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Complexity
Volume 2017, Article ID 7520590, 6 pages
https://doi.org/10.1155/2017/7520590
Research Article

Analysis of a Generalized Lorenz–Stenflo Equation

1College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China
2Mathematical Postdoctoral Station, College of Mathematics and Statistics, Southwest University, Chongqing 400716, China
3Information Office, Chongqing Technology and Business University, Chongqing 400067, China

Correspondence should be addressed to Rui Chen; moc.qq@5386779721

Received 20 September 2017; Accepted 26 November 2017; Published 13 December 2017

Academic Editor: Giacomo Innocenti

Copyright © 2017 Fuchen Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 [214]. 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]:

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]:where , , , and are state variables and , , , , , and 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 , , , , , and with the initial state System (2) has Lyapunov exponents as , , , and 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 , , , , , and , as shown in Figures 14.

Figure 1: Projection of hyperchaotic attractor of system (2) onto the xOyz space with , , , , , and .
Figure 2: Projection of hyperchaotic attractor of system (2) onto the space with , , , , , and .
Figure 3: Projection of hyperchaotic attractor of system (2) onto the space with , , , , , and .
Figure 4: Projection of hyperchaotic attractor of system (2) onto the space with , , , , , and .

In this paper, all the simulations are carried out by using fourth-order Runge-Kutta Method with time-step

The rest of this paper is organized as follows. In Section 2, 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 any , , , , , , there exists a positive number , such thatis the ultimate bound set of system (2), where

Proof. Define the following Lyapunov-like function:where , and , are arbitrary constants.
And we can getLet Then, we can get the surface :is an ellipsoid in for Outside , , while inside , Thus, the ultimate boundedness for system (2) can only be reached on Since the Lyapunov-like function is a continuous function and is a bounded closed set, then the function (4) can reach its maximum value on the surface that is defined in (6). Obviously, contains solutions of system (2). It is obvious that the set is the ultimate bound set for system (2).
This completes the proof.

Theorem 2. Suppose that
Let be an arbitrary solution of system (2) and Then the estimationholds for system (2), and thus is the globally exponential attractive set of system (2); that is,

Proof. Define the following functions:then we can getConstruct the Lyapunov-like functionDifferentiating the above Lyapunov-like function in (11) with respect to time along the trajectory of system (2) yieldsThus, we havewhich clearly shows that is the globally exponential attractive set of system (2).
The proof is complete.

Remark 3. (i) In particular, let us take , in Theorem 2, we can get the conclusions below.

Suppose that

Let be an arbitrary solution of system (2) and Then the estimationholds for system (2), and thus is the globally exponential attractive set and positive invariant set of system (2); that is,

(ii) Let us take ; then we can getas the globally exponential attractive set and positive invariant set of system (2) according to Theorem 2.

When , we can get that as the globally exponential attractive set and positive invariant set of system (2).

Figure 5 shows hyperchaotic attractor of system (2) in the space defined by . Figure 6 shows hyperchaotic attractor of system (2) in the space defined by . Figure 7 shows hyperchaotic attractor of system (2) in the space defined by . Figure 8 shows hyperchaotic attractor of system (2) in the space defined by .

Figure 5: Localization of hyperchaotic attractor of system (2) in the space defined by .
Figure 6: Localization of hyperchaotic attractor of system (2) in the space defined by .
Figure 7: Localization of hyperchaotic attractor of system (2) in the space defined by .
Figure 8: Localization of hyperchaotic attractor of system (2) in the space defined by .

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.

Disclosure

All authors have read and approved the final manuscript.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant no. 11501064), 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), Chongqing Postdoctoral Science Foundation Special Funded Project (Grant no. Xm2017174), and the Research Fund of Chongqing Technology and Business University (Grant no. 1752073). The authors thank Professors Jinhu Lu in Institute of Systems Science, Chinese Academy of Sciences, Gaoxiang Yang in Ankang University, Ping Zhou in Chongqing University of Posts and Telecommunications, and Min Xiao in Nanjing University of Posts and Telecommunications for their help.

