新四维超混沌系统的复杂动力学研究

doi:10.16088/j.issn.1001-6600.2019.03.011

摘要/Abstract

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

洪玲玲,杨启贵. 新四维超混沌系统的复杂动力学研究[J]. 广西师范大学学报（自然科学版）, 2019, 37(3): 96-105.

HONG Lingling, YANG Qigui. Research on Complex Dynamics of a New 4D Hyperchaotic System[J]. Journal of Guangxi Normal University(Natural Science Edition), 2019, 37(3): 96-105.

参考文献

[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.

相关文章 15

[1]	郑涛, 周欣然, 张龙. 三种群捕食-竞争-合作混杂模型的全局渐近稳定性[J]. 广西师范大学学报（自然科学版）, 2020, 38(5): 64-70.
[2]	王俊杰, 温雪岩, 徐克生, 于鸣. 基于局部敏感哈希的改进堆叠算法[J]. 广西师范大学学报（自然科学版）, 2020, 38(4): 21-31.
[3]	张杰, 李晓军. 无界域上非自治随机反应扩散方程一致随机吸引子的存在性[J]. 广西师范大学学报（自然科学版）, 2020, 38(2): 134-143.
[4]	佘连兵, 高云龙. 无界域上非自治Navier-Stokes方程的后向紧动力学[J]. 广西师范大学学报（自然科学版）, 2020, 38(1): 41-46.
[5]	罗兰, 周楠, 司杰. 不确定细胞神经网络鲁棒稳定新的时滞划分法[J]. 广西师范大学学报（自然科学版）, 2019, 37(4): 45-52.
[6]	吴娟,朱宏阳,梅平,陈武,李中宝. 聚甲基丙烯酸甲酯改性纳米SiO₂及其Pickering乳液稳定性[J]. 广西师范大学学报（自然科学版）, 2019, 37(3): 120-131.
[7]	陈思谕, 邹艳丽, 周建, 谭华珍. 电网发电机功率分配及电网负载不均衡发展研究[J]. 广西师范大学学报（自然科学版）, 2019, 37(2): 52-59.
[8]	韩会庆, 蔡广鹏, 尹昌应, 马庚, 张英佳, 陆艺. 2000年和2015年乌江中上游景观稳定性变化研究[J]. 广西师范大学学报（自然科学版）, 2019, 37(1): 197-204.
[9]	苗新艳, 张龙, 罗颜涛, 潘丽君. 一类交替变化的竞争—合作混杂种群模型研究[J]. 广西师范大学学报（自然科学版）, 2018, 36(3): 25-31.
[10]	黄开娇, 肖飞雁. 具有Beddington-DeAngelis型功能性反应的随机捕食—被捕食系统[J]. 广西师范大学学报（自然科学版）, 2018, 36(3): 32-40.
[11]	梅春草,韦笃取*,罗晓曙. 分布式发电系统中感性负载的稳定性研究[J]. 广西师范大学学报（自然科学版）, 2018, 36(2): 50-55.
[12]	冯金明,李遵先. 一类具扩散的传染病模型的稳定性分析[J]. 广西师范大学学报（自然科学版）, 2018, 36(2): 63-68.
[13]	陈春燕, 许志鹏, 邝华. 连续记忆效应的交通流跟驰建模与稳定性分析[J]. 广西师范大学学报（自然科学版）, 2017, 35(3): 14-21.
[14]	邢伟, 高晋芳, 颜七笙, 周其华. 具有非线性传染率及脉冲免疫接种的SIQR传染病模型[J]. 广西师范大学学报（自然科学版）, 2017, 35(2): 58-65.
[15]	傅杰,邹艳丽,谢蓉. 簇网络的同步及稳定性研究[J]. 广西师范大学学报（自然科学版）, 2017, 35(1): 7-15.

Metrics

Viewed

Full text

341

HTML			PDF

Just accepted	Online first	Issue	Just accepted	Online first	Issue
0	0	0	0	0	341

From	Others	local

Times	34	307
Rate	10%	90%

Abstract

200

Just accepted	Online first	Issue

0	0	200

From	Others	local

Times	180	20
Rate	90%	10%

Cited

Web of Science	Crossref	ScienceDirect	Search for Citations in Google Scholar >>


This page requires you have already subscribed to WoS.

Shared

Discussed