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Qualitative behaviors of the high-order Lorenz-Stenflo chaotic system arising in mathematical physics describing the atmospheric acoustic-gravity waves

Advances in Difference Equations20172017:290

https://doi.org/10.1186/s13662-017-1351-7

Received: 7 June 2017

Accepted: 6 September 2017

Published: 15 September 2017

Abstract

The boundedness of chaotic systems plays an important role in investigating the stability of the equilibrium, estimating the Lyapunov dimension of attractors, the Hausdorff dimension of attractors, the existence of periodic solutions, chaos control, and chaos synchronization. However, as far as the authors know, there are only a few papers dealing with bounds of high-order chaotic systems due to their complex algebraic structure. To sort this out, in this paper, we study the bounds of a high-order Lorenz-Stenflo system arising in mathematical physics. Based on Lyapunov stability theory, we show that there exists a globally exponential attractive set for this system. The innovation of the paper is that we not only prove that this system is globally bounded for all the parameters, but also give a family of mathematical expressions of global exponential attractive sets of this system with respect to its parameters. We also study some other dynamical characteristics of this chaotic system such as invariant sets and chaotic behaviors. To justify the theoretical analysis, we carry out detailed numerical simulations.

Keywords

High-order Lorenz-Stenflo systemLyapunov exponentsLyapunov stabilitydomain of attractionnonlinear dynamics

1 Introduction

Chaos phenomena and chaotic systems have been extensively studied by many researchers due to their various applications in the fields of atmospheric dynamics, population dynamics, electric circuits, cryptology, fluid dynamics, lasers, engineering, stock exchanges, chemical reactions, and so on [111]. Most of the complex dynamical phenomena are characterized by chaotic and hyperchaotic systems of nonlinear ordinary differential equations [121].

Stenflo [22] obtained the Lorenz-Stenflo equation from the equations describing the atmospheric acoustic-gravity waves. The Lorenz-Stenflo equation is described by the following equations:
$$ \textstyle\begin{cases} \frac{dx}{dt} = \sigma ( y - x ) + sv, \\ \frac{dy}{dt} = - xz + rx - y, \\ \frac{dz}{dt} = xy - bz, \\ \frac{dv}{dt} = - x - \sigma v, \end{cases} $$
(1)
where x, y, z, v are state variables of the Lorenz-Stenflo equation (1), and σ, s, r, b are positive parameters of the system, σ is the Prandtl number, r is the generalized Rayleigh parameter, b is the geometric parameter, and s is the rotation parameter. The Lorenz-Stenflo equation is regarded as an extended Lorenz equation since it reduces to the Lorenz equation [1] when the rotation parameter s is zero, and it also can be obtained from the rotating thermal convection equations. The Lorenz-Stenflo system is a four-dimensional continuous-time dynamical system, derived to model atmospheric acoustic-gravity waves in a rotating atmosphere. Knowledge about acoustic gravity waves is important because they may be responsible both for minor local weather changes and for large-scale phenomena, for instance, storms. Many dynamical behaviors such as stability [23], bifurcation [24, 25], periodic solutions [26] and chaotic behaviors [27] of the Lorenz-Stenflo equations have been thoroughly studied for decades after Stenflo.
To improve the stability or predictability of the Lorenz-Stenflo system (1), Park et al. [26, 28] introduced the high-order Lorenz-Stenflo equations by including terms with higher vertical wave numbers:
$$ \textstyle\begin{cases} \frac{dx}{dt} = \sigma ( y - x ) + sv, \\ \frac{dy}{dt} = - xz + rx - y, \\ \frac{dz}{dt} = xy - xu - bz, \\ \frac{dv}{dt} = - x - \sigma v, \\ \frac{du}{dt} = xz - 2x\omega - ( 1 + 2b ) u, \\ \frac{d\omega }{dt} = 2xu - 4b\omega , \end{cases} $$
(2)
where x, y, z, v, u, and ω are state variables, σ, s, r, and b are the same positive parameters as in the original Lorenz-Stenflo system (1). Various dynamical behaviors, such as stability, periodic and chaotic solutions, and Lyapunov exponents spectra of the high-order Lorenz-Stenflo equations (2), have been thoroughly studied [28, 29]. When \(\sigma = 10\), \(b = \frac{8}{3}\), \(r = 40\), \(s = 50\), system (2) has a chaotic attractor [28, 29]. When \(\sigma = 10\), \(b = \frac{8}{3}\), \(r = 40\), \(s = 50\), chaotic attractors of system (2) in \(xOyz\) space are shown in Figure 1. Chaotic attractors of system (2) in \(yOz\) plane are shown in Figure 2. Chaotic attractors of system (2) in \(xOy\) plane are shown in Figure 3. Chaotic attractors of system (2) in \(xOz\) plane are shown in Figure 4.
Figure 1

Chaotic attractors of system ( 2 ) in \(\pmb{xOyz}\) space.

