广义Lorenz-Stenflo混沌系统的动力学分析与全局指数同步

张付臣, 陈松, 周雯静, 肖敏

doi:10.13306/j.1672-3813.2025.03.010

复杂系统与复杂性科学 >

2025 , Vol. 22 >Issue 3: 73 - 81

DOI: https://doi.org/10.13306/j.1672-3813.2025.03.010

研究论文

广义Lorenz-Stenflo混沌系统的动力学分析与全局指数同步

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1.重庆工商大学 a.数学与统计学院;b.统计智能计算与监测重庆市重点实验室,重庆 400067;
2.南京邮电大学 a.自动化学院;b.人工智能学院,南京 210023

张付臣(1983-),男,山东临沂人,博士,教授,主要研究方向为混沌系统稳定性分析与控制。

收稿日期: 2023-09-05

修回日期: 2023-11-15

基金资助

重庆市自然科学基金面上项目(CSTB2022NSCQ-MSX1548);“成渝地区双城经济圈建设”科技创新专项项目(KJCX2020037);重庆市教委科技项目(KJQN202100813,KJQN201800818);国家自然科学基金项目(62073172);江苏省自然科学基金(BK20221329);重庆市社会经济与应用统计重点实验室项目(ZDPTTD201909);重庆工商大学校内科技项目-青年项目(1952012);重庆工商大学研究生创新项目(yjscxx2024-284-55);重庆工商大学研究生教育教学改革项目(24YJG307);重庆市高等教育教学改革研究项目(253156)

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Dynamical Analysis and Global Exponential Synchronization of the Generalized Lorenz-Stenflo Chaotic System

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1. a. School of Mathematics and Statistics; b. Chongqing Key Laboratory of Statistical and Intelligent Computing and Monitoring, Chongqing Technology and Business University, Chongqing 400067, China;
2. a. College of Automation; b. College of Artificial Intelligence, Nanjing University of Posts and Telecommunications, Nanjing 210023, China

Received date: 2023-09-05

Revised date: 2023-11-15

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摘要

为发现更多的新混沌系统,在Lorenz-Stenflo混沌系统的基础上通过添加扰动参数构建了广义Lorenz-Stenflo混沌系统。通过耗散性、平衡点及其稳定性、分岔图和Lyapunov指数谱分析了该系统的基本混沌动力学特征。基于Lyapunov稳定性理论给出了系统全局指数吸引集的解析表达式;利用系统解有界性的估计式对两个广义Lorenz-Stenflo系统添加线性反馈控制使达到全局指数同步;最后对同步过程进行数值仿真,计算机仿真结果证实了全局指数同步的理论可行性。

本文引用格式

张付臣, 陈松, 周雯静, 肖敏 . 广义Lorenz-Stenflo混沌系统的动力学分析与全局指数同步[J]. 复杂系统与复杂性科学, 2025 , 22(3) : 73 -81 . DOI: 10.13306/j.1672-3813.2025.03.010

Abstract

In order to discover new chaotic systems, the generalized Lorenz-Stenflo chaotic system is constructed based on the Lorenz-Stenflo chaotic system by adding the disturbance parameters. The basic chaotic characteristics of this system are analyzed by means of the dissipation, equilibrium point and stability, bifurcation diagram and Lyapunov exponential spectrum. Based on Lyapunov stability theory, the analytical expression of the global exponential attractive set of the system is given. The global exponential synchronization is achieved by adding linear feedback control to two generalized Lorenz-Stenflo systems by using the estimator of system solution boundness. Finally, the numerical simulation of the synchronization process is carried out, and the computer simulation results confirm the theoretical feasibility of global exponential synchronization.

