Diverse dynamical behaviors and predefined-time synchronization of a simple memristive chaotic system

LAI Qiang; WANG Jun; HUANG Daxun

doi:10.7498/aps.74.20250954

Abstract
The rich dynamical analysis and predefined-time synchronization of simple memristive chaotic systems are of great significance in fully understanding the mechanism of dynamics formation and expanding the potential applications of chaotic systems. A four-dimensional memristive chaotic system with only a single nonlinear term is proposed to reveal various dynamic behaviors under the change of parameters and initial conditions, and to realize effective synchronization control. Based on dissipativity analysis and Lyapunov exponent computation, and combined with bifurcation analysis and multi steady state exploration, it is shown that the system possesses an infinite number of unstable equilibrium points and exhibits homogeneous and heterogeneous multistability, including point attractors, periodic attractors, and chaotic attractors. Moreover, it is found that amplitude modulation of the output signals of the system can be precisely achieved by adjusting internal parameters of the memristor, thus providing a theoretical basis for achieving effective amplitude modulation of periodic and chaotic signals. A predefined-time sliding mode surface with linear and bidirectional power-law nonlinear decay terms is constructed to address synchronization. Sufficient conditions for predefined-time convergence of synchronization errors are derived using Lyapunov stability theory, and a double-stage sliding mode controller with an adjustable upper bound on synchronization time is designed. The resulting control law ensures an adjustable synchronization time bound and rapid error suppression under arbitrary disturbances. Numerical simulations further confirm the effectiveness and robustness of the proposed control scheme, indicating that even under external disturbances or significant variations in initial conditions, the error variables can still converge precisely within the predefined time.

Keywords:
memristive chaotic system /

multistability /

amplitude modulation /

sliding mode control /

predefined-time synchronization
FullText HTML

Cited By
图 1 系统(2)的混沌吸引子　(a) x-y; (b) x-z; (c) y-z; (d) x-w; (e) y-w; (f) z-w

Figure 1. Chaotic attractors of system (2): (a) x-y; (b) x-z; (c) y-z; (d) x-w; (e) y-w; (f) z-w.

DownLoad: Full-Size Img PowerPoint

图 2 系统(2)关于参数$ a \in [0.8, 0.95] $的数值结果　(a) 分岔图; (b) Lyapunov 指数谱

Figure 2. Numerical results of system (2) with parameter $ a \in [0.8, 0.95] $: (a) Bifurcation diagram; (b) Lyapunov exponent spectrum.

DownLoad: Full-Size Img PowerPoint

图 3 系统(2)在不同参数a时的吸引子　(a) $ a = 0.8 $; (b) $ a = 0.89 $; (c) $ a = 0.95 $

Figure 3. Attractors of system (2) under different values of parameter $ a $: (a) $ a = 0.8 $; (b) $ a = 0.89 $; (c) $ a = 0.95 $.

DownLoad: Full-Size Img PowerPoint

图 4 系统(2)关于初值$ [0.1, 0.1, 0.1, w(0)] $, $ w(0) \in [ - 4, 4] $的数值结果　(a) 分岔图; (b) Lyapunov 指数谱

Figure 4. Numerical results of system (2) with initial values $ [0.1, 0.1, 0.1, w(0)] $, where $ w(0) \in [ - 4, 4] $: (a) Bifurcation diagram; (b) Lyapunov exponent spectrum.

DownLoad: Full-Size Img PowerPoint

图 5 系统(2)的异构多稳态　(a) 一个点吸引子、两个混沌吸引子和三个周期吸引子; (b) 两个周期吸引子和四个混沌吸引子

Figure 5. Heterogeneous multistability of system (2): (a) One point attractor, two chaotic attractors, and three periodic attractors; (b) two periodic attractors and four chaotic attractors.

DownLoad: Full-Size Img PowerPoint

图 6 系统(2)的混沌信号调幅控制　(a) $ \beta \in [0.1, 0.95] $时各状态变量的最大幅值; (b) 不同$ \beta $对应的混沌吸引子

Figure 6. Amplitude modulation control of chaotic signals in system (2): (a) Maximum amplitudes of state variables for $ \beta \in [0.1, 0.95] $; (b) chaotic attractors corresponding to different values of $ \beta $.

DownLoad: Full-Size Img PowerPoint

图 7 系统(2)的周期信号调幅控制　(a) $ \beta \in [0.1, 0.95] $时各状态变量的最大幅值; (b) 不同$ \beta $对应的周期吸引子

Figure 7. Amplitude modulation control of periodic signals in system (2): (a) Maximum amplitudes of state variables for $ \beta \in [0.1, 0.95] $; (b) periodic attractors corresponding to different values of $ \beta $.

DownLoad: Full-Size Img PowerPoint

图 8 不同状态变量收敛轨迹

Figure 8. Convergence trajectories of different state variables.

DownLoad: Full-Size Img PowerPoint

图 9 误差变量演变轨迹

Figure 9. Evolution trajectories of the error variables.