Figure 2

Chaotic attractors of system ( 2 ) in \(\pmb{yOz}\) plane.

Figure 3

Chaotic attractors of system ( 2 ) in \(\pmb{xOy}\) plane.

Figure 4

Chaotic attractors of system ( 2 ) in \(\pmb{xOz}\) plane.

Remark 1

An oscillation in a dynamical system can be localized numerically if the initial conditions from its open neighborhood lead to the long-time behavior that approaches the oscillation. Such an oscillation (or a set of oscillations) is called an attractor, and its attracting set is called a basin of attraction. Thus, from a point of view of the numerical analysis of nonlinear dynamical models, it is essential to classify an attractor as a self-excited or hidden attractor depending on simplicity of finding its basin of attraction [14, 30, 31]: An attractor is called a self-excited attractor if its basin of attraction intersects with an arbitrarily small open neighborhood of an unstable equilibrium; otherwise, it is called a hidden attractor (see [14] and [30, 31] for a detailed discussion of the attractors in dynamical systems). Although the authors in [28, 29] point out that system (2) has a attractor for \(\sigma = 10\), \(b = \frac{8}{3}\), \(r = 40\), \(s = 50\), they do not point out which type of attractor system (2) has. It is necessary and interesting to discuss the classification of the attractors in system (2) in the future.

2 Some dynamics of high-order Lorenz-Stenflo system

2.1 Invariance

The positive z-axis, u-axis, and ω-axis are invariant under the flow, that is, they are positively invariant under the flow generated by system (2). However, this is not the case on the positive x-axis, y-axis, and v-axis for system (2) since they are all not positively invariant under the flow generated by system (2).

2.2 Ultimate bound set and domain of attraction

In this section, we further investigate the ultimate bound set and global domain of attraction of the high-order Lorenz-Stenflo system (2). The main result is described by the following theorems, Theorems 1 and 2.

Theorem 1

For any \(\lambda_{1} > 0\), \(m > 0\), \(\sigma > 0\), \(s > 0\), \(r > 0\), \(b > 0\), there exists a positive number \(M > 0\) such that
$$ \Psi = \bigl\{ X \mid \lambda_{1} ( x - m_{2} ) ^{2} + my^{2} + m ( z - 2 \lambda_{2} ) ^{2} + \lambda_{1}s ( v - m_{3} ) ^{2} + mu^{2} + m ( \omega - \lambda_{2} ) ^{2} \le M \bigr\} $$
is the ultimate bound and positively invariant set of the high-order Lorenz-Stenflo system (2), where \(X ( t ) = ( x ( t ) ,y ( t ) ,z ( t ) ,v ( t ) ,u ( t ) ,\omega ( t ) )\).