References

  1. E. N. Lorenz, “Deterministic nonperiodic flow,” Journal of the Atmospheric Sciences, vol. 20, no. 2, pp. 130–141, 1963. View at Publisher · View at Google Scholar
  2. G. A. Leonov and N. V. Kuznetsov, “On differences and similarities in the analysis of Lorenz, Chen, and LU systems,” Applied Mathematics and Computation, vol. 256, pp. 334–343, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. X. Wang and M. Wang, “A hyperchaos generated from Lorenz system,” Physica A: Statistical Mechanics and its Applications, vol. 387, no. 14, pp. 3751–3758, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. G. A. Leonov, “General existence conditions of homoclinic trajectories in dissipative systems. Lorenz, Shimizu-Morioka, LU and Chen systems,” Physics Letters A, vol. 376, no. 45, pp. 3045–3050, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. F. Zhang and G. Zhang, “Further results on ultimate bound on the trajectories of the Lorenz system,” Qualitative Theory of Dynamical Systems, vol. 15, no. 1, pp. 221–235, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. G. A. Leonov, “Bounds for attractors and the existence of homoclinic orbits in the lorenz system,” Journal of Applied Mathematics and Mechanics, vol. 65, no. 1, pp. 19–32, 2001. View at Publisher · View at Google Scholar · View at Scopus
  7. G. A. Leonov, A. I. Bunin, and N. Koksch, “Attraktorlokalisierung des Lorenz-Systems,” Zeitschrift für Angewandte Mathematik und Mechanik. Ingenieurwissenschaftliche Forschungsarbeiten, vol. 67, no. 12, pp. 649–656, 1987. View at Publisher · View at Google Scholar · View at MathSciNet
  8. F. Zhang, C. Mu, S. Zhou, and P. Zheng, “New results of the ultimate bound on the trajectories of the family of the Lorenz systems,” Discrete and Continuous Dynamical Systems - Series B, vol. 20, no. 4, pp. 1261–1276, 2015. View at Publisher · View at Google Scholar · View at Scopus
  9. G. A. Leonov and N. V. Kuznetsov, “Hidden attractors in dynamical systems: from hidden oscillations in Hilbert-Kolmogorov, Aizerman, and KALman problems to hidden chaotic attractor in Chua circuits,” International Journal of Bifurcation and Chaos, vol. 23, no. 1, Article ID 1330002, 69 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. G. A. Leonov, N. V. Kuznetsov, N. A. Korzhemanova, and D. V. Kusakin, “Lyapunov dimension formula for the global attractor of the Lorenz system,” Communications in Nonlinear Science and Numerical Simulation, vol. 41, pp. 84–103, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. G. A. Leonov, N. V. Kuznetsov, and T. N. Mokaev, “Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion,” The European Physical Journal Special Topics, vol. 224, no. 8, pp. 1421–1458, 2015. View at Publisher · View at Google Scholar · View at Scopus
  12. N. V. Kuznetsov, T. N. Mokaev, and P. A. Vasilyev, “Numerical justification of Leonov conjecture on LYApunov dimension of Rossler attractor,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 4, pp. 1027–1034, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. N. V. Kuznetsov, G. A. Leonov, M. V. Yuldashev, and R. V. Yuldashev, “Hidden attractors in dynamical models of phase-locked loop circuits: limitations of simulation in MATLAB and SPICE,” Communications in Nonlinear Science and Numerical Simulation, vol. 51, pp. 39–49, 2017. View at Publisher · View at Google Scholar · View at Scopus
  14. E. M. Elsayed and A. M. Ahmed, “Dynamics of a three-dimensional systems of rational difference equations,” Mathematical Methods in the Applied Sciences, vol. 39, no. 5, pp. 1026–1038, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. L. Stenflo, “Generalized Lorenz equations for acoustic-gravity waves in the atmosphere,” Physica Scripta, vol. 53, no. 1, pp. 83-84, 1996. View at Publisher · View at Google Scholar · View at Scopus
  16. G. A. Leonov, “Generalized Lorenz equations for acoustic-gravity waves in the atmosphere. Attractors dimension, convergence and homoclinic trajectories,” Communications on Pure and Applied Analysis, vol. 16, no. 6, pp. 2253–2267, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  17. Y.-M. Chen and H.-H. Liang, “Zero-zero-Hopf bifurcation and ultimate bound estimation of a generalized Lorenz-Stenflo hyperchaotic system,” Mathematical Methods in the Applied Sciences, vol. 40, no. 10, pp. 3424–3432, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. P. Frederickson, J. L. Kaplan, E. D. Yorke, and J. A. Yorke, “The Liapunov dimension of strange attractors,” Journal of Differential Equations, vol. 49, no. 2, pp. 185–207, 1983. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D: Nonlinear Phenomena, vol. 16, no. 3, pp. 285–317, 1985. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus