Complex Dynamical Behaviors of Lorenz-Stenflo Equations

Abstract

A mathematical chaos model for the dynamical behaviors of atmospheric acoustic-gravity waves is considered in this paper. Boundedness and globally attractive sets of this chaos model are studied by means of the generalized Lyapunov function method. The innovation of this paper is that it not only proves this system is globally bounded but also provides a series of global attraction sets of this system. The rate of trajectories entering from the exterior of the trapping domain to its interior is also obtained. Finally, the detailed numerical simulations are carried out to justify theoretical results. The results in this study can be used to study chaos control and chaos synchronization of this chaos system.

1. Introduction

In 1963, E.N. Lorenz [1] obtained the famous Lorenz chaotic system to describe weather changes. The Lorenz chaotic system stimulated the interest of researchers to study chaotic systems and chaotic phenomena. Since then, some new chaotic systems, which are not equivalent to the Lorenz system, were found, i.e., Rössler system [2,3], Chua system [4], Chen system [5], Lu system [6], Shimizu-Morioka system [7], hyperchaos Lorenz system [7], etc. Chaotic systems have potential applications in different scientific areas, such as secure communications [8], image encryption [9], laser technology [10], physical systems [11], circuits system [12] and chaos synchronization [13,14,15,16,17].

In 1996, the physicist Lennart Stenflo [18] obtained a new four-dimensional continuous-time dynamical chaotic system by adding a new variable $w$ to the Lorenz system to describe the complex dynamical behaviors of the atmospheric acoustic-gravity waves. The Lorenz-Stenflo system can be described by the following ordinary differential equations [18]:

$$\{\begin{array}{l}\frac{dx}{dt}=a\left(y-x\right)+rw,\\ \frac{dy}{dt}=cx-y-xz,\\ \frac{dz}{dt}=xy-bz,\\ \frac{dw}{dt}=-x-aw,\end{array}$$

(1)

where $a,b,c,r$ are parameters of system (1) and $a>0,b>0,c>0,r>0.$ $a$ is the Prandtl number of system (1), $c$ is the Rayleigh number of system (1), $b$ is the geometric parameter of system (1), and $r$ is the rotation parameter of system (1). The Lorenz-Stenflo system is a four-dimensional continuous-time dynamical system that can describe the dynamical behaviors of the atmospheric acoustic-gravity waves in a rotating atmosphere [18]. It is important for us to study acoustic gravity waves because they are associated with minor weather changes and large-scale phenomena. Dynamical behaviors of the Lorenz-Stenflo system, such as periodicity [19,20], bifurcation phenomenon [21,22], synchronization behaviors [23], and chaotic behaviors, have been studied by many researchers. System (1) has chaotic attractors when $a=1,b=0.7,$ $c=26,r=1.5.$ Figure 1 shows the chaotic attractors of the Lorenz-Stenflo system (1) in three-dimensional (3D) spaces.

Boundedness of chaotic systems is an important concept in dynamical systems [24,25], which plays an important role in chaos control and chaos synchronization. Boundedness of the Lorenz system has been investigated by Leonov et al. in a series of articles [26,27,28]. One can show that there is a bounded ellipsoid in $R^3$, which all orbits of the Lorenz system with an exponential rate will eventually enter.

Despite the fact that many qualitative results on the Lorenz-Stenflo system have been obtained [19,20,21,22,23], there is a fundamental question that has not been completely answered so far: Is there a global trapping region for the Lorenz-Stenflo system? How to get the boundedness of a chaotic system is particularly significant both for theoretical research and engineering applications [13,24,29,30]. Motivated by the above discussion, we will investigate the ultimate bound set and the global attractive sets of the Lorenz-Stenflo system. The novelty of this paper is that this paper not only proves the Lorenz-Stenflo system is globally bounded for all parameters of the system by means of the generalized Lyapunov function method, but it also gives a family of mathematical expressions of globally exponential attractive sets for the Lorenz-Stenflo system, with respect to all parameters of this system. The innovation of this paper is that this paper not only proves the Lorenz-Stenflo system is globally bounded but also provides a series of global attraction sets of this Lorenz-Stenflo system. The rate of trajectories entering from the exterior of the trapping domain to its interior is also obtained.

