation conditions a3(F-a-1)=G2 and 2G2>a4. Figure 3(a) shows the phase plane of model (1) with G=1.9, F=-4, and a=-1. It can be seen from Figure 3(a) that there is a stable limit cycle. Figure 3(b) shows the phase plane of model (1) with G=2.1, F=-4, and a=-1. It can be seen from Figure 3(b) that there is a stable equilibrium point. Therefore, we have verified that model (1) exists a Hopf bifurcation. In addition, we can calculate μ2(a,b)=1/7, according to Theorem 5, and Hopf bifurcation is supercritical. Also, we can calculate β2(a,b)=-94/7<0, according to Theorem 5, and the periodic solution is stable.(a) Phase plane of system (1) with parameter G=1.9. (b) Phase plane of system (1) with parameters G=2.1.
(a)
(b)


## 3. Chaotic Behavior of the Lorenz-84 Model

In this section, the chaotic behavior of model (1) with the parameters G, b, a, and F are discussed, and the complex dynamic behaviors are analyzed by Lyapunov exponents spectrum, bifurcation diagram, Poincaré section, and power spectrum. Especially, the topological horseshoe is given to rigorous approaches to study chaos in model (1). For the sake of simplicity, the initial condition in model (1) is chosen as [1,1,1].


### 3.1. Dynamical Behaviors of Model (1) by Varying Parameters G and b

This model has been found to be chaotic over a wide range of parameters and has many interesting complex dynamical behaviors by varying parametersG and b, respectively. It is well known that Lyapunov exponents measure the exponential rates of divergence or convergence of nearby trajectories in phase space. The dynamical behaviors of model (1) by varying parameters G and b are listed in Table 1.Table 1
The dynamic behaviors of model (1) about the parameters G, b, a, and F.
The parameterG
Dynamic behaviors
The parameterb
Dynamic behaviors
(a=0.25,b=4 and F=8)
(a=0.25,F=8 and G=1)
0
<
G
<
0.277
Limit cycle
0
<
b
<
1.07
Limit point
0.277
≤
G
<
0.797
Limit cycle period 2
1.07
≤
b
<
1.73
Limit cycle
0.797
≤
G
<
0.801
Limit cycle period 4
1.73
≤
b
<
2.53
Chaos
0.801
≤
G
<
0.819
Limit cycle period 2
2.53
≤
b
<
3.93
Limit cycle
0.819
≤
G
<
0.831
Limit cycle period 4
3.93
≤
b
<
4.16
Chaos
0.831
≤
G
<
1.188
Chaos
4.16
≤
b
<
5.53
Limit cycle
1.188
≤
G
<
1.212
Limit cycle period 4
5.53
≤
b
<
5.9
Chaos
1.212
≤
G
<
1.352
Chaos
5.9
≤
b
<
8.53
Limit cycle
1.352
≤
G
<
1.364
Limit cycle period 2
8.53
≤
b
<
8.60
Chaos
1.364
≤
G
<
1.85
Limit point
8.60
≤
b
<
10
Limit cycle
The parametera
Dynamic behaviors
The parameterF
Dynamic behaviors
(b=4,F=8 and G=1)
(a=0.25,b=4 and G=1)
0
<
a
<
0.134
Limit point
0
<
F
<
4.313
Limit point
0.134
≤
a
<
0.152
Chaos
4.313
≤
F
<
4.518
Limit cycle
0.152
≤
a
<
0.207
Limit cycle
4.518
≤
F
<
5.159
Chaos
0.207
≤
a
<
0.240
Chaos
5.159
≤
F
<
6.948
Limit cycle
0.240
≤
a
<
0.243
Limit cycle
6.948
≤
F
<
8.763
Chaos
0.243
≤
a
<
0.280
Chaos
8.763
≤
F
<
15
Limit cycle
0.280
≤
a
<
0.3
Limit cycleDynamical behaviors of model (1) by varying parameter G are first discussed. Suppose that the parameters a=0.25, F=8, and b=4 are fixed. Let the parameter G vary in interval [0,1.4]. The Lyapunov exponents spectrum is shown in Figure 4. It is shown from Figure 4 that when G∈[0.831,1.188] and G∈[0.831,1.188], the max Lyapunov exponents are greater than zero; that is to say, model (1) has chaotic state. Figure 5(a) shows the bifurcation diagram of model (1) about the parameter G. Figure 5(b) shows the part of Figure 5(a) with G>0. It is shown from Figure 5 that when G∈[0,1.4], model (1) can evolve into chaotic state. Moreover, there are some periodic windows in the chaotic regions. It is shown from Figure 5(a) that the dynamical behaviors of model (1) have the symmetry for the parameter G; that is, to say, when the parameters G>0 and G<0, model (1) has the same dynamical behaviors.Figure 4
Lyapunov exponents aboutG for the parameter F=8.The bifurcation diagram aboutG.
(a)
(b)To illustrate the chaotic behavior, the parameterG is chosen as G=1. The corresponding phase portrait of chaotic attractor is shown in Figure 6(a) and its spectrum is continuous as shown in Figure 6(b). From Figure 6, model (1) has chaotic state. Next, the topological horseshoe will be used to study chaos in model (1). The topological horseshoe is well recognized as one of the most rigorous approaches to study chaos with computer. It has been successfully applied in many chaotic systems and hyperchaotic systems [32–40]. As a basic and striking result in chaotic dynamics, topological horseshoe provides a powerful tool in the rigorous study of chaos and dynamical systems obtaining the topological entropy, verifying the existence of chaos, showing the structure of chaotic attractors, revealing the mechanism inside chaotic phenomena, and so on. Some definitions and properties about the topological horseshoe [33–36] are given at first.(a) Phase portraits of the chaos attractor. (b) Spectrum ofx.
(a)
(b)LetD be a compact connected region of Rn, and let Di, i=1,2,…,m be disjoint compact connected subsets of D homeomorphic to the unit square. Let f:D→Rn be a piecewise continuous map which is a diffeomorphism on each compact set Di.Definition 9 (see [33, 36]).
