# Analysis of a Generalized Lorenz–Stenflo Equation

Authors: Fuchen Zhang; Rui Chen; Xiusu Chen
Journal: Complexity
 (2017)

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## Abstract

Although the globally attractive sets of a hyperchaotic system have important applications in the fields of engineering, science, and technology, it is often a difficult task for the researchers to obtain the globally attractive set of the hyperchaotic systems due to the complexity of the hyperchaotic systems. Therefore, we will study the globally attractive set of a generalized hyperchaotic Lorenz–Stenflo system describing the evolution of finite amplitude acoustic gravity waves in a rotating atmosphere in this paper. Based on Lyapunov-like functional approach combining some simple inequalities, we derive the globally attractive set of this system with its parameters. The effectiveness of the proposed methods is illustrated via numerical examples.


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## Body

## 1. Introduction

In 1963, Lorenz found the well-known three-dimensional Lorenz model when he studied the dynamics of the atmosphere [1]. Since then, various complex dynamical behaviors of the Lorenz system have been studied by mathematicians, physicists, and engineers from various fields due to various applications in the fields of engineering, science, and technology [2–14]. In order to improve the stability or predictability of the Lorenz system, Stenflo and Leonov derived the following four-dimensional Lorenz–Stenflo system with four parameters to describe the dynamics of the atmosphere [15, 16]:(1)dxdt=ay-x+dw,dydt=cx-y-xz,dzdt=xy-bz,dwdt=-x-aw.In order to give a better description of the atmosphere, Chen and Liang propose a generalized Lorenz–Stenflo system with six parameters according to the Lorenz–Stenflo system [17]:(2)dxdt=ay-x+sw,dydt=cx-dy-xz,dzdt=xy-bz,dwdt=-x-rw,where x, y, z, and w are state variables and a, b, c, d, r, and s are positive parameters of system (2). System (2) can describe the dynamic behavior of finite amplitude acoustic gravity waves in a rotating atmosphere.The Lyapunov exponents of the dynamical system (2) are calculated numerically for the parameter values a=19.42, b=1.91, c=29.45, d=2.86, r=0.23, and s=9.64 with the initial state x0,y0,z0,w0=2.2,2.0,10.5,20. System (2) has Lyapunov exponents as λLE1=0.0696, λLE2=0.0359, λLE3=0.0002, and λLE4=-24.5176 for the parameters listed above (see [18, 19] for a detailed discussion of Lyapunov exponents of strange attractors in dynamical systems). Thus, system (2) has two positive Lyapunov exponents and the strange attractor, which means system (2) can exhibit a variety of interesting and complex chaotic behaviors. System (2) has a hyperchaotic attractor with a=19.42, b=1.91, c=29.45, d=2.86, r=0.23, and s=9.64, as shown in Figures 1–4.Figure 1
Projection of hyperchaotic attractor of system (2) onto thexOyz space with a=19.42, b=1.91, c=29.45, d=2.86, r=0.23, and s=9.64.Figure 2
Projection of hyperchaotic attractor of system (2) onto the xOyw space with a=19.42, b=1.91, c=29.45, d=2.86, r=0.23, and s=9.64.Figure 3
Projection of hyperchaotic attractor of system (2) onto the xOzw space with a=19.42, b=1.91, c=29.45, d=2.86, r=0.23, and s=9.64.Figure 4
Projection of hyperchaotic attractor of system (2) onto the yOzw space with a=19.42, b=1.91, c=29.45, d=2.86, r=0.23, and s=9.64.In this paper, all the simulations are carried out by using fourth-order Runge-Kutta Method with time-steph=0.005.The rest of this paper is organized as follows. In Section2, the globally attractive set for the chaotic attractors in (2) is studied using Lyapunov stability theory. To validate the ultimate bound estimation, numerical simulations are also provided. Finally, the conclusions are drawn in Section 3.


