搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

含有偏置电压源的非齐次分数阶忆阻混沌电路动力学分析与实验研究

吴朝俊 方礼熠 杨宁宁

引用本文:
Citation:

含有偏置电压源的非齐次分数阶忆阻混沌电路动力学分析与实验研究

吴朝俊, 方礼熠, 杨宁宁

Dynamic analysis and experiment of chaotic circuit of non-homogeneous fractional memristor with bias voltage source

Wu Chao-Jun, Fang Li-Yi, Yang Ning-Ning
PDF
HTML
导出引用
  • 物理忆阻器具有不对称的紧磁滞回线, 为了更加简便地模拟物理忆阻器的不对称紧磁滞曲线, 本文提出了一种含有偏置电压源的分数阶二极管桥忆阻器模型, 其具有可连续调控磁滞回线的能力. 首先, 结合分数阶微积分理论, 建立了含有偏置电压源的二极管桥忆阻器的分数阶模型, 并对其电气特性进行分析. 其次, 将其与Jerk混沌电路相融合, 建立了含有偏置电压源的非齐次分数阶忆阻混沌电路模型, 研究了偏置电压对其系统动态行为的影响. 再次, 在PSpice中搭建了分数阶的等效电路模型, 并对其进行电路仿真验证, 实验结果与数值仿真基本一致. 最后, 在LabVIEW中完成了电路实验, 验证了理论分析的正确性与可行性. 结果表明, 含有偏置电压源的分数阶忆阻器, 可以通过调控偏置电压源的电压, 连续获得不对称紧磁滞回线. 随着偏置电源电压的改变, 非齐次分数阶忆阻混沌系统由于对称性的破环, 表现出由倍周期分岔进入混沌的行为.
    A physical memristor has an asymmetric tight hysteresis loop. In order to simulate the asymmetric tight hysteresis curve of the physical memristor more conveniently, a fractional-order diode bridge memristor model with a bias voltage source is proposed in this paper, which can continuously regulate the hysteresis loop. Firstly, based on fractional calculus theory, a fractional order model of a diode bridge memristor with a bias voltage source is established, and its electrical characteristics are analyzed. Secondly, by integrating it with the Jerk chaotic circuit, a non-homogeneous fractional order memristor chaotic circuit model with a bias voltage source is established, and the influence of bias voltage on its system dynamic behavior is studied. Once again, a fractional-order equivalent circuit model is built in PSpice and validated through circuit simulation. The experimental results are basically consistent with the numerical simulation results. Finally, the experiments on the circuit are completed in LabVIEW to validate the correctness and feasibility of the theoretical analysis. The results indicate that the fractional order memristor with bias voltage source can continuously obtain asymmetric tight hysteresis loop by adjusting the voltage of the bias voltage source. As the bias power supply voltage changes, the non-homogeneous fractional order memristor chaotic system exhibits that the period doubling bifurcation turns into chaos due to the symmetry breaking.
      通信作者: 方礼熠, 3148130199@qq.com
    • 基金项目: 国家自然科学基金(批准号: 51507134)和陕西省自然科学基金(批准号: 2018JM5068, 2021JM-449)资助的课题.
      Corresponding author: Fang Li-Yi, 3148130199@qq.com
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 51507134) and the Natural Science Foundation of Shaanxi Province, China (Grant Nos. 2018JM5068, 2021JM-449).
    [1]

    Chua L O 1971 IEEE Trans. Circuit. Theory 18 507Google Scholar

    [2]

    Strukov D B, Snider G S, Stewart D R, Williams R S 2008 Nature 453 80Google Scholar

    [3]

    Wen S P, Wei H Q, Yan Z, Guo Z Y, Yang, Y, Huang T W, Chen Y R 2019 IEEE Trans. Netw. Sci. Eng. 7 1431Google Scholar

    [4]

    Liu S J, Wang Y Z, Fardad M, Varshney P K 2018 IEEE Circ. Syst. Mag. 18 29Google Scholar

    [5]

    Yao P, Wu H Q, Gao B, Tang J S, Zhang Q T, Zhang W Q, Yang J J, Qian H 2020 Nature 577 641Google Scholar

    [6]

    包涵, 包伯成, 林毅, 王将, 武花干 2016 物理学报 65 180501Google Scholar

    Bao H, Bao B C, Lin Y, Wang J, Wu H G 2016 Acta Phys. Sin. 65 180501Google Scholar

    [7]

    郑广超, 刘崇新, 王琰 2018 物理学报 67 050502Google Scholar

    Zheng G C, Liu C X, Wang Y 2018 Acta Phys. Sin. 67 050502Google Scholar

    [8]

    Li C B, Wang R, Ma X, Jiang Y C, Liu Z H 2021 Chin. Phys. B 30 201Google Scholar

    [9]