参考文献

[1] LORENZ E N. Deterministic non-periods flows[J]. Journal of the Atmospheric Science, 1963, 20, 130-141.
[2] CHEN G R, UETA T. Yet another chaotic attractor[J]. International Journal of Bifurcation and Chaos, 1999, 9(7): 1465-1466.
[3] LU J H, CHEN G R, CHENG D Z, et al. Bridge the gap between the Lorenz system and the Chen system[J]. International Journal of Bifurcation and Chaos, 2002, 12(12): 2917-2926.
[4] ROSSLER O E. An equation for hyperchaos[J]. Physics Letters A, 1979, 71(2/3): 155-157
[5] 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: 2-19.
[6] 韩秀萍, 陆君安. 从环状网络到链状网络同步能力的变化[J]. 中国科学(E辑: 信息科学), 2007,37(6): 748-756.
HAN X P, LU J N. Change of synchronization capability from ring network to chain network[J]. Science in China (Series E: Information Sciences), 2007,37(6): 748-756.
[7] HIROKAZU F, TOMOJI Y. Stability theory of synchronized motion in coupled-oscillator systems[J]. Progress of Theoretical Physics, 1983, 69(1): 32-47.
[8] PECORA L M, CARROLL T L. Synchronization in chaotic systems[J]. Physical Review Letters, 1990, 64(8): 821-824.
[9] VALENCIA-PONCE M A, CASTANEDA-AVINA P R, TLELO-CUAUTLE E, et al. CMOS OTA-based filters for designing fractional-order chaotic oscillators[J]. Fractal and Fractional, 2021, 5(3): 122-137.
[10] 祝晓静, 李科赞, 丁勇. 相继投影同步及其在保密通信中的应用[J]. 复杂系统与复杂性科学, 2022, 19(1): 27-33.
ZHU X J, LI K Z, DING Y. Successive Projective synchronization and its application in secure communication[J]. Complex Systems and Complexity Science, 2022, 19(1): 27-33.
[11] YANG X S, CAO J D, YANG Z C. Synchronization of coupled reaction-diffusion neural neworks with time-varying delays via pinning-impulsive controller[J]. SIAM Journal on Control and Optimization, 2013, 51(5): 3486-3510.
[12] 祝大伟, 涂俐兰. 随机扰动下Lorenz混沌系统的自适应同步与参数识别[J]. 物理学报, 2013, 62(5): 106-111.
ZHU D W, TU L L. Adaptive synchronization and parameter identification of Lorenz chaotic systems with random perturbations[J]. Acta Physica Sinica, 2013, 62(5): 106-111.
[13] 王铁邦, 曹天德, 雷勇,等. Lorenz系统的混沌同步[J]. 南京师大学报(自然科学版), 2009, 32(1): 57-61.
WANG T B, CAO T D, LEI Y, et al. Chaotic synchronization of Lorenz system[J]. Journal of Nanjing Normal University (Natural Science Edition), 2009, 32(1): 57-61.
[14] 牛玉俊, 徐伟, 戎海武,等. 随机脉冲微分方程的p阶矩稳定性和参激白噪声作用下Lorenz系统的脉冲同步[J]. 物理学报, 2009, 58(5): 2983-2988.
NIU Y J, XU W, RONG H W, et al. P-order moment stability of random impulsive differential equations and pulse synchronization of Lorenz system under parametric white noise[J]. Acta Physica Sinica, 2009, 58(5): 2983-2988.
[15] SOLIS-PÉREZ J E, GÓMEZ-AGUILAR J F, HERNÁNDEZ-PÉREZ J A, et al. Dynamical analysis of the M-Complex Lorenz system and its anti-synchronization via M-Sliding mode control[J]. Results in Physics, 2022, 43: 106080.
[16] 张芳芳, 刘树堂, 余卫勇. 时滞复Lorenz混沌系统特性及其自时滞同步[J]. 物理学报, 2013, 62(22): 64-72.
ZHANG F F, LIU S T, YU W Y. Characteristics and self-delay synchronization of complex Lorenz chaotic systems with delay[J]. Acta Physica Sinica, 2013, 62(22): 64-72.
[17] LIN L X. Projective synchronization of two coupled Lorenz chaotic systems in predefined time[J]. International Journal of Dynamics and Control, 2022, 10(3): 879-889.