Proof

Define the following Lyapunov-like function
$$ V ( X ) = \lambda_{1} ( x - m_{2} ) ^{2} + my^{2} + m ( z - 2\lambda_{2} ) ^{2} + \lambda_{1}s ( v - m_{3} ) ^{2} + mu^{2} + m ( \omega - \lambda_{2} ) ^{2}, $$
(3)
where \(\forall \lambda_{1} > 0\), \(\forall m > 0\), \(\lambda_{2} = \frac{ \lambda_{1}\sigma + mr}{2m}\), \(X ( t ) = ( x ( t ) , y ( t ) , z ( t ) , v ( t ) , u ( t ) , \omega ( t ) ) \), and \(m_{2} \in R\), \(m_{3} \in R\) are arbitrary constants.
We have
$$\begin{aligned}& \frac{dV( X( t ) ) }{dt}\bigg| _{(2)} \\& \quad = 2\lambda_{1} ( x - m_{2} ) \frac{dx}{dt} + 2my \frac{dy}{dt} + 2m ( z - 2\lambda_{2} ) \frac{dz}{dt} + 2 \lambda_{1}s ( v - m_{3} ) \frac{dv}{dt} \\& \quad \quad {}+ 2mu\frac{du}{dt} + 2m ( \omega - \lambda_{2} ) \frac{d \omega }{dt}, \\& \quad = 2\lambda_{1} ( x - m_{2} ) ( \sigma y - \sigma x + sv ) + 2my ( - xz + rx - y ) + 2m ( z - 2\lambda_{2} ) ( xy - xu - bz ) \\& \quad \quad {}+ 2\lambda_{1}s ( v - m_{3} ) ( - x - \sigma v ) + 2mu \bigl[ xz - 2x\omega - ( 1 + 2b ) u \bigr] + 2m ( \omega - \lambda_{2} ) ( 2xu - 4b\omega ) \\& \quad = - 2\lambda_{1}\sigma x^{2} + 2\lambda_{1} \sigma m_{2}x + 2\lambda _{1}sm_{3}x - 2my^{2} - 2\lambda_{1}\sigma m_{2}y - 2bmz^{2} + 4bm \lambda_{2}z - 2\lambda_{1}s\sigma v^{2} \\& \quad \quad {}- 2\lambda_{1}sm_{2}v + 2\lambda_{1}sm_{3} \sigma v - 2m ( 1 + 2b ) u ^{2} - 8bm\omega^{2} + 8bm \lambda_{2}\omega. \end{aligned}$$
Let \(\frac{dV ( X ( t ) ) }{dt} = 0\). Then, we get that the surface
$$ \Gamma : \left \{ \textstyle\begin{array}{l} X \mid - 2\lambda_{1}\sigma x^{2} + 2\lambda_{1}\sigma m _{2}x + 2\lambda_{1}sm_{3}x - 2my^{2} \\ \quad {}- 2\lambda_{1}\sigma m_{2}y - 2bmz ^{2} + 4bm\lambda_{2}z - 2\lambda_{1}s\sigma v^{2} \\ \quad {}- 2\lambda_{1}sm_{2}v + 2\lambda_{1}sm_{3}\sigma v - 2m ( 1 + 2b ) u ^{2} - 8bm\omega^{2} + 8bm\lambda_{2}\omega = 0 \end{array}\displaystyle \right \} $$
(4)
is an ellipsoid in \(R^{6}\) \(\forall \lambda_{1} > 0\), \(m > 0\), \(\sigma > 0\), \(s > 0\), \(r > 0\), \(b > 0\). Outside Γ, \(\frac{dV ( X ( t ) ) }{dt} < 0\), whereas inside Γ, \(\frac{dV ( X ( t ) ) }{dt} > 0\). Thus, the ultimate boundedness for system (2) can only be reached on Γ. Since the Lyapunov-like function \(V ( X ) \) is a continuous function and Γ is a bounded closed set, the function (3) can reach its maximum value \(\max_{X \in \Gamma } V ( X ) = M\) on the surface Γ. Obviously, \(\{ X \mid V ( X ) \le \max_{X \in \Gamma } V ( X ) = M,X\in \Gamma \} \) contains solutions of system (2). It is obvious that the set Ψ is the ultimate bound set and positively invariant set for system (2).

This completes the proof. □

Theorem 1 points that the trajectories of system (2) are ultimately bounded. However, Theorem 1 does not give the rate of the trajectories of system (2) going from the exterior of the trapping set to the interior of the trapping set. The rate of the trajectories rate of system (2) is studied in the next theorem, Theorem 2.

In the following section, we further investigate the globally attractive set of the high-order Lorenz-Stenflo system (2). We use the following Lyapunov-like function
$$ V ( X ) = \lambda_{1} ( x - m_{2} ) ^{2} + my^{2} + m ( z - 2\lambda_{2} ) ^{2} + \lambda_{1}s ( v - m_{3} ) ^{2} + mu^{2} + m ( \omega - \lambda_{2} ) ^{2} $$
(5)
which is obviously positive definite and radially unbounded. Here, \(\forall \lambda_{1} > 0\), \(\forall m > 0\), \(\lambda_{2} = \frac{\lambda_{1}\sigma + mr}{2m}\), and \(m_{2} \in R\), \(m_{3} \in R\) are arbitrary constants.

Let \(X ( t ) = ( x ( t ) ,y ( t ) ,z ( t ) ,v ( t ) ,u ( t ) ,\omega ( t ) ) \) be an arbitrary solution of system (2). We have the following results for system (2).