    秦铭宏, 赖强, 吴永红 2022 物理学报 71 160502Google Scholar

    Qing M H, Lai Q, Wu Y H 2022 Acta Phys. Sin. 71 160502Google Scholar

    [10]

    Chen M, Ren X, Wu H G, Xu Q, Bao B C 2019 Front. Inform. Tech. El. 20 1706Google Scholar

    [11]

    Wu H G, Ye Y, Chen M, Xu Q, Bao B C 2019 IEEE Access 7 145022Google Scholar

    [12]

    Wang N, Zhang G S, Kuznetsov N V, Bao H 2021 Commun. Nonlinear Sci. Numer. Simul. 92 105494Google Scholar

    [13]

    Wu H G, Bao B C, Liu Z, Xu Q, Jiang P 2016 Nonlinear Dyn. 83 893Google Scholar

    [14]

    俞亚娟, 王在华 2015 物理学报 64 238401Google Scholar

    Yu Y J, Wang Z H 2015 Acta Phys. Sin. 64 238401Google Scholar

    [15]

    Ramakrishnan B, Durdu A, Rajagopal K, Akgul A 2020 AEU-Int. J. Electron. Commun. 123 153319Google Scholar

    [16]

    Kengne J, Tabekoueng Z N, Tamba V K, Negou A N 2015 Chaos 25 103126Google Scholar

    [17]

    Hu W P, Wang Z, Zhao Y P, Deng Z C 2020 Appl. Math. Lett. 103 106207Google Scholar

    [18]

    Kengne L K, Kengne J, Telem N A K, Pone J R M 2021 J. Circuit. Syst. Comp. 30 2150077Google Scholar

    [19]

    Kengne J, Mogue R L T, Fozin T F, Telem A N K 2019 Chaos Solitons Fractals 121 63Google Scholar

    [20]

    Cao H, Seoane J M, Sanjuán M A F 2007 Chaos Solitons Fractals 34 197Google Scholar

    [21]

    Kengne L K, Pone J R M, Tagne H T K, Kengne J 2020 AEU-Int. J. Electron. Commun. 118 153146Google Scholar

    [22]

    Wu H, Zhou J, Chen M, Xu Q, Bao B C 2022 Chaos, Solitons Fractals 154 111624Google Scholar

    [23]

    Yang N N, Xu C, Wu C J, Jia R, Liu C X 2018 Nonlinear Dyn. 97 33Google Scholar

  • 图 1  含偏置电压源忆阻器等效电路及忆阻器符号[22]

    Fig. 1.  Equivalent circuit of memristor with bias voltage source and symbol of memristor[22].

    图 2  含偏置电压源的分数阶忆阻器等效电路及忆阻器符号

    Fig. 2.  Equivalent circuit of fractional memristor with bias voltage source and symbol of memristor.

    图 3  含偏置电压源的分数阶忆阻器磁滞回线 (a) q1 = 0.98, Em = 0 V, 频率改变; (b) f = 200 Hz, Em = 0.1 V, 分数阶次改变; (c) q1 = 0.98, f = 200 Hz, 偏置电压改变

    Fig. 3.  Hysteresis loop of fractional memristor with bias voltage source: (a) q1 = 0.98, Em = 0 V, frequency change; (b) f = 200 Hz, Em = 0.1 V, fractional order change; (c) q1 = 0.98, f = 200 Hz, bias voltage change.

    图 4  V-I曲线的不对称程度

    Fig. 4.  Degree of asymmetry of the V-I curve.

    图 5  非齐次分数阶忆阻混沌系统

    Fig. 5.  Non homogeneous fractional order memristor chaotic system.

    图 6  参数d变化的平衡点在x1-x4平面上的分布

    Fig. 6.  Distribution of equilibrium points for parameter d changes on the x1-x4 plane.

    图 7  整数阶系统分岔图, 其中红色(+)和绿色(–)分别为系统初值取(±0.45 V, 0 V, 0 V, 0 V)

    Fig. 7.  Bifurcation diagram of integer order system, where red (+) and green (–) represent system initial values of (±0.45 V, 0 V, 0 V, 0 V).

    图 8  不同偏置电压Em下, 整数阶系统由周期到混沌相图, 其中红色(+)和绿色(–)分别为系统初值取(±0.45 V, 0 V, 0 V, 0 V)(a) 0 V; (b) 0.2 V; (c) 0.3 V; (d) 0.32 V; (e) 0.35 V; (f) 0.6 V

    Fig. 8.  Phase diagram of integer order systems from period to chaos at different bias voltage Em, where red (+) and green (–) represent initial values of (±0.45 V, 0 V, 0 V, 0 V) for the system: (a) 0 V; (b) 0.2 V; (c) 0.3 V; (d) 0.32 V; (e) 0.35 V; (f) 0.6 V.