2. Boundedness and Global Attraction

The ultimate bound set and global domain of attraction of the Lorenz-Stenflo system will be studied in the following part. We can get the following Theorem 1 and Theorem 2.

Theorem 1.

For any$a>0,b>0,c>0,r>0,$there exists a positive number$L>0$, such that

$${\mathsf{\Psi }}_{\lambda ,m}=\{\left(x,y,z,w\right)|\lambda {\left(x-m_1\right)}^2+m{y}^2+m{\left(z-\frac{\lambda a+mc}{m}\right)}^2+\lambda r{\left(w-m_3\right)}^2\le L\}$$

(2)

is the ultimate bound set and positively invariant set of the Lorenz-Stenflo system (1), where$X\left(t\right)=\left(x\left(t\right),y\left(t\right),z\left(t\right),w\left(t\right)\right),\forall \lambda >0,\forall m>0,$and$m_1,m_3$are arbitrary real numbers.

Proof. 

Define the generalized Lyapunov function

$$V\left(X\right)=\lambda {\left(x-m_1\right)}^2+m{y}^2+m{\left(z-\frac{\lambda a+mc}{m}\right)}^2+\lambda r{\left(w-m_3\right)}^2,$$

(3)

where $\forall \lambda >0,\forall m>0,X\left(t\right)=\left(x\left(t\right),y\left(t\right),z\left(t\right),w\left(t\right)\right),$ and $m_1,m_3$ are arbitrary real numbers.

And we can get

$$\begin{array}{lll}{\frac{dV\left(X\left(t\right)\right)}{dt}|}_{(1)}& =& 2\lambda \left(x-m_1\right)\frac{dx}{dt}+2my\frac{dy}{dt}+2m\left(z-\frac{\lambda a+mc}{m}\right)\frac{dz}{dt}+2\lambda r\left(w-m_3\right)\frac{dw}{dt},\\ & =& 2\lambda \left(x-m_1\right)\left(ay-ax+rw\right)+2my\left(cx-y-xz\right)+2m\left(z-\frac{\lambda a+mc}{m}\right)\left(xy-bz\right)\\ & & +2\lambda r\left(w-m_3\right)\left(-x-aw\right),\\ & =& -2a\lambda x^2-2m{y}^2-2mb{z}^2-2a\lambda r{w}^2+2\lambda \left(r{m}_3+a{m}_1\right)x-2a\lambda m_{1}y\\ & & +2b\left(\lambda a+mc\right)z+2\lambda r\left(a{m}_3-m_1\right)w.\end{array}$$

Obviously, the set $\mathsf{\Gamma }$ defined by

$$\left\{\begin{array}{l}X|-2a\lambda x^2-2m{y}^2-2mb{z}^2-2a\lambda r{w}^2+2\lambda \left(r{m}_3+a{m}_1\right)x-2a\lambda m_{1}y+2b\left(\lambda a+mc\right)z\\ +2\lambda r\left(a{m}_3-m_1\right)w=0.\end{array}\right\}$$

(4)

is an ellipsoid in $R^4$ for $\forall a>0,b>0,c>0,r>0,\lambda >0,m>0.$ Outside $\mathsf{\Gamma }$, $\frac{dV\left(X\left(t\right)\right)}{dt}<0,$ while inside $\mathsf{\Gamma }$, $\frac{dV\left(X\left(t\right)\right)}{dt}>0.$ Thus, boundedness of system (1) can only be reached on $\mathsf{\Gamma }.$ Obviously, the continuous function (3) can reach its maximum value $\underset{X\in \mathsf{\Gamma }}{\max }V\left(X\right)=L$ on the bounded closed set $\mathsf{\Gamma }$. Clearly, $\{X|V\left(X\right)\le \underset{X\in \mathsf{\Gamma }}{\max }V\left(X\right)=L,X\in \mathsf{\Gamma }\}$ contains solutions to system (1). It is easy to prove that ${\mathsf{\Psi }}_{\lambda ,m}$ is the ultimate bound set and positively invariant set for system (1).