## 2. Bounds for the Chaotic Attractors in System (2)

Theorem 1.
For anya>0, b>0, c>0, d>0, r>0, s>0, there exists a positive number M>0, such that(3)Ωλ,m=X∣λx-m12+my2+mz-aλ+cmm2+λsw-m32≤M,∀λ>0,∀m>0is the ultimate bound set of system (2), where Xt=xt,yt,zt,wt.Proof.
Define the following Lyapunov-like function:(4)Vλ,mX=Vλ,mx,y,z,w=λx-m12+my2+mz-aλ+cmm2+λsw-m32,where ∀λ>0,∀m>0,Xt=xt,yt,zt,vt,ut,ωt, and m1∈R, m3∈R are arbitrary constants.
And we can get(5)dVλ,mXtdt2=2λx-m1dxdt+2mydydt+2mz-aλ+cmmdzdt+2λsw-m3dwdt=2λx-m1ay-ax+sw+2mycx-dy-xz+2mz-aλ+cmmxy-bz+2λsw-m3-x-rw=-2aλx2+2aλm1+λsm3x-2dmy2-2aλm1y-2bmz2+2bmm2z-2λsrw2-2λsm1-λsrm3w.Let dVXt/dt=0. Then, we can get the surface Γ:(6)X∣-aλx2+aλm1+λsm3x-dmy2-aλm1y-bmz2+bmm2z-λsrw2-λsm1-λsrm3w=0is an ellipsoid in R4 for ∀λ>0,∀m>0,a>0,b>0,c>0,d>0,r>0,s>0. Outside Γ, dVλ,mXt/dt<0, while inside Γ, dVλ,mXt/dt>0. Thus, the ultimate boundedness for system (2) can only be reached on Γ. Since the Lyapunov-like function Vλ,mX is a continuous function and Γ is a bounded closed set, then the function (4) can reach its maximum value maxX∈Γ⁡Vλ,mX=M on the surface Γ that is defined in (6). Obviously, X∣Vλ,mX≤maxX∈Γ⁡Vλ,mX=M,X∈Γ contains solutions of system (2). It is obvious that the set Ωλ,m is the ultimate bound set for system (2).
This completes the proof.Theorem 2.
Suppose that∀a>0,b>0,d>0,r>0,c>0,s>0,λ>0,m>0.
Letxt,yt,zt,wt be an arbitrary solution of system (2) and (7)Lλ,m2=1θa2λ2md+λsr+aλm12+bmm22+λs2a+λsrm32,θ=min⁡a,b,d,r>0,Vλ,mX=Vλ,mx,y,z,w=λx-m12+my2+mz-aλ+cmm2+λsw-m32,∀λ>0,∀m>0,∀m1∈R,∀m3∈R.Then the estimation(8)Vλ,mXt-Lλ,m2≤Vλ,mXt0-Lλ,m2e-θt-t0holds for system (2), and thus Ωλ,m=X∣Vλ,mX≤Lλ,m2 is the globally exponential attractive set of system (2); that is, lim-t→+∞⁡Vλ,mXt≤Lλ,m2.Proof.
Define the following functions:(9)fx=-aλx2+2λsm3x,hy=-dmy2-2aλm1y,gw=-λsrw2-2λsm1w.then we can get(10)maxx∈R⁡fx=λs2m32a,maxy∈R⁡hy=a2λ2m12dm,maxw∈R⁡gw=λsm12r.Construct the Lyapunov-like function(11)Vλ,mX=Vλ,mx,y,z,w=λx-m12+my2+mz-aλ+cmm2+λsw-m32,∀λ>0,∀m>0,∀m1∈R,∀m3∈R.Differentiating the above Lyapunov-like function Vλ,mX in (11) with respect to time t along the trajectory of system (2) yields(12)dVλ,mXtdt2=2λx-m1dxdt+2mydydt+2mz-aλ+cmmdzdt+2λsw-m3dwdt=2λx-m1ay-ax+sw+2mycx-dy-xz+2mz-aλ+cmmxy-bz+2λsw-m3-x-rw=-2aλx2+2aλm1+λsm3x-2dmy2-2aλm1y-2bmz2+2bmm2z-2λsrw2-2λsm1-λsrm3w=-aλx2+2aλm1x-aλx2+2λsm3x-dmy2-dmy2-2aλm1y-bmz2-bmz2+2bmm2z-λsrw2-2λsm1w-λsrw2+2λsrm3w=-aλx2-2m1x-aλx2+2λsm3x-dmy2-dmy2-2aλm1y-bmz2-2m2z-bmz2-λsrw2-2m3w-λsrw2-2λsm1w=-aλx-m12+aλm12+fx-dmy2+hy-bmz-m22+bmm22-bmz2-λsrw-m32+λsrm32+gw=-aλx-m12-dmy2-bmz-m22-λsrw-m32+fx+hy+gw-bmz2+aλm12+bmm22+λsrm32≤-θVλ,mX+maxx∈R⁡fx+maxy∈R⁡hy+maxw∈R⁡gw+aλm12+bmm22+λsrm32=-θVλ,mX+λs2m32a+a2λ2m12md+λsm12r+aλm12+bmm22+λsrm32=-θVλ,mX-Lλ,m2.Thus, we have(13)Vλ,mXt-Lλ,m2≤Vλ,mXt0-Lλ,m2e-θt-t0,limt→+∞¯⁡Vλ,mXt≤Lλ,m2,which clearly shows that Ωλ,m=X∣Vλ,mX≤Lλ,m2 is the globally exponential attractive set of system (2).