    图 9  整数阶系统各平面相图 (a) v1-v2; (b) v1-v3; (c) v1-v4; (d) v2-v3

    Fig. 9.  Phase diagrams of various planes in integer order systems: (a) v1-v2; (b) v1-v3; (c) v1-v4; (d) v2-v3.

    图 10  分数阶系统分岔图, 其中红色(+)和绿色(–)分别为系统初值取(±0.45 V, 0 V, 0 V, 0 V)

    Fig. 10.  Bifurcation diagram of fractional order system, where red (+) and green (–) represent system initial values of (±0.45 V, 0 V, 0 V, 0 V).

    图 11  不同偏置电压Em下, 分数阶系统由周期到混沌相图, 其中红色(+)和绿色(–)分别为系统初值取(±0.45 V, 0 V, 0 V, 0 V)(a) 0 V; (b) 0.2 V; (c) 0.3 V; (d) 0.32 V; (e) 0.4 V; (f) 0.6 V

    Fig. 11.  Phase diagram of fractional order systems from Period to chaos at different bias voltage Em, where red (+) and green (–) represent initial values of (±0.45 V, 0 V, 0 V, 0 V) for the system: (a) 0 V; (b) 0.2 V; (c) 0.3 V; (d) 0.32 V; (e) 0.4 V; (f) 0.6 V.

    图 12  分数阶系统各平面相图 (a) v1-v2; (b) v1-v3; (c) v1-v4; (d) v2-v3

    Fig. 12.  Phase diagrams of fractional order systems in various planes: (a) v1-v2; (b) v1-v3; (c) v1-v4; (d) v2-v3.

    图 13  系统随阶次变化分岔图 (a) 系统初值为(0.45 V, 0 V, 0 V, 0 V); (b) 系统初值为(–0.45 V, 0 V, 0 V, 0 V)

    Fig. 13.  Bifurcation diagram of system with order variation: (a) Initial value of the system is (0.45 V, 0 V, 0 V, 0 V); (b) initial value of the system is (–0.45 V, 0 V, 0 V, 0 V).

    图 14  含有偏置电压源的分数阶系统吸引盆

    Fig. 14.  Fractional order system suction basin with bias voltage source.

    图 15  整数阶系统电路仿真原理图

    Fig. 15.  Schematic diagram of integer order system circuit simulation.

    图 16  不同偏置电压Em下, 整数阶系统电路仿真由周期到混沌相图, 其中红色(+)和绿色(–)分别为系统初值取(±0.45 V, 0 V, 0 V, 0 V) (a) 0 V; (b) 0.2 V; (c) 0.3 V; (d) 0.32 V; (e) 0.35 V; (f) 0.6 V

    Fig. 16.  Circuit simulation of integer order system from period to chaos phase diagram at different bias voltage Em, where red (+) and green (–) represent initial values of (±0.45 V, 0 V, 0 V, 0 V) for the system: (a) 0 V; (b) 0.2 V; (c) 0.3 V; (d) 0.32 V; (e) 0.35 V; (f) 0.6 V.

    图 17  整数阶系统电路仿真各平面相图 (a) v1-v2; (b) v1-v3; (c) v1-v4; (d) v2-v3

    Fig. 17.  Fractional order system circuit simulation phase diagrams of each plane: (a) v1-v2; (b) v1-v3; (c) v1-v4; (d) v2-v3.

    图 18  分数阶电容等效电路

    Fig. 18.  Fractional order capacitor equivalent circuit.

    图 19  分数阶系统电路仿真原理图

    Fig. 19.  Schematic diagram of fractional order system circuit simulation.

    图 20  不同偏置电压Em下, 分数阶系统由周期到混沌电路仿真相图, 其中红色(+)和绿色(–)分别为系统初值取(±0.45 V, 0 V, 0 V, 0 V) (a) 0 V; (b) 0.2 V; (c) 0.3 V; (d) 0.32 V; (e) 0.4 V; (f) 0.6 V

    Fig. 20.  Simulation phase diagram of fractional order system from period to chaos circuit at different bias voltage Em, where red (+) and green (–) represent initial values of (±0.45 V, 0 V, 0 V, 0 V) for the system: (a) 0 V; (b) 0.2 V; (c) 0.3 V; (d) 0.32 V; (e) 0.4 V; (f) 0.6 V.

    图 21  分数阶系统电路仿真各平面相图 (a) v1-v2; (b) v1-v3; (c) v1-v4; (d) v2-v3

    Fig. 21.  Fractional order system circuit simulation phase diagrams of each plane: (a) v1-v2; (b) v1-v3; (c) v1-v4; (d) v2-v3.