The proof is completed. □

Although Theorem 1 claims that there exist the ultimate bound set and positively invariant set for system (1), it is very difficult to calculate the mathematical analytic expression of the positive number $L>0$, since there are too many parameters in (2).We can get the expression of (2) easily for the special case $m_1=0,m_3=0.$ We can get the following theorem for the special case $m_1=0,m_3=0.$

Theorem 2.

For any$a>0,b>0,c>0,r>0,$such that

$${\mathsf{\Omega }}_{\lambda ,m}=\{\left(x,y,z,w\right)|\lambda x^2+m{y}^2+m{\left(z-\frac{\lambda a+mc}{m}\right)}^2+\lambda r{w}^2\le R^2\}$$

(5)

is the ultimate bound set and positively invariant set of Lorenz-Stenflo system (1), where$\forall \lambda >0,\forall m>0,$

$$R^2=\{\begin{array}{ll}\frac{b^2{\left(\lambda \alpha +mc\right)}^2}{4ma\left(b-a\right)},& a\le 1,b\ge 2a,\\ \frac{b^2{\left(\lambda a+mc\right)}^2}{4m\left(b-1\right)},& a>1,b\ge 2,\\ \frac{{\left(\lambda \alpha +mc\right)}^2}{m},& b<2a,b<2.\end{array}$$

Proof. 

Define the generalized Lyapunov function

$$V_1\left(X\right)=\lambda x^2+m{y}^2+m{\left(z-\frac{\lambda a+mc}{m}\right)}^2+\lambda r{w}^2,$$

(6)

where $\forall \lambda >0,\forall m>0,X\left(t\right)=\left(x\left(t\right),y\left(t\right),z\left(t\right),w\left(t\right)\right).$

And we can get

$$\begin{array}{lll}{\frac{d{V}_1\left(X\left(t\right)\right)}{dt}|}_{(1)}& =& 2\lambda x\frac{dx}{dt}+2my\frac{dy}{dt}+2m\left(z-\frac{\lambda a+mc}{m}\right)\frac{dz}{dt}+2\lambda rw\frac{dw}{dt},\\ & =& 2\lambda x\left(ay-ax+rw\right)+2my\left(cx-y-xz\right)+2m\left(z-\frac{\lambda a+mc}{m}\right)\left(xy-bz\right)\\ & & +2\lambda rw\left(-x-aw\right),\\ & =& -2a\lambda x^2-2m{y}^2-2mb{z}^2-2a\lambda r{w}^2+2b\left(\lambda a+mc\right)z.\end{array}$$

Obviously,

$${\mathsf{\Gamma }}_1=\{\left(x,y,z,w\right)|-2a\lambda x^2-2m{y}^2-2mb{z}^2-2a\lambda r{w}^2+2b\left(\lambda a+mc\right)z=0\}$$

(7)

is an ellipsoid in $R^4$ for $\forall a>0,b>0,c>0,r>0,\lambda >0,m>0.$ Outside ${\mathsf{\Gamma }}_1$, $\frac{d{V}_1\left(X\left(t\right)\right)}{dt}<0,$ while inside ${\mathsf{\Gamma }}_1$, $\frac{d{V}_1\left(X\left(t\right)\right)}{dt}>0.$ Thus, the ultimate boundedness for system (1) can only be reached on the bounded closed set ${\mathsf{\Gamma }}_1.$ Obviously, the continuous function $V_1\left(X\right)$ can reach its maximum value $\underset{X\in \mathsf{\Gamma }}{\max }{V}_1\left(X\right)=R^2$ on the bounded closed set ${\mathsf{\Gamma }}_1$. In order to calculate mathematical analytic expression of $\underset{X\in \mathsf{\Gamma }}{\max }{V}_1\left(X\right)=R^2$, we have to solve the following optimization problem:

$$\{\begin{array}{l}\max V_1\left(X\right)=\max \left\{\lambda x^2+m{y}^2+m{\left(z-\frac{\lambda a+mc}{m}\right)}^2+\lambda r{w}^2\right\},\\ s.t.-2a\lambda x^2-2m{y}^2-2mb{z}^2-2a\lambda r{w}^2+2b\left(\lambda a+mc\right)z=0,\end{array}$$

(8)

The above optimization problem is equivalent to

$$\{\begin{array}{l}\max V_1\left(X\right)=\max \left\{\lambda x^2+m{y}^2+m{\left(z-\frac{\lambda a+mc}{m}\right)}^2+\lambda r{w}^2\right\},\\ s.t.\frac{\lambda x^2}{\frac{b{\left(\lambda a+mc\right)}^2}{4am}}+\frac{m{y}^2}{\frac{b{\left(\lambda a+mc\right)}^2}{4m}}+\frac{m{\left(z-\frac{\lambda a+mc}{2m}\right)}^2}{\frac{{\left(\lambda a+mc\right)}^2}{4m}}+\frac{\lambda r{w}^2}{\frac{b{\left(\lambda a+mc\right)}^2}{4am}}=1.\end{array}$$

(9)

Let us take

$$\sqrt{\lambda }x=x_1,\sqrt{m}y=y_1,\sqrt{m}z=z_1,\sqrt{\lambda r}w=w_1,$$

Then, the above optimization problem (9) is changed into

$$\{\begin{array}{l}\max V_1\left(X\right)=\max \left\{x_1^2+y_1^2+{\left(z_1-\frac{\lambda a+mc}{\sqrt{m}}\right)}^2+w_1^2\right\},\\ s.t.\frac{x_1^2}{\frac{b{\left(\lambda a+mc\right)}^2}{4am}}+\frac{y_1^2}{\frac{b{\left(\lambda a+mc\right)}^2}{4m}}+\frac{{\left(z_1-\frac{\lambda a+mc}{2\sqrt{m}}\right)}^2}{\frac{{\left(\lambda a+mc\right)}^2}{4m}}+\frac{w_1^2}{\frac{b{\left(\lambda a+mc\right)}^2}{4am}}=1,\end{array}$$

(10)

We can get the solution of optimization problem (10) as

$$R^2=\{\begin{array}{ll}\frac{b^2{\left(\lambda \alpha +mc\right)}^2}{4ma\left(b-a\right)},& a\le 1,b\ge 2a,\\ \frac{b^2{\left(\lambda a+mc\right)}^2}{4m\left(b-1\right)},& a>1,b\ge 2,\\ \frac{{\left(\lambda \alpha +mc\right)}^2}{m},& b<2a,b<2.\end{array}$$

The proof is completed. □

Remark 1.

(i) Let us take$\forall \lambda >0,\forall m>0$in Theorem 2, then we can get a series of ultimate bound and positively invariant sets of the Lorenz-Stenflo system (1), according to Theorem 2;

(ii) Theorem 2 gives a family of mathematical expressions of the ultimate bound sets of the Lorenz-Stenflo system with respect to the parameters of the system. Particularly, we can get the following conclusion of Theorem 2for the special case$\lambda =1,m=1$:

For any$a>0,b>0,c>0,r>0,$such that

$${\mathsf{\Omega }}_2=\{\left(x,y,z,w\right)|x^2+y^2+{\left(z-a-c\right)}^2+r{w}^2\le l^2\}$$

