一种具有多对称同质吸引子的四维混沌系统的超级多稳定性研究

黄丽莲; 姚文举; 项建弘; 王霖郁

doi:10.11999/JEIT201095

一种具有多对称同质吸引子的四维混沌系统的超级多稳定性研究

doi: 10.11999/JEIT201095

1.
哈尔滨工程大学信息与通信工程学院哈尔滨 150001
2.
哈尔滨工程大学先进船舶通信与信息技术重点实验室哈尔滨 150001

基金项目: 国家自然科学基金(61203004)，黑龙江省自然科学基金(F201220)，黑龙省自然科学基金联合引导项目(LH2020F022)

详细信息

作者简介:
黄丽莲：女，1972年生，教授，硕士生导师，副博士生导师，研究方向为非线性系统的混沌控制与同步

姚文举：男，1994年生，硕士生，研究方向为非线性系统的混沌控制与同步

项建弘：男，1977年生，副教授，研究方向为5G无线通信、人工智能与深度学习、自适应信号处理等

王霖郁：女，1977年生，副教授，研究方向为电路与系统

通讯作者:
项建弘　xiangjianhong@hrbeu.edu.cn

中图分类号: TP271
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出版历程
- 收稿日期: 2020-12-30
- 修回日期: 2021-06-02
- 网络出版日期: 2021-08-26
- 刊出日期: 2022-01-10

Extreme Multi-stability of a Four-dimensional Chaotic System with Infinitely Many Symmetric Homogeneous Attractors

1.
College of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China
2.
The Key Laboratory of Advanced Ship Communication and Information Technology, Harbin Engineering University, Harbin 150001, China

Funds: The National Natural Science Foundation of China (61203004), The Natural Science Foundation of Heilongjiang Province (F201220)，The Heilongjiang Natural Science Foundation Joint Guide Project (LH2020F022)

摘要

摘要: 该文在一个经典3维混沌系统的基础上提出一个新的具有超级多稳定性的4维混沌系统。新系统具有一个线平衡点，可以产生无限多对称的同质吸引子。通过相轨图和庞加莱截面等方法分析了系统的混沌特性。重点利用相轨图、分岔图和Lyapunov指数谱等方法分析了初始条件对系统超级多稳定性的影响，分析表明该系统具有很大的初值变化范围，除零点外恒定的Lyapunov指数谱，中心对称的离散分岔图。进一步地，该文研究了系统初值对称性与吸引子对称性的关系，不同于现有混沌系统中的对称吸引子，该系统可以产生无限多对称的同质吸引子。最后，利用电路仿真软件搭建模拟电路捕捉该系统的混沌吸引子，其结果验证了数值仿真的正确性。
- 超级多稳定性 /
- 无限多对称的同质吸引子 /
- 中心对称的离散分岔图
Abstract: A new four-dimensional chaotic system with extreme multi-stability based on a classic three-dimensional chaotic system is proposed. The new system has a line equilibrium point, which can generate an infinite number of symmetrical homogeneous attractors. The chaotic characteristics of the system are analyzed by phase orbit diagram and Poincaré section methods. Using phase orbit diagrams, bifurcation diagrams and Lyapunov exponent spectrum methods, the influence of initial conditions on the extreme multi-stability of the system is analyzed. The analysis shows that the system has a large initial value variation range, and the Lyapunov exponent spectrum is constant except for the zero point. In addition, the system also has centrally symmetrical discrete bifurcation diagrams. Furthermore, the relationship between the initial symmetry of the system and the symmetry of the attractor is studied, which is different from the symmetrical attractor in the existing chaotic system, which can generate an infinite number of symmetrical homogeneous attractors. Finally, circuit simulation software is used to build an analog circuit to capture the chaotic attractor of the system, and the result verifies the correctness of the numerical simulation.
- Extreme multi-stability /
- An infinite number of symmetrical homogeneous attractors /
- Centrally symmetrical discrete bifurcation diagrams

HTML全文

图 1 混沌吸引子的相轨图

下载: 全尺寸图片幻灯片

图 2 混沌吸引子的庞加莱截面

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图 3 初值$w(0)$在$[ - {\rm{200}},{\rm{200}}]$区间内变化的分岔图和Lyapunov指数谱

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图 4 无限多共存吸引子的相图

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图 5 初值$x(0)$在$[ - {10^3},{10^3}]$区间内变化的分岔图和Lyapunov指数谱

下载: 全尺寸图片幻灯片

图 6 初值$y(0)$在$[ - {10^3},{10^3}]$区间内变化的分岔图和Lyapunov指数谱

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图 7 无限多同质吸引子的相图

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图 8 初值$z(0)$在$[ - {10^3},{10^3}]$区间内变化的分岔图和Lyapunov指数谱

