Journal of Guangxi Normal University(Natural Science Edition) ›› 2019, Vol. 37 ›› Issue (3): 96-105.doi: 10.16088/j.issn.1001-6600.2019.03.011

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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
  • Online:2019-07-12 Published:2019-07-12

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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

CLC Number: 

  • 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.
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