(11)

is the ultimate bound set and positively invariant set of the Lorenz-Stenflo system (1), where

$$l^2=\{\begin{array}{ll}\frac{b^2{\left(\alpha +c\right)}^2}{4a\left(b-a\right)},& a\le 1,b\ge 2a,\\ \frac{b^2{\left(a+c\right)}^2}{4\left(b-1\right)},& a>1,b\ge 2,\\ {\left(\alpha +c\right)}^2,& b<2a,b<2.\end{array}$$

Figure 2 shows the chaotic attractor of the Lorenz-Stenflo system (1) in xOyz space, defined by ${\mathsf{\Omega }}_2$ in (11) with $a=1,b=0.7,c=26,r=1.5.$

Remark 2.

From Figure 2, we can see that numerical simulation is consistent with theoretical analysis results (11).

Though Theorem 1 and Theorem 2 give the ultimate bound and positively invariant set of the Lorenz-Stenflo system (1), it does not give the rate of the trajectories entering from the exterior of the trapping domain to its interior. The rate of trajectories entering from the exterior of the trapping domain to its interior of the Lorenz-Stenflo system (1) is described by the following Theorem 3:

Theorem 3.

For any$a>0,b>0,c>0,r>0,$and let

$$M^2=\frac{1}{\theta }\left[a\lambda {\left(m_1\right)}^2+\frac{\lambda r^2{\left(m_3\right)}^2}{a}+\frac{a^2{\lambda }^2{\left(m_1\right)}^2}{m}+\frac{b{\left(\lambda a+mc\right)}^2}{m}+a\lambda r{\left(m_3\right)}^2+\frac{\lambda r{\left(m_1\right)}^2}{a}\right],$$

$$V\left(X\right)=\lambda {\left(x-m_1\right)}^2+m{y}^2+m{\left(z-\frac{\lambda a+mc}{m}\right)}^2+\lambda r{\left(w-m_3\right)}^2,$$

$$\forall \lambda >0,\forall m>0,m_1\in R,m_3\in R,$$

$$\theta =\min \left(a,b,1\right)>0,X\left(t\right)=\left(x\left(t\right),y\left(t\right),z\left(t\right),w\left(t\right)\right).$$

Then we can obtain the following exponential estimate for the trajectory of system (1)

$$\left[V\left(X\left(t\right)\right)-M^2\right]\le \left[V\left(X\left(t_0\right)\right)-M^2\right]e^{-\theta \left(t-t_0\right)}.$$

(12)

Thus,${\Delta }_{\lambda ,m}=\left\{X|V\left(X\right)\le M^2\right\}$is the globally exponential attractive set of system (1), i.e.,

$$\underset{t\to +\infty }{\overline{\lim }}V\left(X\left(t\right)\right)\le M^2.$$

Proof. 

Define

$$f\left(x\right)=-a\lambda x^2+2\lambda r{m}_{3}x,h\left(y\right)=-m{y}^2-2a\lambda m_{1}y,g\left(w\right)=-a\lambda r{w}^2-2\lambda r{m}_{1}w,$$

then we can get

$$\underset{x\in R}{\max }f\left(x\right)=\frac{\lambda r^2{\left(m_3\right)}^2}{a},\underset{y\in R}{\max }h\left(y\right)=\frac{a^2{\lambda }^2{m}_1^2}{m},\underset{w\in R}{\max }g\left(w\right)=\frac{\lambda r{\left(m_1\right)}^2}{a}.$$

Define

$$V\left(X\right)=\lambda {\left(x-m_1\right)}^2+m{y}^2+m{\left(z-\frac{\lambda a+mc}{m}\right)}^2+\lambda r{\left(w-m_3\right)}^2,$$

(13)

where $\forall \lambda >0,\forall m>0,X\left(t\right)=\left(x\left(t\right),y\left(t\right),z\left(t\right),w\left(t\right)\right),$ and $m_1,m_3$ are arbitrary real numbers.