下载: 全尺寸图片幻灯片

图 9 对称的同质吸引子的相图

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图 10 对称吸引子在$t = 30\;{\rm{s}}$内的时域波形图

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图 11 系统式(2)的模拟电路图

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图 12 混沌吸引子的V_x–V_z平面电路仿真结果

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参考文献(36)

[1]	GLEICK J and HILBORN R C. Chaos, making a new science[J]. American Journal of Physics, 1988, 56(11): 1053–1054. doi: 10.1119/1.15345
[2]	陈关荣, 吕金虎. Lorenz系统族的动力学分析、控制与同步[M]. 北京: 科学出版社, 2003: 278. CHEN Guanrong and LÜ Jinhu. Dynamics Analysis, Control and Synchronization of Lorenz System Family[M]. Beijing: Science Press, 2003: 278.
[3]	HASLER M. Engineering chaos for encryption and broadband communication[J]. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1995, 353(1701): 115–126.
[4]	OPPENHEIM A V, WORNELL G W, ISABELLE S H, et al. Signal processing in the context of chaotic signals[C]. ICASSP-92: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing, San Francisco, USA, 1992: 117–120.
[5]	GRASSI G and MASCOLO S. A system theory approach for designing cryptosystems based on hyperchaos[J]. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 1999, 46(9): 1135–1138. doi: 10.1109/81.788815
[6]	WANG B, ZOU F C, and CHENG J. A memristor-based chaotic system and its application in image encryption[J]. Optik, 2018, 154: 538–544. doi: 10.1016/j.ijleo.2017.10.080
[7]	HASSAN M F. A new approach for secure communication using constrained hyperchaotic systems[J]. Applied Mathematics and Computation, 2014, 246: 711–730. doi: 10.1016/j.amc.2014.08.029
[8]	FILALI R L, BENREJEB M, and BORNE P. On observer-based secure communication design using discrete-time hyperchaotic systems[J]. Communications in Nonlinear Science and Numerical Simulation, 2014, 19(5): 1424–1432. doi: 10.1016/j.cnsns.2013.09.005
[9]	ELBERT T, RAY W J, KOWALIK Z J, et al. Chaos and physiology: Deterministic chaos in excitable cell assemblies[J]. Physiological Reviews, 1994, 74(1): 1–47. doi: 10.1152/physrev.1994.74.1.1
[10]	YANG Lijiang and CHEN Tianlun. Application of chaos in genetic algorithms[J]. Communications in Theoretical Physics, 2002, 38(2): 168–172. doi: 10.1088/0253-6102/38/2/168
[11]	MASLOV V P. Theory of chaos and its application to the crisis of debts and the origin of inflation[J]. Russian Journal of Mathematical Physics, 2009, 16(1): 103–120. doi: 10.1134/S1061920809010087
[12]	LORENZ E N. Deterministic nonperiodic flow[J]. Journal of the Atmospheric Sciences, 1963, 20(2): 130–141. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
[13]	CHUA L, KOMURO M, and MATSUMOTO T. The double scroll family[J]. IEEE Transactions on Circuits and Systems, 1986, 33(11): 1072–1118. doi: 10.1109/TCS.1986.1085869
[14]	MADAN R N. Chua’s Circuit: A Paradigm for Chaos[M]. Singapore: World Scientific, 1993: 1042.
[15]	LÜ Jinhu, CHEN Guanrong, and ZHANG Suochun. Dynamical analysis of a new chaotic attractor[J]. International Journal of Bifurcation and Chaos, 2002, 12(5): 1001–1015. doi: 10.1142/S0218127402004851
[16]	STRUKOV D B, SNIDER G S, STEWART D R, et al. The missing memristor found[J]. Nature, 2008, 453(7191): 80–83. doi: 10.1038/nature06932
[17]	ITOH M and CHUA L O. Memristor oscillators[J]. International Journal of Bifurcation and Chaos, 2008, 18(11): 3183–3206. doi: 10.1142/S0218127408022354
[18]	BAO Bocheng, LIU Zhong, and XU Jianping. Transient chaos in smooth memristor oscillator[J]. Chinese Physics B, 2010, 19(3): 030510. doi: 10.1088/1674-1056/19/3/030510
[19]	闵富红, 王珠林, 王恩荣, 等. 新型忆阻器混沌电路及其在图像加密中的应用[J]. 电子与信息学报, 2016, 38(10): 2681–2688. MIN Fuhong, WANG Zhulin, WANG Enrong, et al. New memristor chaotic circuit and its application to image encryption[J]. Journal of Electronics &Information Technology, 2016, 38(10): 2681–2688.