[18] 廖晓昕. Lorenz混沌族中若干数学问题新研究[M]. 武汉: 华中科技大学出版社, 2017: 1-16, 74-79.
[19] LIN D, ZHANG F C, LIU J M. Symbolic dynamics-based error analysis on chaos synchronization via noisy channels[J]. Physical Review E, 2014, 90(1): 012908.
[20] 时帅帅, 刘立才, 杜传红. 新混沌系统分析及其同步控制[J]. 工程数学学报, 2022, 39(5): 709-724.
SHI S S, LIU L C, DU C H. Analysis and synchronization control of new chaotic systems[J]. Chinese Journal of Engineering Mathematics, 2022, 39(5): 709-724.
[21] 廖晓昕. 论Lorenz混沌系统全局吸引集和正向不变集的新结果及对混沌控制与同步的应用[J]. 中国科学(E辑: 信息科学), 2004, 34(12): 1404-1419.
LIAO X X. New Results on global attractive set and positive invariant set of Lorenz chaotic systems and its applications to chaos control and synchronization[J]. Science in China (Series E: Information Sciences), 2004, 34(12): 1404-1419.
[22] 廖晓昕, 罗海庚, 傅予力,等. 论Lorenz系统族的全局指数吸引集和正向不变集[J]. 中国科学(E辑: 信息科学), 2007, 37(6): 757-769.
LIAO X X, LUO H G, FU Y L, et al. On the global exponential attractive set and positive invariant Set of Lorenz family of systems[J]. Science in China (Series E: Information Sciences), 2007, 37(6): 757-769.
[23] 付文芳, 江明辉. 一个新混沌系统的全局指数吸引集及反馈同步[J]. 武汉理工大学学报, 2008(7): 103-106.
FU W F, JIANG M H. Global exponential Attraction set and feedback synchronization of a new chaotic system[J]. Journal of Wuhan University of Technology, 2008(7): 103-106.
[24] 刘莹, 王贺元, 陈荟颖. 强迫布鲁塞尔振子动力学行为和全局指数同步的数值仿真[J]. 动力学与控制学报, 2017, 15(5): 423-429.
LIU Y, WANG H Y, CHEN H Y. Numerical simulation of forced dynamic behavior of Brussels oscillator and global exponential synchronization[J]. Journal of Dynamics and Control, 2017, 15(5): 423-429.
[25] ZHANG F C, LIAO X F, ZHANG G Y. On the global boundedness of the Lü system[J]. Applied Mathematics and Computation, 2016, 284: 332-339.
[26] ZHANG F C, LIAO X F, MU C L, et al. On global boundedness of the Chen system[J]. Discrete and Continuous Dynamical Systems-Series B, 2017, 22(4): 1673-1681.
[27] QIN W X, CHEN G R. On the boundedness of solutions of the Chen system[J]. Journal of Mathematical Analysis and Applications, 2007, 329(1): 445-451.
[28] ZHANG F C, LIAO X F, ZHANG G Y, et al. Dynamical behaviors of a generalized Lorenz family[J]. Discrete and Continuous Dynamical Systems-Series B, 2017, 22(10): 3707-3720.
[29] ZHANG F C, LIAO X F, ZHANG G Y, et al. Dynamical analysis of the generalized Lorenz systems[J]. Journal of Dynamical and Control Systems, 2017, 23(2): 349-362.
[30] ZHANG X, CHEN G R. Boundedness of the complex Chen system[J]. Discrete and Continuous Dynamical Systems-Series B, 2023, 27(10): 5673-5700.
[31] CHIEN F S, INC M, YOSEFZADE H, et al. Predicting the chaos and solution bounds in a complex dynamical system[J]. Chaos, Solitons and Fractals, 2021, 153: 111474.
[32] STENFLO L. Generalized Lorenz equations for acoustic-gravity waves in the atmosphere[J]. Physica Scripta, 1996, 53(1): 83-84.
[33] 王贺元, 白晨. 四模Lorenz-Stenflo系统动力学行为分析及其数值仿真[J]. 沈阳师范大学学报(自然科学版), 2022, 40(5): 421-425.
WANG H Y, BAI C. Dynamic behavior analysis and numerical simulation of four-mode lorenz-stenflo system[J]. Journal of Shenyang Normal University (Natural Science Edition), 2022, 40(5): 421-425.

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