Theorem 2

Suppose that \(\forall \sigma > 0\), \(s > 0\), \(r > 0\), \(b > 0\), and let
$$\begin{aligned}& L^{2} = \frac{1}{\theta } \biggl[ \frac{\lambda_{1}s^{2} ( m_{3} ) ^{2}}{\sigma } + \frac{ ( \lambda_{1}\sigma m_{2} ) ^{2}}{m} + \frac{\lambda_{1}s ( m_{2} ) ^{2}}{\sigma } + \lambda_{1} \sigma ( m_{2} ) ^{2} + 8bm ( \lambda_{2} ) ^{2} + \lambda_{1}s\sigma ( m_{3} ) ^{2} \biggr] , \\& \theta = \min ( \sigma ,b ) > 0. \end{aligned}$$
Then, for system (2), we have the estimate
$$ \bigl[ V \bigl( X ( t ) \bigr) - L^{2} \bigr] \le \bigl[ V \bigl( X ( t_{0} ) \bigr) - L^{2} \bigr] e^{ - \theta ( t - t_{0} ) }. $$
(6)
Thus \(\Omega = \{ X \mid V ( X ) \le L^{2} \} \) is a globally exponential attractive set of system (2), that is, \(\overline{\lim }_{t \to + \infty } V ( X ( t ) ) \le L^{2}\).