[20]	STANKEVICH N and VOLKOV E. Multistability in a three-dimensional oscillator: Tori, resonant cycles and chaos[J]. Nonlinear Dynamics, 2018, 94(4): 2455–2467. doi: 10.1007/s11071-018-4502-9
[21]	RAJAGOPAL K, JAFARI S, KARTHIKEYAN A, et al. Hyperchaotic memcapacitor oscillator with infinite equilibria and coexisting attractors[J]. Circuits, Systems, and Signal Processing, 2018, 37(9): 3702–3724. doi: 10.1007/s00034-018-0750-7
[22]	ZHANG Sen, ZENG Yicheng, LI Zhijun, et al. A novel simple no-equilibrium chaotic system with complex hidden dynamics[J]. International Journal of Dynamics and Control, 2018, 6(4): 1465–1476. doi: 10.1007/s40435-018-0413-3
[23]	WU Huagan, BAO Han, XU Quan, et al. Abundant coexisting multiple attractors’ behaviors in three-dimensional sine chaotic system[J]. Complexity, 2019, 2019: 3687635.
[24]	NJITACKE Z T, MOGUE R L T, LEUTCHO G D, et al. Heterogeneous multistability in a novel system with purely nonlinear terms[J]. International Journal of Electronics, 2021, 108(7): 1166–1182. doi: 10.1080/00207217.2020.1833371
[25]	包涵, 包伯成, 林毅, 等. 忆阻自激振荡系统的隐藏吸引子及其动力学特性[J]. 物理学报, 2016, 65(18): 180501. doi: 10.7498/aps.65.180501 BAO Han, BAO Bocheng, LIN Yi, et al. Hidden attractor and its dynamical characteristic in memristive self-oscillating system[J]. Acta Physica Sinica, 2016, 65(18): 180501. doi: 10.7498/aps.65.180501
[26]	KENGNE J, TABEKOUENG Z N, TAMBA V K, et al. Periodicity, chaos, and multiple attractors in a memristor-based Shinriki’s circuit[J]. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2015, 25(10): 103126. doi: 10.1063/1.4934653
[27]	BAO Bocheng, BAO Han, WANG Ning, et al. Hidden extreme multistability in memristive hyperchaotic system[J]. Chaos, Solitons & Fractals, 2017, 94: 102–111. doi: 10.1016/j.chaos.2016.11.016
[28]	BAO Han, JIANG Tao, CHU Kaibin, et al. Memristor-based canonical Chua’s circuit: Extreme multistability in voltage-current domain and its controllability in flux-charge domain[J]. Complexity, 2018, 2018: 5935637.
[29]	CHEN Mo, SUN Mengxia, BAO Bocheng, et al. Controlling extreme multistability of memristor emulator-based dynamical circuit in flux-charge domain[J]. Nonlinear Dynamics, 2018, 91(2): 1395–1412. doi: 10.1007/s11071-017-3952-9
[30]	WANG Guangyi, SHI Chuanbao, WANG Xiaowei, et al. Coexisting oscillation and extreme multistability for a memcapacitor-based circuit[J]. Mathematical Problems in Engineering, 2017, 2017: 6504969.
[31]	AHMADI A, WANG X, NAZARIMEHR F, et al. Coexisting infinitely many attractors in a new chaotic system with a curve of equilibria: Its extreme multi-stability and Kolmogorov–Sinai entropy computation[J]. Advances in Mechanical Engineering, 2019, 11(11): 1–8.
[32]	GONG Lihua, WU Rouging, and ZHOU Nanrun. A new 4D chaotic system with coexisting hidden chaotic attractors[J]. International Journal of Bifurcation and Chaos, 2020, 30(10): 2050142. doi: 10.1142/S0218127420501424
[33]	HUANG Lilian, YAO Wenju, XIANG Jianhong, et al. Heterogeneous and homogenous multistabilities in a novel 4D memristor-based chaotic system with discrete bifurcation diagrams[J]. Complexity, 2020, 2020: 2408460.
[34]	LIU Wenbo and CHEN Guanrong. A new chaotic system and its generation[J]. International Journal of Bifurcation and Chaos, 2003, 13(1): 261–267. doi: 10.1142/S0218127403006509
[35]	LIU Wenbo and CHEN Guanrong. Can a three-dimensional smooth autonomous quadratic chaotic system generate a single four-scroll attractor?[J]. International Journal of Bifurcation and Chaos, 2004, 14(4): 1395–1403. doi: 10.1142/S0218127404009880
[36]	包伯成. 混沌电路导论[M]. 北京: 科学出版社, 2013: 45–46. BAO Bocheng. An Introduction to Chaotic Circuits[M]. Beijing: Science Press, 2013: 45–46.