For eachDi,1≤i≤m, let Di1 and Di2 be two fixed disjoint compact subsets of Di contained in the boundary ∂Di. A connected subset l of Di is said to connect Di1 and Di2, if l∩Di1≠⌀ and l∩Di2≠⌀.Definition 10 (see [33, 36]).
Letl⊂Di be a connection of Di1 and Di2. We say that f(l) is acrossing Dj, if l contains a connected subset l′ such that f(l′)⊂Dj is a connection of Dj1 and Dj2. In this case, we denote it by f↦Dj. Furthermore, if f↦Dj holds true for every connection l of Di1 and Di2, then f(Di) is said to be acrossing Dj and is denoted by f(Di)⊂Dj with respect to two pairs, (Di1,Di2) and (Dj1,Dj2).Theorem 11 (see [34–36]).
If the relationf(Di)⊂Dj holds for every pair with i,j, taken from 1≤i, j≤m, then there exists a compact invariant set K⊂D, such that f∣K is semiconjugate to the full m-shift dynamics σ∣∑m and topological entropy ent(f)≥log⁡m.Here, them-shift is also called the Bernoulli m-shif. The symbolic series space ∑m‍ is compact, totally disconnected, and perfect. More definitions and properties about the topological horseshoe can be found in [32–40]. To find horseshoes in model (1), we use the method proposed in [34] with an efficient and powerful toolbox in MATLAB called “A toolbox for finding horseshoes in 2D map” (Li Q D. HSTOOL-A MATLAB toolbox for finding horseshoes in 2D maps, 2007–2011. http://www.mathworks.cn/matlabcentral/fileexchange/14075). When G=1, according to the Matlab toolbox, we can get Poincaré map of model (1) at the z=0 (Figure 7(a)). We numerically find two quadrilateral D1 and D2 in the x-y (Poincaré map) plane. The four vertices of D1 are
(39)0.727230483,1.444887781,0.666821561,1.360099751,0.843401487,1.245386534,0.913104089,1.330174564,
and the four vertices of D2 are
(40)1.029275093,1.245386534,0.964219331,1.165586035,1.085037175,1.040897756,1.164033457,1.130673317.
Based on the theoretical result [32–40] and Theorem 11.(a) Poincare mapping on the crossingz=0. (b) The topological horseshoe at G=1.
(a)
(b)Theorem 12.
There exists a compact invariant setK⊂D1∪D2, such that H5∣K is semiconjugate to 2-shift dynamics and the topological entropy of H is ent(H)≥1/5log⁡2.Proof.
According to Theorem11, we need to show that the relations
(41)H5D1⟼D1,H5D1⟼D2,H5D2⟼D1,H5D2⟼D2
hold true. For the above four relations, it is easy to see from Figure 7(b). Furthermore, it also follows from Theorem 11 that the topological entropy of H is ent(H)≥1/5log⁡2. Clearly, ent(H)>0, which means that model (1) is chaotic indeed when G=1. The proof is completed.In [17], when the parameters a and b are, respectively, chosen as a=0.25 and b=4, Lorenz pointed out that F and G should be allowed to vary periodically during one year. In particular, F should be larger in winter than in summer. In the numerical study of  [17], the author pointed that (F,G)=(6,1) and (F,G)=(8,1) represent the summer conditions and the winter conditions, respectively. With the winter conditions (F,G)=(8,1), model (1) has chaotic behavior. In this paper, a and b are, respectively, chosen as 0.25 and 4, and F is chosen as 6,7,8,9, and 10, respectively. According to the numerical study, model (1) appears to be of more complex dynamics behavior when F=8 than when F=6,7,9,10 by varying parameter G. Moreover, model (1) has been found to be chaotic over more wider range of parameter G when F=8 than when F=6,7,9,10, respectively. When F=8, the Lyapunov exponent spectra have been already shown in Figure 4. In the following, we consider the cases of F=6 and F=10, respectively. The cases of F=7 and F=9 are similar to the cases of F=6 and F=10, therefore, they are omitted here. When the parameter F=6 and F=10, the Lyapunov exponents about G are shown in Figures 8(a) and 8(b), respectively. It is shown that from Figure 8(a) that model (1) has no chaotic behavior when F=6. And from Figure 8(b), when F=10, model (1) has the chaotic behaviors with 1.473≤G<1.538.Lyapunov exponents aboutG.
(a)
Lyapunov exponents aboutG for the parameter F=6
(b)
Lyapunov exponents aboutG for the parameter F=10Dynamical behaviors of model (1) by varying parameters b are also considered. When parameters a=0.25, F=8, and G=1 are fixed, let the parameter b vary in interval [0,10]. Similarly, the Lyapunov exponents spectrum is shown in Figure 9. It is shown from Figure 9 that when b∈[1.73,2.53], b∈[3.93,4.16], b∈[5.53,5.90], and b∈[8.53,8.60], the max Lyapunov exponents are greater than zero; that is to say, model (1) has chaotic state. The corresponding bifurcation diagram is displayed in Figure 10. It is shown from Figure 10(a) that the dynamical behaviors of model (1) have the symmetry for the parameter b;, that is, to say, when the parameter b>0 and b<0, model (1) has the same dynamical behaviors. Figure 10(b) shows the part of Figure 10(a) with b>0. This is similar to the parameter G, and the dynamical behaviors of model (1) about parameter b are more keenly aware of the change than the parameter G.Figure 9
Lyapunov exponents aboutb.The bifurcation diagram aboutb.
(a)
(b)