Proof

Define the following functions:
$$ f ( x ) = - \lambda_{1}\sigma x^{2} + 2\lambda_{1}sm_{3}x,\qquad h( y ) = - my^{2} - 2\lambda_{1}\sigma m_{2}y,\qquad g ( v ) = - \lambda_{1}s\sigma v^{2} - 2\lambda_{1}m_{2}sv. $$
Then we have
$$ \max_{x \in R}f ( x ) = \frac{\lambda_{1}s^{2} ( m_{3} ) ^{2}}{\sigma },\qquad \max _{y \in R}h ( y ) = \frac{ ( \lambda _{1}\sigma m_{2} ) ^{2}}{m},\qquad \max _{v \in R}g ( v ) = \frac{ \lambda_{1}s ( m_{2} ) ^{2}}{\sigma }. $$
Differentiating the Lyapunov-like function \(V ( X ) \) in (5) with respect time t along the trajectory of system (2) yields
$$\begin{aligned}& \frac{dV ( X ( t ) ) }{dt} \bigg| _{(2)} \\& \quad = 2 \lambda_{1} ( x - m_{2} ) \frac{dx}{dt} + 2my\frac{dy}{dt} + 2m ( z - 2 \lambda_{2} ) \frac{dz}{dt} + 2\lambda_{1}s ( v - m_{3} ) \frac{dv}{dt} \\& \quad \quad {} + 2mu\frac{du}{dt} + 2m ( \omega - \lambda_{2} ) \frac{d \omega }{dt} \\& \quad = 2\lambda_{1} ( x - m_{2} ) ( \sigma y - \sigma x + sv ) + 2my ( - xz + rx - y ) + 2m ( z - 2\lambda_{2} ) ( xy - xu - bz ) \\& \quad \quad {}+ 2\lambda_{1}s ( v - m_{3} ) ( - x - \sigma v ) + 2mu \bigl[ xz - 2x\omega - ( 1 + 2b ) u \bigr] + 2m ( \omega - \lambda_{2} ) ( 2xu - 4b\omega ) \\& \quad = - 2\lambda_{1}\sigma x^{2} + 2\lambda_{1}\sigma m_{2}x \\& \quad \quad {}+ 2\lambda _{1}sm_{3}x - 2my^{2} - 2\lambda_{1}\sigma m_{2}y - 2bmz^{2} + 4bm \lambda_{2}z - 2\lambda_{1}s\sigma v^{2} \\& \quad \quad {}- 2\lambda_{1}sm_{2}v + 2\lambda_{1}sm_{3} \sigma v - 2m ( 1 + 2b ) u ^{2} - 8bm\omega^{2} + 8bm \lambda_{2}\omega \\& \quad = - \lambda_{1}\sigma x^{2} + 2\lambda_{1}\sigma m_{2}x - \lambda_{1} \sigma x^{2} \\& \quad \quad {} + 2 \lambda_{1}sm_{3}x - my^{2} - my^{2} - 2 \lambda_{1} \sigma m_{2}y - 2bmz^{2} + 4bm \lambda_{2}z \\& \quad \quad {}- \lambda_{1}s\sigma v^{2} + 2\lambda_{1}sm_{3} \sigma v - \lambda_{1}s \sigma v^{2} - 2\lambda_{1}sm_{2}v - 2m ( 1 + 2b ) u^{2} - 8bm \omega^{2} + 8bm \lambda_{2}\omega \\& \quad = - \lambda_{1}\sigma x^{2} + 2\lambda_{1}\sigma m_{2}x + f ( x ) - my^{2} + h ( y ) - 2bmz^{2} + 4bm\lambda_{2}z - \lambda_{1}s\sigma v^{2} + 2 \lambda_{1}sm_{3}\sigma v \\& \quad \quad {}+ g ( v ) - 2m ( 1 + 2b ) u^{2} - 8bm\omega^{2} + 8bm \lambda_{2}\omega \\& \quad \le - \lambda_{1}\sigma x^{2} + 2\lambda_{1}\sigma m_{2}x + f ( x ) - my^{2} + h ( y ) - bmz^{2} + 4bm \lambda_{2}z - \lambda_{1}s\sigma v^{2} + 2 \lambda_{1}sm_{3}\sigma v \\& \quad \quad {}+ g ( v ) - 2m ( 1 + 2b ) u^{2} - 8bm\omega^{2} + 8bm \lambda_{2}\omega \\& \quad = - \lambda_{1}\sigma ( x - m_{2} ) ^{2} + \lambda_{1} \sigma ( m_{2} ) ^{2} + f ( x ) - my^{2} + h ( y ) - bm ( z - 2\lambda_{2} ) ^{2} + 4bm ( \lambda _{2} ) ^{2} \\& \quad \quad {}- \lambda_{1}s\sigma ( v - m_{3} ) ^{2} + \lambda_{1}s \sigma ( m_{3} ) ^{2} + g ( v ) - 2m ( 1 + 2b ) u^{2} - 8bm\omega^{2} + 8bm\lambda_{2} \omega \\& \quad = - \lambda_{1}\sigma ( x - m_{2} ) ^{2} -my^{2} - bm ( z - 2\lambda_{2} ) ^{2} \\& \quad \quad {}- \lambda_{1}s\sigma ( v - m_{3} ) ^{2} - 2m ( 1 + 2b ) u^{2} - 8bm\omega^{2} + 8bm\lambda_{2} \omega \\& \quad \quad {}+ f ( x ) + h ( y ) + g ( v ) + \lambda _{1}\sigma ( m_{2} ) ^{2} + 4bm ( \lambda_{2} ) ^{2} + \lambda_{1}s\sigma ( m_{3} ) ^{2} \\& \quad \le - \lambda_{1}\sigma ( x - m_{2} ) ^{2} - my^{2} - bm ( z - 2\lambda_{2} ) ^{2} - \lambda_{1}s\sigma ( v - m _{3} ) ^{2} - mu^{2} - 4bm\omega^{2} + 8bm\lambda_{2}\omega \\& \quad \quad {}+ f ( x ) + h ( y ) + g ( v ) + \lambda _{1}\sigma ( m_{2} ) ^{2} + 4bm ( \lambda_{2} ) ^{2} + \lambda_{1}s\sigma ( m_{3} ) ^{2} \\& \quad = - \lambda_{1}\sigma ( x - m_{2} ) ^{2} -my^{2} - bm ( z - 2\lambda_{2} ) ^{2} - \lambda_{1}s\sigma ( v - m_{3} ) ^{2} - mu^{2} - 4bm ( \omega - \lambda_{2} ) ^{2} \\& \quad \quad {}+ f ( x ) + h ( y ) + g ( v ) + \lambda _{1}\sigma ( m_{2} ) ^{2} + 4bm ( \lambda_{2} ) ^{2} + \lambda_{1}s\sigma ( m_{3} ) ^{2} + 4bm ( \lambda _{2} ) ^{2} \\& \quad \le - \lambda_{1}\sigma ( x - m_{2} ) ^{2} - my^{2} - bm ( z - 2\lambda_{2} ) ^{2} - \lambda_{1}s\sigma ( v - m _{3} ) ^{2} - mu^{2} - bm ( \omega - \lambda_{2} ) ^{2} \\& \quad \quad {}+ f ( x ) + h ( y ) + g ( v ) + \lambda _{1}\sigma ( m_{2} ) ^{2} + 8bm ( \lambda_{2} ) ^{2} + \lambda_{1}s\sigma ( m_{3} ) ^{2} \\& \quad \le - \theta V ( X ) + \max_{x \in R}f ( x ) + \max _{y \in R}h ( y ) + \max_{v \in R}g ( v ) + \lambda_{1}\sigma ( m_{2} ) ^{2} + 8bm ( \lambda_{2} ) ^{2} + \lambda_{1}s\sigma ( m_{3} ) ^{2} \\& \quad = - \theta V ( X ) + \frac{\lambda_{1}s^{2} ( m_{3} )^{2}}{\sigma } + \frac{ ( \lambda_{1}\sigma m_{2} ) ^{2}}{m} + \frac{\lambda_{1}s ( m_{2} ) ^{2}}{\sigma } \\& \quad \quad {} + \lambda_{1} \sigma ( m_{2} ) ^{2} + 8bm ( \lambda_{2} ) ^{2} + \lambda_{1}s\sigma ( m_{3} ) ^{2} \\& \quad = - \theta \bigl[ V \bigl( X ( t ) \bigr) - L^{2} \bigr] . \end{aligned}$$
Thus, we have
$$ \bigl[ V \bigl( X ( t ) \bigr) - L^{2} \bigr] \le \bigl[ V \bigl( X ( t_{0} ) \bigr) - L^{2} \bigr] e^{ - \theta ( t - t_{0} ) }. $$
Therefore,
$$ \mathop{\overline{\lim}}_{t \to + \infty } V \bigl( X ( t ) \bigr) \le L^{2}, $$
which clearly shows that \(\Omega = \{ X \mid V ( X ) \le L^{2} \} \) is a globally exponential attractive set of system (2). The proof is complete. □