DownLoad: Full-Size Img PowerPoint

图 10 初始条件改变的状态变量收敛轨迹

Figure 10. Convergence trajectories of state variables under changed initial conditions.

DownLoad: Full-Size Img PowerPoint

图 11 初始条件改变的误差演变轨迹

Figure 11. Evolution trajectories of error variables under changed initial conditions.

DownLoad: Full-Size Img PowerPoint

图 12 多组不同初始条件下收敛时间分析

Figure 12. Convergence time analysis under multiple initial conditions.

DownLoad: Full-Size Img PowerPoint

图 13 鲁棒性分析

Figure 13. Robustness analysis.

DownLoad: Full-Size Img PowerPoint

map

[1]	Wang M J, Ding J, Li J H, He S B, Zhang X N, Iu H H C, Li Z J 2024 Eur. Phys. J. Plus 139 64 Google Scholar
[2]	Lai Q, Liu Y J, Fortuna L 2024 IEEE Trans. Circuits Syst. I: Reg. Pap. 71 4665 Google Scholar
[3]	Luo Y R, Fan C L, Xu C B, Li X Y 2024 Chaos Soliton. Fract. 183 114951 Google Scholar
[4]	Lai Q, Liu Y 2025 Sci. China Technol. Sci. 68 1320401 Google Scholar
[5]	赖强, 秦铭宏 2025 贵州师范大学学报(自然科学版) 43 1 Lai Q, Qin M H 2025 J. Guizhou Normal Univ. (Nat. Sci. Ed. ) 43 1
[6]	Yuan Y, Yu F, Tan B H, Huang Y Y, Yao W, Cai S, Lin H R 2025 Chaos 35 013121 Google Scholar
[7]	Liu Y, Chen L, Li C D, Liu X, Zhou W H, Li K 2023 Soft Comput. 27 18403-18418 Google Scholar
[8]	Ding L N, Xuan M T 2025 Entropy 27 481 Google Scholar
[9]	张贵重, 全旭, 刘嵩 2022 71 240502 Google Scholar Zhang G Z, Quan X, Liu S 2022 Acta Phys. Sin. 71 240502 Google Scholar
[10]	He Y, Zhang Y Q, He X, Wang X Y 2021 Sci. Rep. 11 6398 Google Scholar
[11]	Zhang B, Liu L 2023 Mathematics 11 2585 Google Scholar
[12]	Madani Y A, Aldwoah K, Younis B, Alsharafi M, Osman O, Muflh B 2025 Sci. Rep. 15 14104 Google Scholar
[13]	Han J P, Zhou L Q 2024 Neurocomputing 610 128612 Google Scholar
[14]	赵智鹏, 周双, 王兴元 2021 70 230502 Google Scholar Zhao Z P, Zhou S, Wang X Y 2021 Acta Phys. Sin. 70 230502 Google Scholar
[15]	Ramamoorthy R, Rajagopal K, Leutcho G D, Krejcar O, Namazi H, Hussain I 2022 Chaos Soliton. Fract. 156 111834 Google Scholar
[16]	Beyhan S 2024 Chaos Soliton. Fract. 180 114578 Google Scholar
[17]	Geng X T, Feng J W, Zhao Y, Li N, Wang J Y 2023 Electron. Res. Arch. 31 3291 Google Scholar
[18]	赖强, 王君 2024 73 180503 Google Scholar Lai Q, Wang J 2024 Acta Phys. Sin. 73 180503 Google Scholar
[19]	Tino N, Niamsup P 2021 Front. Appl. Math. Stat. 7 589406 Google Scholar
[20]	Al-Saggaf U M, Bettayeb M, Djennoune S 2022 Eur. J. Control 63 164 Google Scholar
[21]	Zhang M J, Zang H Y, Bai L Y 2022 Chaos Soliton. Fract. 164 112745 Google Scholar
[22]	Yan S H, Wu X Y, Jiang J W 2025 Chaos Soliton. Fract. 196 116337 Google Scholar
[23]	贾美美, 曹家伟, 白明明 2024 73 170502 Google Scholar Jia M M, Cao J W, Bai M M 2024 Acta Phys. Sin. 73 170502 Google Scholar
[24]	Liu Y, Zhu F 2022 Comput. Intel. Neurosc. 2022 3264936
[25]	Zheng W, Qu S C, Tang Q, Du X N 2025 J. Vib. Control 31 528-538 Google Scholar
[26]	包伯成, 胡文, 许建平, 刘中, 邹凌 2011 60 120502 Google Scholar Bao B C, Hu W, Xu J P, Liu Z, Zou L 2011 Acta Phys. Sin. 60 120502 Google Scholar
[27]	Li C B, Sprott J C 2016 Optik 127 10389 Google Scholar
[28]	赖强, 秦铭宏 2025 电子与信息学报 47 1 Lai Q, Qin M H 2025 J. Electron. Inf. Technol. 47 1