### 3.2. Dynamical Behaviors of Model (1) by Varying Parameters a and F

This section mainly focuses on the dynamical behaviors of model (1) by varying parameters a and F, respectively. The dynamical behaviors of model (1) by varying parameters a and F are also listed in Table 1. Dynamical behaviors of model (1) by varying parameters a are first considered. When parameters b=4, F=8, and G=1 are fixed, let the parameter a varies in interval [0.1,0.3], the Lyapunov exponents spectrum is shown in Figure 11(a). It is shown from Figure 11(a) that when a∈[0.1345,0.1526], a∈[0.207,0.2406], and a∈[0.2435,0.2806] and the max Lyapunov exponents are greater than zero, model (1) has chaotic state. The corresponding bifurcation diagram is displayed in Figure 12(a).(a) Lyapunov exponents abouta. (b) Lyapunov exponents about F.
(a)
(b)(a) The bifurcation diagram abouta. (b) The bifurcation diagram about F.
(a)
(b)Similarly, when parametersa=0.25, b=4, and G=1 are fixed, let the parameter F vary in interval [4,9], the Lyapunov exponents spectrum is shown in Figure 11(b). When F∈[4.518,5.159] and F∈[6.948,8.763], the max Lyapunov exponents are greater than zero, and model (1) has chaotic state. Figure 12(b) shows the bifurcation diagram of model (1) about the parameter F. From the Lyapunov exponents spectrum and the bifurcation diagram, shown in Figures 11 and 12, the dynamical behaviors of model (1) about a and F are very similar.