The proof is complete.Remark 3.
(i) In particular, let us takem1=0, m3=0 in Theorem 2, we can get the conclusions below.Suppose that∀a>0,b>0,d>0,r>0,c>0,s>0,λ>0,m>0.Letxt,yt,zt,wt be an arbitrary solution of system (2) and (14)Mλ,m2=baλ+cm2θm,θ=min⁡a,b,d,r>0,Vλ,mX=Vλ,mx,y,z,w=λx2+my2+mz-aλ+cmm2+λsw2,∀λ>0,∀m>0.Then the estimation(15)Vλ,mXt-Mλ,m2≤Vλ,mXt0-Mλ,m2e-θt-t0holds for system (2), and thus (16)Σλ,m=x,y,z,w∣λx2+my2+mz-aλ+cmm2+λsw2≤Mλ,m2,∀λ>0,∀m>0is the globally exponential attractive set and positive invariant set of system (2); that is, (17)limt→+∞¯⁡Vλ,mXt≤Mλ,m2.(ii) Let us takem1=0,m3=0,λ=1,m=1; then we can get(18)Σ1,1=x,y,z,w∣x2+y2+z-a-c2+sw2≤ba+c2min⁡a,b,d,ras the globally exponential attractive set and positive invariant set of system (2) according to Theorem 2.Whena=19.42,b=1.91,c=29.45,d=2.86,r=0.23,s=9.64, we can get that (19)Ω1,1=x,y,z,w∣x2+y2+z-48.872+9.64w2≤140.82as the globally exponential attractive set and positive invariant set of system (2).Figure5 shows hyperchaotic attractor of system (2) in the xOyz space defined by Ω1,1. Figure 6 shows hyperchaotic attractor of system (2) in the xOyw space defined by Ω1,1. Figure 7 shows hyperchaotic attractor of system (2) in the xOzw space defined by Ω1,1. Figure 8 shows hyperchaotic attractor of system (2) in the yOzw space defined by Ω1,1.Figure 5
Localization of hyperchaotic attractor of system (2) in the xOyz space defined by Ω1,1.Figure 6
Localization of hyperchaotic attractor of system (2) in the xOyw space defined by Ω1,1.Figure 7
Localization of hyperchaotic attractor of system (2) in the xOzw space defined by Ω1,1.Figure 8
Localization of hyperchaotic attractor of system (2) in the yOzw space defined by Ω1,1.


## 3. Conclusions

In this paper, we have investigated some global dynamics of a generalized Lorenz–Stenflo system describing the evolution of finite amplitude acoustic gravity waves in a rotating atmosphere. Based on the Lyapunov method, the globally attractive sets were formulated combining simple inequalities. Finally, numerical examples were presented to show the effectiveness of the proposed method.



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