And we have

$$\begin{array}{l}{\frac{dV\left(X\left(t\right)\right)}{dt}|}_{(1)}=2\lambda \left(x-m_1\right)\frac{dx}{dt}+2my\frac{dy}{dt}+2m\left(z-\frac{\lambda a+mc}{m}\right)\frac{dz}{dt}+2\lambda r\left(w-m_3\right)\frac{dw}{dt},\\ =2\lambda \left(x-m_1\right)\left(ay-ax+rw\right)+2my\left(cx-y-xz\right)+2m\left(z-\frac{\lambda a+mc}{m}\right)\left(xy-bz\right)\\ +2\lambda r\left(w-m_3\right)\left(-x-aw\right),\\ =-2a\lambda x^2-2m{y}^2-2mb{z}^2-2a\lambda r{w}^2+2\lambda \left(r{m}_3+a{m}_1\right)x-2a\lambda m_{1}y\\ +2b\left(\lambda a+mc\right)z+2\lambda r\left(a{m}_3-m_1\right)w,\\ =-2a\lambda x^2+2\lambda \left(r{m}_3+a{m}_1\right)x-2m{y}^2-2a\lambda m_{1}y-2mb{z}^2+2b\left(\lambda a+mc\right)z\\ -2a\lambda r{w}^2+2\lambda r\left(a{m}_3-m_1\right)w,\\ =-a\lambda x^2+2\lambda a{m}_{1}x-a\lambda x^2+2\lambda r{m}_{3}x-2m{y}^2-2a\lambda m_{1}y-2mb{z}^2+2b\left(\lambda a+mc\right)z\\ -a\lambda r{w}^2+2\lambda ra{m}_{3}w-a\lambda r{w}^2-2\lambda r{m}_{1}w,\\ =-a\lambda x^2+2\lambda a{m}_{1}x+f\left(x\right)-2m{y}^2-2a\lambda m_{1}y-2mb{z}^2+2b\left(\lambda a+mc\right)z\\ -a\lambda r{w}^2+2\lambda ra{m}_{3}w+g\left(w\right),\\ \le -a\lambda x^2+2\lambda a{m}_{1}x+\underset{x\in R}{\max }f\left(x\right)-2m{y}^2-2a\lambda m_{1}y-2mb{z}^2+2b\left(\lambda a+mc\right)z\\ -a\lambda r{w}^2+2\lambda ra{m}_{3}w+\underset{w\in R}{\max }g\left(w\right),\\ =-a\lambda x^2+2\lambda a{m}_{1}x+\frac{\lambda r^2{\left(m_3\right)}^2}{a}-2m{y}^2-2a\lambda m_{1}y-2mb{z}^2+2b\left(\lambda a+mc\right)z\\ -a\lambda r{w}^2+2\lambda ra{m}_{3}w+\frac{\lambda r{\left(m_1\right)}^2}{a},\\ =-a\lambda {\left(x-m_1\right)}^2+a\lambda m_1^2+\frac{\lambda r^2{\left(m_3\right)}^2}{a}-m{y}^2+h\left(y\right)-2mb{z}^2+2b\left(\lambda a+mc\right)z\\ -a\lambda r{w}^2+2\lambda ra{m}_{3}w+\frac{\lambda r{\left(m_1\right)}^2}{a},\\ \le -a\lambda {\left(x-m_1\right)}^2+a\lambda m_1^2+\frac{\lambda r^2{\left(m_3\right)}^2}{a}-m{y}^2+\underset{y\in R}{\max }h\left(y\right)-mb{z}^2+2b\left(\lambda a+mc\right)z\\ -a\lambda r{\left(w-m_3\right)}^2+a\lambda r{m}_3^2+\frac{\lambda r{\left(m_1\right)}^2}{a},\\ =-a\lambda {\left(x-m_1\right)}^2+a\lambda m_1^2+\frac{\lambda r^2{\left(m_3\right)}^2}{a}-m{y}^2+\frac{a^2{\lambda }^2{m}_1^2}{m}-mb{z}^2+2b\left(\lambda a+mc\right)z\\ -a\lambda r{\left(w-m_3\right)}^2+a\lambda r{m}_3^2+\frac{\lambda r{\left(m_1\right)}^2}{a},\\ =-a\lambda {\left(x-m_1\right)}^2+a\lambda m_1^2+\frac{\lambda r^2{\left(m_3\right)}^2}{a}-m{y}^2+\frac{a^2{\lambda }^2{m}_1^2}{m}-mb{\left(z-\frac{\lambda a+mc}{m}\right)}^2+\frac{b{\left(\lambda a+mc\right)}^2}{m}\\ -a\lambda r{\left(w-m_3\right)}^2+a\lambda r{m}_3^2+\frac{\lambda r{\left(m_1\right)}^2}{a},\\ =-a\lambda {\left(x-m_1\right)}^2-m{y}^2-mb{\left(z-\frac{\lambda a+mc}{m}\right)}^2-a\lambda r{\left(w-m_3\right)}^2+a\lambda m_1^2+\frac{\lambda r^2{\left(m_3\right)}^2}{a}\\ +\frac{a^2{\lambda }^2{\left(m_1\right)}^2}{m}+\frac{b{\left(\lambda a+mc\right)}^2}{m}+a\lambda r{\left(m_3\right)}^2+\frac{\lambda r{\left(m_1\right)}^2}{a},\\ =-\theta V\left(X\right)+a\lambda m_1^2+\frac{\lambda r^2{\left(m_3\right)}^2}{a}+\frac{a^2{\lambda }^2{\left(m_1\right)}^2}{m}+\frac{b{\left(\lambda a+mc\right)}^2}{m}+a\lambda r{\left(m_3\right)}^2+\frac{\lambda r{\left(m_1\right)}^2}{a},\\ =-\theta \left[V\left(X\left(t\right)\right)-M^2\right].\end{array}$$