Remark 2

(i) Let us take \(\lambda_{1} = 1\), \(m = 1\), \(m_{2} = 0\), \(m_{3} = 0\) in Theorem 2. Then we get that
$$ \Delta = \biggl\{ ( x,y,z,v,u,w ) \mid x^{2} + y^{2} + ( z - \sigma - r ) ^{2} + sv^{2} + u^{2} + \biggl( \omega - \frac{\sigma + r}{2} \biggr) ^{2} \le \frac{2b ( \sigma + r ) ^{2}}{\min ( \sigma ,b ) } \biggr\} $$
(7)
is a globally exponential attractive set of system (2) according to Theorem 2.
(ii) Taking \(\sigma = 10\), \(b = \frac{8}{3}\), \(r = 40\), \(s = 50\), we get that
$$ \Delta = \bigl\{ ( x,y,z,v,u,w ) \mid x^{2} + y ^{2} + ( z - 50 ) ^{2} + 50v^{2} + u^{2} + ( \omega - 25 ) ^{2} \le ( 50\sqrt{2} ) ^{2} \bigr\} $$
(8)
is a globally exponential attractive set of system (2) according to Theorem 2. Figure 5 shows chaotic attractors of system (2) in the \(( x,y,z ) \) space defined by Δ in (8). Figure 6 shows chaotic attractors of system (2) in the \(( x,y,v ) \) space defined by Δ in (8). Figure 7 shows chaotic attractors of system (2) in the \(( x,y,u ) \) space defined by Δ in (8). Figure 8 shows chaotic attractors of system (2) in the \(( x,y, \omega ) \) space defined by Δ in (8).
Figure 5

Chaotic attractors of ( 2 ) in the \(\pmb{( x,y,z ) }\) space defined byΔ.

Figure 6

Chaotic attractors of ( 2 ) in the \(\pmb{( x,y,v ) }\) space defined byΔ.

Figure 7

Chaotic attractors of ( 2 ) in the \(\pmb{( x,y,u ) }\) space defined byΔ.

Figure 8

Chaotic attractors of ( 2 ) in the \(\pmb{( x,y,\omega ) }\) space defined byΔ.

3 Conclusions

By means of Lyapunov-like functions, we have studied some dynamical behaviors of a high-order Lorenz-Stenflo system using theoretical analysis and numerical simulations. The obtained results show that this system has complex dynamics and this system deserves a further detailed investigation. The results of this paper are useful in many engineering applications such as chaos synchronization, chaos cryptology, coding information, and information compression.