## 3.1. Dynamical Behaviors of Model (1) by Varying Parameters G and b

This model has been found to be chaotic over a wide range of parameters and has many interesting complex dynamical behaviors by varying parametersG and b, respectively. It is well known that Lyapunov exponents measure the exponential rates of divergence or convergence of nearby trajectories in phase space. The dynamical behaviors of model (1) by varying parameters G and b are listed in Table 1.Table 1
The dynamic behaviors of model (1) about the parameters G, b, a, and F.
The parameterG
Dynamic behaviors
The parameterb
Dynamic behaviors
(a=0.25,b=4 and F=8)
(a=0.25,F=8 and G=1)
0
<
G
<
0.277
Limit cycle
0
<
b
<
1.07
Limit point
0.277
≤
G
<
0.797
Limit cycle period 2
1.07
≤
b
<
1.73
Limit cycle
0.797
≤
G
<
0.801
Limit cycle period 4
1.73
≤
b
<
2.53
Chaos
0.801
≤
G
<
0.819
Limit cycle period 2
2.53
≤
b
<
3.93
Limit cycle
0.819
≤
G
<
0.831
Limit cycle period 4
3.93
≤
b
<
4.16
Chaos
0.831
≤
G
<
1.188
Chaos
4.16
≤
b
<
5.53
Limit cycle
1.188
≤
G
<
1.212
Limit cycle period 4
5.53
≤
b
<
5.9
Chaos
1.212
≤
G
<
1.352
Chaos
5.9
≤
b
<
8.53
Limit cycle
1.352
≤
G
<
1.364
Limit cycle period 2
8.53
≤
b
<
8.60
Chaos
1.364
≤
G
<
1.85
Limit point
8.60
≤
b
<
10
Limit cycle
The parametera
Dynamic behaviors
The parameterF
Dynamic behaviors
(b=4,F=8 and G=1)
(a=0.25,b=4 and G=1)
0
<
a
<
0.134
Limit point
0
<
F
<
4.313
Limit point
0.134
≤
a
<
0.152
Chaos
4.313
≤
F
<
4.518
Limit cycle
0.152
≤
a
<
0.207
Limit cycle
4.518
≤
F
<
5.159
Chaos
0.207
≤
a
<
0.240
Chaos
5.159
≤
F
<
6.948
Limit cycle
0.240
≤
a
<
0.243
Limit cycle
6.948
≤
F
<
8.763
Chaos
0.243
≤
a
<
0.280
Chaos
8.763
≤
F
<
15
Limit cycle
0.280
≤
a
<
0.3
Limit cycleDynamical behaviors of model (1) by varying parameter G are first discussed. Suppose that the parameters a=0.25, F=8, and b=4 are fixed. Let the parameter G vary in interval [0,1.4]. The Lyapunov exponents spectrum is shown in Figure 4. It is shown from Figure 4 that when G∈[0.831,1.188] and G∈[0.831,1.188], the max Lyapunov exponents are greater than zero; that is to say, model (1) has chaotic state. Figure 5(a) shows the bifurcation diagram of model (1) about the parameter G. Figure 5(b) shows the part of Figure 5(a) with G>0. It is shown from Figure 5 that when G∈[0,1.4], model (1) can evolve into chaotic state. Moreover, there are some periodic windows in the chaotic regions. It is shown from Figure 5(a) that the dynamical behaviors of model (1) have the symmetry for the parameter G; that is, to say, when the parameters G>0 and G<0, model (1) has the same dynamical behaviors.Figure 4
Lyapunov exponents aboutG for the parameter F=8.The bifurcation diagram aboutG.
(a)
(b)To illustrate the chaotic behavior, the parameterG is chosen as G=1. The corresponding phase portrait of chaotic attractor is shown in Figure 6(a) and its spectrum is continuous as shown in Figure 6(b). From Figure 6, model (1) has chaotic state. Next, the topological horseshoe will be used to study chaos in model (1). The topological horseshoe is well recognized as one of the most rigorous approaches to study chaos with computer. It has been successfully applied in many chaotic systems and hyperchaotic systems [32–40]. As a basic and striking result in chaotic dynamics, topological horseshoe provides a powerful tool in the rigorous study of chaos and dynamical systems obtaining the topological entropy, verifying the existence of chaos, showing the structure of chaotic attractors, revealing the mechanism inside chaotic phenomena, and so on. Some definitions and properties about the topological horseshoe [33–36] are given at first.(a) Phase portraits of the chaos attractor. (b) Spectrum ofx.
(a)
(b)LetD be a compact connected region of Rn, and let Di, i=1,2,…,m be disjoint compact connected subsets of D homeomorphic to the unit square. Let f:D→Rn be a piecewise continuous map which is a diffeomorphism on each compact set Di.Definition 9 (see [33, 36]).
For eachDi,1≤i≤m, let Di1 and Di2 be two fixed disjoint compact subsets of Di contained in the boundary ∂Di. A connected subset l of Di is said to connect Di1 and Di2, if l∩Di1≠⌀ and l∩Di2≠⌀.Definition 10 (see [33, 36]).
Letl⊂Di be a connection of Di1 and Di2. We say that f(l) is acrossing Dj, if l contains a connected subset l′ such that f(l′)⊂Dj is a connection of Dj1 and Dj2. In this case, we denote it by f↦Dj. Furthermore, if f↦Dj holds true for every connection l of Di1 and Di2, then f(Di) is said to be acrossing Dj and is denoted by f(Di)⊂Dj with respect to two pairs, (Di1,Di2) and (Dj1,Dj2).Theorem 11 (see [34–36]).
If the relationf(Di)⊂Dj holds for every pair with i,j, taken from 1≤i, j≤m, then there exists a compact invariant set K⊂D, such that f∣K is semiconjugate to the full m-shift dynamics σ∣∑m and topological entropy ent(f)≥log⁡m.Here, them-shift is also called the Bernoulli m-shif. The symbolic series space ∑m‍ is compact, totally disconnected, and perfect. More definitions and properties about the topological horseshoe can be found in [32–40]. To find horseshoes in model (1), we use the method proposed in [34] with an efficient and powerful toolbox in MATLAB called “A toolbox for finding horseshoes in 2D map” (Li Q D. HSTOOL-A MATLAB toolbox for finding horseshoes in 2D maps, 2007–2011. http://www.mathworks.cn/matlabcentral/fileexchange/14075). When G=1, according to the Matlab toolbox, we can get Poincaré map of model (1) at the z=0 (Figure 7(a)). We numerically find two quadrilateral D1 and D2 in the x-y (Poincaré map) plane. The four vertices of D1 are
(39)0.727230483,1.444887781,0.666821561,1.360099751,0.843401487,1.245386534,0.913104089,1.330174564,
and the four vertices of D2 are
(40)1.029275093,1.245386534,0.964219331,1.165586035,1.085037175,1.040897756,1.164033457,1.130673317.
Based on the theoretical result [32–40] and Theorem 11.(a) Poincare mapping on the crossingz=0. (b) The topological horseshoe at G=1.
(a)
(b)Theorem 12.
There exists a compact invariant setK⊂D1∪D2, such that H5∣K is semiconjugate to 2-shift dynamics and the topological entropy of H is ent(H)≥1/5log⁡2.Proof.
According to Theorem11, we need to show that the relations
(41)H5D1⟼D1,H5D1⟼D2,H5D2⟼D1,H5D2⟼D2
hold true. For the above four relations, it is easy to see from Figure 7(b). Furthermore, it also follows from Theorem 11 that the topological entropy of H is ent(H)≥1/5log⁡2. Clearly, ent(H)>0, which means that model (1) is chaotic indeed when G=1. The proof is completed.In [17], when the parameters a and b are, respectively, chosen as a=0.25 and b=4, Lorenz pointed out that F and G should be allowed to vary periodically during one year. In particular, F should be larger in winter than in summer. In the numerical study of  [17], the author pointed that (F,G)=(6,1) and (F,G)=(8,1) represent the summer conditions and the winter conditions, respectively. With the winter conditions (F,G)=(8,1), model (1) has chaotic behavior. In this paper, a and b are, respectively, chosen as 0.25 and 4, and F is chosen as 6,7,8,9, and 10, respectively. According to the numerical study, model (1) appears to be of more complex dynamics behavior when F=8 than when F=6,7,9,10 by varying parameter G. Moreover, model (1) has been found to be chaotic over more wider range of parameter G when F=8 than when F=6,7,9,10, respectively. When F=8, the Lyapunov exponent spectra have been already shown in Figure 4. In the following, we consider the cases of F=6 and F=10, respectively. The cases of F=7 and F=9 are similar to the cases of F=6 and F=10, therefore, they are omitted here. When the parameter F=6 and F=10, the Lyapunov exponents about G are shown in Figures 8(a) and 8(b), respectively. It is shown that from Figure 8(a) that model (1) has no chaotic behavior when F=6. And from Figure 8(b), when F=10, model (1) has the chaotic behaviors with 1.473≤G<1.538.Lyapunov exponents aboutG.
(a)
Lyapunov exponents aboutG for the parameter F=6
(b)
Lyapunov exponents aboutG for the parameter F=10Dynamical behaviors of model (1) by varying parameters b are also considered. When parameters a=0.25, F=8, and G=1 are fixed, let the parameter b vary in interval [0,10]. Similarly, the Lyapunov exponents spectrum is shown in Figure 9. It is shown from Figure 9 that when b∈[1.73,2.53], b∈[3.93,4.16], b∈[5.53,5.90], and b∈[8.53,8.60], the max Lyapunov exponents are greater than zero; that is to say, model (1) has chaotic state. The corresponding bifurcation diagram is displayed in Figure 10. It is shown from Figure 10(a) that the dynamical behaviors of model (1) have the symmetry for the parameter b;, that is, to say, when the parameter b>0 and b<0, model (1) has the same dynamical behaviors. Figure 10(b) shows the part of Figure 10(a) with b>0. This is similar to the parameter G, and the dynamical behaviors of model (1) about parameter b are more keenly aware of the change than the parameter G.Figure 9
Lyapunov exponents aboutb.The bifurcation diagram aboutb.
(a)
(b)