Thus, we have the exponential estimate for the trajectory of system (1):

$$\left[V\left(X\left(t\right)\right)-M^2\right]\le \left[V\left(X\left(t_0\right)\right)-M^2\right]e^{-\theta \left(t-t_0\right)}.$$

and

$$\underset{t\to +\infty }{\overline{\lim }}V\left(X\left(t\right)\right)\le M^2.$$

Hence,

$${\Delta }_{\lambda ,m}=\left\{X|V\left(X\right)\le M^2\right\}$$

is the globally exponential attractive set of system (1).

The proof is completed. □

Remark 3.

(i) Let us take$\forall \lambda >0,\forall m>0$in Theorem 3, then we can get a series of global exponential attractive sets of the Lorenz-Stenflo system (1), according to Theorem 3;

(ii) Theorem 3 not only gives a family of mathematical expressions of global exponential attractive sets for the Lorenz-Stenflo system with respect to the parameters of this system, but also provides an explicit exponential rate of trajectories entering from the exterior of the trapping domain to its interior.

3. Conclusions

The main goal of this paper is to study the Lorenz-Stenflo chaos system, describing the dynamical behavior of atmospheric acoustic-gravity waves. A family of hyperelliptic estimates of the ultimate bound and positively invariant set for the Lorenz-Stenflo system is obtained by means of the generalized Lyapunov function method. Furthermore, the rate of trajectories entering from the exterior of the trapping domain to its interior is also obtained. This generalized Lyapunov function method can be extended to study other chaotic systems. The results in this study provide a new and in-depth understanding of the Lorenz-Stenflo chaotic system. The results in this paper can be used to study the Lyapunov dimension of attractors, the Hausdorff dimension of attractors, chaos control, and chaos synchronization of this Lorenz-Stenflo chaos system in the future.