[1]	Lai Qiang, Wang Jun. Finite and fixed-time synchronization of memristive chaotic systems based on sliding mode reaching law. Acta Physica Sinica, doi: 10.7498/aps.73.20241013
[2]	Ding Da-Wei, Lu Xiao-Qi, Hu Yong-Bing, Yang Zong-Li, Wang Wei, Zhang Hong-Wei. Multistability of fractional-order memristor-coupled heterogeneous neurons and its hardware realization. Acta Physica Sinica, doi: 10.7498/aps.71.20221525
[3]	Zhang Gui-Zhong, Quan Xu, Liu Song. Analysis and FPGA implementation of memristor chaotic system with extreme multistability. Acta Physica Sinica, doi: 10.7498/aps.71.20221423
[4]	Zheng Lian-Qing, Peng Yi. Chaos control of voltage mode controlled buck-boost converter. Acta Physica Sinica, doi: 10.7498/aps.65.220502
[5]	Wu Hua-Gan, Chen Sheng-Yao, Bao Bo-Cheng. Impulsive synchronization and initial value effect for a memristor-based chaotic system. Acta Physica Sinica, doi: 10.7498/aps.64.030501
[6]	Xue Kai-Jia, Wang Cong-Qing. Sliding mode control of fractional order chaotic system based on an online error correction adaptive SVR. Acta Physica Sinica, doi: 10.7498/aps.64.070502
[7]	Pan Guang, Wei Jing. Design of an adaptive sliding mode controller for synchronization of fractional-order chaotic systems. Acta Physica Sinica, doi: 10.7498/aps.64.040505
[8]	Wang Xi, Wang Yu-Hong, Li Xing-Yuan, Miao Miao. Design of the static var compensator adaptive sliding mode controller considering model uncertainty and time-delay. Acta Physica Sinica, doi: 10.7498/aps.63.238407
[9]	Li An-Liang, Cai Hong, Zhang She-Feng, Bai Xi-Bin. Control and modeling of suspending stabilization problom for floated inertial platform. Acta Physica Sinica, doi: 10.7498/aps.62.150203
[10]	Wang Bin, Wu Chao, Zhu De-Lan. A new double-wing fractional-order chaotic system and its synchronization by sliding mode. Acta Physica Sinica, doi: 10.7498/aps.62.230506
[11]	Hu Wen, Li Jun-Ping, Zhang Gong, Liu Wen-Bo, Zhao Guang-Hao. The chaotic self-FM system and its FM code coupled synchronization. Acta Physica Sinica, doi: 10.7498/aps.61.010504
[12]	Lu Yong-Kun. Active adaptive fuzzy integral sliding mode control for unified chaotic system with uncertainties and disturbance. Acta Physica Sinica, doi: 10.7498/aps.61.220504
[13]	Li Yu-San, Wei Lin-Ling, Yu Miao, Zhang Meng. Chaos synchronization of regular network based on sliding mode control. Acta Physica Sinica, doi: 10.7498/aps.61.120504
[14]	Guo Hui-Jun, Liu Ding, Zhao Guang-Zhou. Active radial basis function sliding mode controller for unified chaotic system with disturbance and uncertainties. Acta Physica Sinica, doi: 10.7498/aps.60.010510
[15]	Li Hua-Qing, Liao Xiao-Feng, Huang Hong-Yu. Synchronization of uncertain chaotic systems based on neural network and sliding mode control. Acta Physica Sinica, doi: 10.7498/aps.60.020512
[16]	Zhang Ruo-Xun, Cao He-Fei. Adaptive synchronization of fractional-order chaotic system via sliding mode control. Acta Physica Sinica, doi: 10.7498/aps.60.050510
[17]	Fu Shi-Hui, Pei Li-Jun. Chaotic synchronization of Chua’s circuits with nonlinear control. Acta Physica Sinica, doi: 10.7498/aps.59.5985
[18]	Liu Ding, Yan Xiao-Mei. Projective synchronization of fractional-order chaotic systems based on sliding mode control. Acta Physica Sinica, doi: 10.7498/aps.58.3747
[19]	Liu Fu-Cai, Song Jia-Qiu. Anti-synchronizing different chaotic systems using active sliding mode control. Acta Physica Sinica, doi: 10.7498/aps.57.4729
[20]	XUE YUE-JU, FENG RU-PENG. CONTROL OF SPATIO-TEMPORAL CHAOS IN CONTINUOUS TIME COUPLED SYSTEMS BY USING ADAPTIVE FUZZY CONTROL. Acta Physica Sinica, doi: 10.7498/aps.50.440

Metrics

Abstract views: 382
PDF Downloads: 9
Cited By: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

Search

Article

留言板

Diverse dynamical behaviors and predefined-time synchronization of a simple memristive chaotic system

Abstract

FullText HTML

Cited By

Metrics

Authors

Referees

About Journal

About

Search

Article

留言板

Diverse dynamical behaviors and predefined-time synchronization of a simple memristive chaotic system

Abstract

FullText HTML

Cited By

Metrics

Publishing process

Authors

Referees

About Journal

About