《广西师范大学学报》（哲学社会科学版） ›› 2019, Vol. 37 ›› Issue (3): 96-105.doi: 10.16088/j.issn.1001-6600.2019.03.011

• • 上一篇    

新四维超混沌系统的复杂动力学研究

洪玲玲,杨启贵*   

  1. 华南理工大学数学学院,广东广州510640
  • 发布日期:2019-07-12
  • 通讯作者: 杨启贵(1965—), 男(土家族), 重庆市人, 华南理工大学教授, 博导。E-mail:qgyang@scut.edu.cn
  • 基金资助:
    国家自然科学基金(11671149);广东省自然科学基金(2017A030312006)

Research on Complex Dynamics of a New 4D Hyperchaotic System

HONG Lingling, YANG Qigui*   

  1. School of Mathematics, South China University of Technology, Guangzhou Guangdong 510640, China
  • Published:2019-07-12

PDF (PC)

44

摘要： 本文基于三维Lorenz-like混沌系统, 设计线性反馈控制器, 提出了一个仅有2个二次非线性项的新四维超混沌系统。此系统具有简单的代数结构, 但却展现复杂的动力学行为, 并理论证明它与超混沌Li系统是不等价的。为了研究系统的复杂动力学, 本文详细探讨了系统在双曲和非双曲平衡点时的稳定性,且严格分析Hopf分岔, 获得Hopf分岔所产生周期轨的近似表达式和稳定性。进一步借助现代数学软件进行数值仿真, 得到系统的Lyapunov指数谱、Poincaré映射和分岔图, 验证系统超混沌吸引子的存在性。

关键词: 超混沌系统, 稳定性, Hopf分岔, 复杂动力学, 吸引子

Abstract: In this paper, based on the 3D Lorenz-like chaotic system, a linear feedback controller is designed and a new four-dimensional hyperchaos system with only two times nonlinear terms is proposed. This system has simple algebraic structure, but shows complex dynamic behavior, and it is proved theoretically that it is not equivalent to hyperchaotic Li system. In order to study the complex dynamics of the system, the stability of the system at the hyperbolic and non-hyperbolic equilibrium points is discussed in detail, and the Hopf bifurcation is strictly analyzed. The approximate expression and stability of the periodic orbit generated by the Hopf bifurcation are obtained. Furthermore, the Lyapunov exponent spectrum, Poincaré map and bifurcation diagram of the system are obtained by numerical simulation with the help of modern mathematical software, and the existence of the hyperchaotic attractor is verified.

Key words: hyperchaotic system, stability, Hopf bifurcation, complex dynamics, attractor

中图分类号: 

  • O415.5
[1] LORENZ E N. Deterministic non-periodic flow[J]. Journal of the Atmospheric Sciences,1963,20:130-141.DOI:10.1177/0309133308091948.
[2] RSSLER O E. An equation for hyperchaos[J]. Physics Letters A, 1979, 71A(2):155-157. DOI:10.1016/03759601(79)90150-6.
[3] KAPITANIAK T, CHUA L O, ZHONG Guoqun. Experimental hyperchaos in coupled Chua’s circuits[J]. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Application,1994,41(7):499-503. DOI:10.1109/81.298367.
[4] YANG Qigui, LIU Yongjian. A hyperchaotic system from a chaotic system with one saddle and two stable node-foci[J]. Journal of Mathematical Analysis and Applications,2009, 360(1):293-306. DOI:10.1016/jmaa.2009.06.051.
[5] LI Yuxia, LIU Xuezhen, CHEN Guanrong. A new hyperchaotic Lorenz-type system: Generation, analysis, and implementation[J]. International Journal of Circuit Theory and Applications, 2011,39(8):865-879. DOI:10.1002/cta.673.
[6] LI Chunbiao, SPROTT J C. Coexisting hidden attractors in a 4-D simplified Lorenz system[J]. International Journal of Bifurcation and Chaos,2014,24(3):1450034. DOI:10.1142/S0218127414500345.
[7] CHEN Yuming, YANG Qigui. A new Lorenz-type hyperchaotic system with a curve of equilibria[J]. Mathematics and Computers in Simulation, 2015,112:40-55. DOI:10.1016/j.matcom.2014.11.006.
[8] CHEN Lijuan, TANG Song, LI Qingdu, et al. A new 4D hyperchaotic system with high complexity[J]. Mathematics and Computers in Simulation, 2018,146:44-56. DOI:10.1016/j.matcom.2017.10.002.
[9] YANG Qigui, CHEN Yuming. Complex dynamics in the unified Lorenz-type system[J]. International Journal of Bifurcation and Chaos, 2014,24(4):1450055. DOI:10.1142/S0218127414500552.
[10]YANG Qigui, BAI Meili. A new 5D hyperchaotic system based on modified generalized Lorenz system[J]. Nonlinear Dynamic, 2017,88:189-221. DOI:10.1007/s11071-016-3238-7.
[11]WOLF A, SWIFT J B. SWINNEY H L,et al. Determining Lyapunov exponents from a time series[J]. Physica D Nonlinear Phenomena, 1985,16(3):285-317. DOI:10.1016/0167-2789(85)90011-9.
[12]KUZNETSOV Y A. Elements of applied bifurcation theory[M]. New York: Springer-Verlag, 2004.
[13]杨路,张景中,侯晓荣.非线性代数方程组与定理机器证明[M].上海:上海科学与技术出版社,1996:23-25.
[14]WIGGINS S. Introduction to applied nonlinear dynamical systems and chaos[M]. 2nd ed. New York: Springer-Verlag, 2003.
[15]张锦炎,冯贝叶.常微分方程几何理论与分支问题[M].北京:北京大学出版社,2000.
No related articles found!
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
版权所有 © 广西师范大学学报（哲学社会科学版）编辑部
地址：广西桂林市三里店育才路15号　邮编：541004
电话：0773-5857325　E-mail: xbgj@mailbox.gxnu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发