## 3.2. Dynamical Behaviors of Model (1) by Varying Parameters a and F

This section mainly focuses on the dynamical behaviors of model (1) by varying parameters a and F, respectively. The dynamical behaviors of model (1) by varying parameters a and F are also listed in Table 1. Dynamical behaviors of model (1) by varying parameters a are first considered. When parameters b=4, F=8, and G=1 are fixed, let the parameter a varies in interval [0.1,0.3], the Lyapunov exponents spectrum is shown in Figure 11(a). It is shown from Figure 11(a) that when a∈[0.1345,0.1526], a∈[0.207,0.2406], and a∈[0.2435,0.2806] and the max Lyapunov exponents are greater than zero, model (1) has chaotic state. The corresponding bifurcation diagram is displayed in Figure 12(a).(a) Lyapunov exponents abouta. (b) Lyapunov exponents about F.
(a)
(b)(a) The bifurcation diagram abouta. (b) The bifurcation diagram about F.
(a)
(b)Similarly, when parametersa=0.25, b=4, and G=1 are fixed, let the parameter F vary in interval [4,9], the Lyapunov exponents spectrum is shown in Figure 11(b). When F∈[4.518,5.159] and F∈[6.948,8.763], the max Lyapunov exponents are greater than zero, and model (1) has chaotic state. Figure 12(b) shows the bifurcation diagram of model (1) about the parameter F. From the Lyapunov exponents spectrum and the bifurcation diagram, shown in Figures 11 and 12, the dynamical behaviors of model (1) about a and F are very similar.


## 4. Conclusion

In this paper, the stability and local bifurcation of the Lorenz-84 model have been investigated. The stability and local bifurcation of model (1) with G≠0 have been discussed. The conditions of the supercritical Hopf bifurcation have also been derived. In addition, the chaotic behaviors of Lorenz-84 model have been researched. The bifurcation diagrams and Lyapunov exponents spectrums for every parameter have been discussed and the parameter interval range of limit cycle and chaotic attractor of every parameter have also been calculated. Especially, a computer-assisted proof of the chaoticity of the Lorenz-84 model has been presented by a topological horseshoe theory. Future work will focus on multistability and high codimension bifurcations of the Lorenz-84 model.



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*Source: 296279-2014-11-23.xml*