Dynamical Analysis of the Lorenz-84 Atmospheric Circulation Model

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 computerassisted proof of the chaoticity of the model is presented by a topological horseshoe theory.


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][2][3][4][5][6][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][10][11].Thus, low-order models may well have little to do with higherdimensional 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 2 Journal of Applied Mathematics 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 .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][20][21][22][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][25][26][27][28][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- ( ≥ 2) bifurcation or other nonlinear phenomena.This paper is arranged as follows.In Section 2, 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.

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.

The Lorenz-84
Model.The Lorenz-84 model is a threedimensional system [17] and is given by where  represents the strength of the globally averaged westerly current and  and  are the strength of the cosine and sine phases of a chain of superposed waves.The unit of the variable  is equal to the damping time of the waves that is estimated to be five days. and  represent the thermal forcing terms, and the parameter  represents the advection strength of the waves by the westerly current.Hence, the equilibrium point of model (1) satisfies the following equation: That is, (3) It is well known that when the parameter  = 0, the dynamical behaviors of model (1) are simple which have been discussed in [24], while the parameter  ̸ = 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  ̸ = 0. Note from (2) that it is difficult to obtain the explicit mathematical expression of the equilibrium points on the parameters , , , and .However, we can also view the variable  as a parameter.Then, we can get  = (, , , , ),  = (, , , , ) and  satisfies the equation ( − )(1 − 2 + (1 +  2 ) 2 ) =  2 , that are shown in (3).
Proof.It is easy to obtain the result from ( 8) and is therefore omitted here.
For model (10), taking the following transformation where ) .
Proof.It is easy to obtain the result from the above derivative process in Section 2.3 and is therefore omitted here.

The Stability and Bifurcation Analysis with Parameters
̸ = 0 and  ̸ = 0.In this section, we consider the stability and bifurcation analysis of model (1) when  ̸ = 0 and  ̸ = 0. Without loss of generality, we first discuss the stability and bifurcation analysis of model (1) at two special equilibrium points when  = 0 and  = 1.It is easy to verify that when  = 0 and  = 1, the equilibrium points of model ( 1) are  1 (0, , 0) and  2 (1, 0, /), respectively.
In the following, let us consider the stability and bifurcation analysis of model (1) at the equilibrium point  1 .
Theorem 5.For model (1), if  < 0 and Proof.According to (27), we can easily get the conclusion; therefore, we omit it here.
In the above discussions, we consider the stability and bifurcation analysis of model ( 1) at two special equilibrium points  1 and  2 , respectively.In the following, we will discuss stability and bifurcation analysis of model ( 1) at the general equilibrium point ( 0 ,  0 ,  0 ).Theorem 7. Suppose ( 0 ,  0 ,  0 ) is an equilibrium point in model (1), if  0 is a solution of the equation ( − )(1 − 2 + ( 2 + 1) 2 ) =  2 and satisfies the inequations then the equilibrium point  is stable.
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 Theorems 1, 2, and 3.

Chaotic Behavior of the Lorenz-84 Model
In this section, the chaotic behavior of model ( 1) with the parameters , , , and  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].

Dynamical Behaviors of Model (1) by Varying Parameters
and .This model has been found to be chaotic over a wide range of parameters and has many interesting complex dynamical behaviors by varying parameters  and , 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  and  are listed in Table 1.
Dynamical behaviors of model ( 1 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 ; that is, to say, when the parameters  > 0 and  < 0, model (1) has the same dynamical behaviors.
To illustrate the chaotic behavior, the parameter  is chosen as  = 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][33][34][35][36][37][38][39][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][34][35][36] are given at first.
Let  be a compact connected region of   , and let   ,  = 1, 2, . . .,  be disjoint compact connected subsets of  homeomorphic to the unit square.Let  :  →   be a piecewise continuous map which is a diffeomorphism on each compact set   .Definition 9 (see [33,36]).For each   , 1 ≤  ≤ , let  1  and  2  be two fixed disjoint compact subsets of   contained in the boundary Definition 10 (see [33,36]).Let  ⊂   be a connection of  1   and  2  .We say that () is acrossing   , if  contains a connected subset   such that (  ) ⊂   is a connection of  1  and  2  .In this case, we denote it by   →   .Furthermore, if    →   holds true for every connection  of  1  and  2  , then (  ) is said to be acrossing   and is denoted by (  ) ⊂   with respect to two pairs, ( 1  ,  2  ) and ( 1  ,  2  ).
Proof.According to Theorem 11, we need to show that the relations 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  is ent() ≥ (1/5) log 2.
In [17], when the parameters  and  are, respectively, chosen as  = 0.25 and  = 4, Lorenz pointed out that  and  should be allowed to vary periodically during one year.In particular,  should be larger in winter than in summer.In the numerical study of [17], the author pointed that (, ) = (6, 1) and (, ) = (8, 1) represent the summer conditions and the winter conditions, respectively.With the winter conditions (, ) = (8, 1), model (1) has chaotic

Dynamical Behaviors of Model (1) by Varying Parameters
and .This section mainly focuses on the dynamical behaviors of model ( 1) by varying parameters  and , respectively.The dynamical behaviors of model ( 1) by varying parameters  and  are also listed in Table 1.Dynamical behaviors of model ( 1

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  ̸ = 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.
) by varying parameter  are first discussed.Suppose that the parameters  = 0.25,  = 8, and  = 4 are fixed.Let the parameter  vary in interval [0, 1.4].The Lyapunov exponents spectrum is shown in Figure 4.It is shown from Figure 4 that when  ∈ [0.831, 1.188] and  ∈ [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 .
Figure 5(b) shows the part of Figure 5(a) with  > 0. It is shown from Figure 5 that when  ∈ [0, 1.4], model (
) by varying parameters  are first considered.When parameters  = 4,  = 8, and  = 1 are fixed, let the parameter  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  ∈ [0.1345, 0.1526],  ∈ [0.207, 0.2406], and  ∈ [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).Similarly, when parameters  = 0.25,  = 4, and  = 1 are fixed, let the parameter  vary in interval [4, 9], the Lyapunov exponents spectrum is shown in Figure 11(b).When  ∈ [4.518, 5.159] and  ∈ [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 .From the Lyapunov exponents spectrum and the bifurcation diagram, shown in Figures 11 and 12, the dynamical behaviors of model (1) about  and  are very similar.

Table 1 :
The dynamic behaviors of model (1) about the parameters , , , and .