Declarations

Acknowledgements

Fuchen Zhang is supported by National Natural Science Foundation of China (Grant No. 11501064), the Basic and Advanced Research Project of CQCSTC (Grant No. cstc2014jcyjA00040), the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No. KJ1500605), the Research Fund of Chongqing Technology and Business University (Grant No. 2014-56-11), China Postdoctoral Science Foundation (Grant No. 2016M590850). We thank professors Jinhu Lu in Chinese Academy of Sciences, Xiaofeng Liao in College of Electronic and Information Engineering of Southwest University, Gaoxiang Yang in Ankang University and Ping Zhou in Chongqing University of Posts and Telecommunications for their help. The authors wish to thank the editors and reviewers for their conscientious reading of this paper and numerous comments for improvement, which were extremely useful and helpful in modifying the paper.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, People’s Republic of China
(2)
Mathematical post-doctoral station, College of Mathematics and Statistics, Southwest University, Chongqing, People’s Republic of China
(3)
College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, People’s Republic of China

References

  1. Lorenz, EN: Deterministic non-periods flows. J. Atmos. Sci. 20, 130-141 (1963) View ArticleGoogle Scholar
  2. Zhang, FC, Zhang, GY: Further results on ultimate bound on the trajectories of the Lorenz system. Qual. Theory Dyn. Syst. 15(1), 221-235 (2016) MathSciNetView ArticleMATHGoogle Scholar
  3. Li, TY, Yorke, JA: Period three implies chaos. Am. Math. Mon. 82, 985-992 (1975) MathSciNetView ArticleMATHGoogle Scholar
  4. Bao, H, Park, JH, Cao, JD: Synchronization of fractional-order delayed neural networks with hybrid coupling. Complexity 21(S1), 106-112 (2016) MathSciNetView ArticleGoogle Scholar
  5. Matouk, AE: Chaos synchronization of a fractional-order modified Van der Pol-Duffing system via new linear control, backstepping control and Takagi-Sugeno fuzzy approaches. Complexity 21(S1), 116-124 (2016) MathSciNetView ArticleGoogle Scholar
  6. Elsayed, EM, Ahmed, AM: Dynamics of a three-dimensional systems of rational difference equations. Math. Methods Appl. Sci. 39(5), 1026-1038 (2016) MathSciNetView ArticleMATHGoogle Scholar
  7. Kuznetsov, N, Mokaev, T, Vasilyev, P: Numerical justification of Leonov conjecture on Lyapunov dimension of Rössler attractor. Commun. Nonlinear Sci. Numer. Simul. 19(4), 1027-1034 (2014) MathSciNetView ArticleGoogle Scholar
  8. Leonov, G: Bounds for attractors and the existence of homoclinic orbits in the Lorenz system. J. Appl. Math. Mech. 65(1), 19-32 (2001) MathSciNetView ArticleMATHGoogle Scholar
  9. Leonov, G, Bunin, A, Koksch, N: Attractor localization of the Lorenz system. Z. Angew. Math. Mech. 67, 649-656 (1987) MathSciNetView ArticleMATHGoogle Scholar
  10. Leonov, G: General existence conditions of homoclinic trajectories in dissipative systems. Lorenz, Shimizu-Morioka, Lu and Chen systems. Phys. Lett. A 376, 3045-3050 (2012) MathSciNetView ArticleMATHGoogle Scholar
  11. Leonov, G, Kuznetsov, K: Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits. Int. J. Bifurc. Chaos Appl. Sci. Eng. 23, Article ID 1330002 (2013) MathSciNetView ArticleMATHGoogle Scholar
  12. Leonov, G, Kuznetsov, N, Kiseleva, M, Solovyeva, E, Zaretskiy, A: Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor. Nonlinear Dyn. 77, 277-288 (2014) View ArticleGoogle Scholar
  13. Wang, XY, Wang, MJ: A hyperchaos generated from Lorenz system. Physica A 387(14), 3751-3758 (2008) MathSciNetView ArticleGoogle Scholar
  14. Leonov, G, Kuznetsov, N, Mokaev, T: Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion. Eur. Phys. J. Spec. Top. 224(8), 1421-1458 (2015) View ArticleGoogle Scholar
  15. Leonov, G, Kuznetsov, N: On differences and similarities in the analysis of Lorenz, Chen, and Lu systems. Appl. Math. Comput. 256, 334-343 (2015) MathSciNetMATHGoogle Scholar
  16. Zhang, FC, Liao, XF, Mu, CL: Dynamical analysis of the permanent-magnet synchronous motor chaotic system. Adv. Differ. Equ. 2017(1), Article ID 76 (2017). doi:10.1186/s13662-017-1118-1 MathSciNetView ArticleGoogle Scholar
  17. Zhang, FC, Liao, XF, Zhang, GY: Qualitative behaviors of the continuous-time chaotic dynamical systems describing the interaction of waves in plasma. Nonlinear Dyn. 88(3), 1623-1629 (2017) View ArticleGoogle Scholar
  18. Zhang, FC, Mu, CL, Zhou, SM, Zheng, P: New results of the ultimate bound on the trajectories of the family of the Lorenz systems. Discrete Contin. Dyn. Syst., Ser. B 20(4), 1261-1276 (2015) MathSciNetView ArticleMATHGoogle Scholar
  19. Zhang, FC, Liao, XF, Mu, CL, Zhang, GY, Chen, YA: On global boundedness of the Chen system. Discrete Contin. Dyn. Syst., Ser. B 22(4), 1673-1681 (2017) MathSciNetView ArticleMATHGoogle Scholar
  20. Zhang, FC, Liao, XF, Zhang, GY, Mu, CL: Dynamical analysis of the generalized Lorenz systems. J. Dyn. Control Syst. 23(2), 349-362 (2017) MathSciNetView ArticleMATHGoogle Scholar
  21. Zhang, FC, Wang, XY, Liao, XF, Zhang, GY, Mu, CL: Dynamical behaviors of a modified Lorenz-Stenflo chaotic system. Int. J. Bifurc. Chaos Appl. Sci. Eng. 27(5), Article ID 1750074 (2017) MathSciNetView ArticleMATHGoogle Scholar
  22. Stenflo, L: Generalized Lorenz equations for acoustic-gravity waves in the atmosphere. Phys. Scr. 53(1), 83-84 (1996) View ArticleGoogle Scholar
  23. Yu, MY, Yang, B: Periodic and chaotic solutions of the generalized Lorenz equations. Phys. Scr. 54(2), 140-142 (1996) MathSciNetView ArticleMATHGoogle Scholar
  24. Yu, MY, Zhou, CT, Lai, CH: The bifurcation characteristics of the generalized Lorenz equations. Phys. Scr. 54(4), 321-324 (1996) MathSciNetView ArticleMATHGoogle Scholar
  25. Zhou, C, Lai, CH, Yu, MY: Bifurcation behavior of the generalized Lorenz equations at large rotation numbers. J. Math. Phys. 38(10), 5225-5239 (1997). doi:10.1063/1.531938 MathSciNetView ArticleMATHGoogle Scholar
  26. Park, J, Lee, H, Jeon, YL, Baik, JJ: Periodicity of the Lorenz-Stenflo equations. Phys. Scr. 90(6), Article ID 065201 (2015) View ArticleGoogle Scholar
  27. Paulo, CR: On the dynamics in parameter planes of the Lorenz-Stenflo system. Phys. Scr. 90(11), Article ID 115201 (2015) View ArticleGoogle Scholar
  28. Park, J, Han, BS, Lee, H, Jeon, YL, Baik, JJ: Stability and periodicity of high-order Lorenz-Stenflo equations. Phys. Scr. 91(6), Article ID 065202 (2016) View ArticleGoogle Scholar
  29. Rech, PC: On the dynamics of a high-order Lorenz-Stenflo system. Phys. Scr. 91(12), Article ID 125201 (2016) View ArticleGoogle Scholar
  30. Kuznetsov, NV, Leonov, GA, Yuldashev, MV, Yuldashev, RV: Hidden attractors in dynamical models of phase-locked loop circuits: limitations of simulation in MATLAB and SPICE. Commun. Nonlinear Sci. Numer. Simul. 51, 39-49 (2017) View ArticleGoogle Scholar
  31. Leonov, GA, Kuznetsov, NV, Korzhemanova, NA, Kusakin, DV: Lyapunov dimension formula for the global attractor of the Lorenz system. Commun. Nonlinear Sci. Numer. Simul. 41, 84-103 (2016) MathSciNetView ArticleGoogle Scholar

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