    图 22  混沌系统实验原理图及实验平台

    Fig. 22.  Schematic diagram and experimental platform of integer order system experiment.

    图 23  不同偏置电压Em下, 整数阶实验由周期进入混沌相图, 其中粉色(+)与蓝色(–)分别代表系统初值为(±0.45 V, 0 V, 0 V, 0 V) (a) 0 V; (b) 0.2 V; (c) 0.3 V; (d) 0.32 V; (e) 0.35 V; (f) 0.6 V

    Fig. 23.  Integer order experiment from period to chaotic phase diagram at different bias voltage Em, where pink (+) and blue (–) represent system initial values of (±0.45 V, 0 V, 0 V, 0 V), respectively: (a) 0 V; (b) 0.2 V; (c) 0.3 V; (d) 0.32 V; (e) 0.35 V; (f) 0.6 V.

    图 24  整数阶实验各平面相图 (a) v1-v2; (b) v1-v3; (c) v1-v4; (d) v2-v3

    Fig. 24.  Phase diagrams of various planes in integer order experiments: (a) v1-v2; (b) v1-v3; (c) v1-v4; (d) v2-v3.

    图 25  不同偏置电压Em下, 分数阶实验由周期进入混沌相图, 其中粉色(+)与蓝色(–)分别代表系统初值为(±0.45 V, 0 V, 0 V, 0 V) (a) 0 V; (b) 0.2 V; (c) 0.3 V; (d) 0.32 V; (e) 0.35 V; (f) 0.6 V

    Fig. 25.  Integer order experiment from period to chaotic phase diagram at different bias voltage Em, where pink (+) and blue (–) represent system initial values of (±0.45 V, 0 V, 0 V, 0 V), respectively: (a) 0 V; (b) 0.2 V; (c) 0.3 V; (d) 0.32 V; (e) 0.35 V; (f) 0.6 V.

    图 26  分数阶实验各平面相图 (a) v1-v2; (b) v1-v3; (c) v1-v4; (d) v2-v3

    Fig. 26.  Fractional order experiment phase diagrams of each plane: (a) v1-v2; (b) v1-v3; (c) v1-v4; (d) v2-v3.

    表 1  系统平衡点及其稳定性

    Table 1.  System equilibrium point and its stability.

    d 平衡点 特征值λ1λ4 稳定性
    0, 0.1, 0.3 P0(0, 0, 0, 0) 0.3773, –0.6886 ±5.2293i, –0.0172 指数–1 USF
    0, 0.1, 0.3 P(–19.7558, 0, 0, 1.9755) 1.2573 ±5.8895i, –3.5516, –0.0143 指数–2 USF
    0 P+(19.7558, 0, 0, 1.9755) 1.2573 ±5.8895i, –3.5516, –0.0143 指数–2 USF
    0.1 P+(19.9495, 0, 0, 1.9949) 1.2689 ±5.8991i, –3.5751, –0.0143 指数–2 USF
    0.3 P+(20.3363, 0, 0, 2.0336) 1.2918 ±5.9182i, –3.6215, –0.0143 指数–2 USF
    下载: 导出CSV

    表 2  分数阶电容$C_{\text{m}}^{{q_1}}$和$C_{2}^{{q_2}}$的等效电阻参数

    Table 2.  Equivalent resistance parameters of fractional capacitor $C_{\text{m}}^{{q_1}}$ and $C_{2}^{{q_2}}$.

    RinRo1Ro2/(103 Ω)Ro3/(105 Ω)Ro4/(109 Ω)Ro5/(105 Ω)
    $C_{\text{m}}^{{q_1}} = 5.8 \times {10^3}{\text{ nF}}$0.22735.3271.2032.7051.1581.209
    $C_{2}^{{q_2}} = 10{\text{ nF}}$114.81461348.7828.9730.0837.3
    下载: 导出CSV

    表 3  分数阶电容$C_{\text{m}}^{{q_1}}$和$C_{2}^{{q_2}}$的等效电容参数

    Table 3.  Equivalent capacitance parameters of fractional capacitance $C_{\text{m}}^{{q_1}}$ and $C_{2}^{{q_2}}$.

    Co1/(10–7 F)Co2/(10–7 F)Co3/(10–7 F)Co4/(10–8 F)Co1/(10–5 F)
    $C_{\text{m}}^{{q_1}} = 5.8 \times {10^3}{\text{ nF}}$446.2496.3554.3647.86200
    $C_{2}^{{q_2}} = 10{\text{ nF}}$1.6721.7601.8601.0579.214
    下载: 导出CSV
  • [1]

    Chua L O 1971 IEEE Trans. Circuit. Theory 18 507Google Scholar

    [2]

    Strukov D B, Snider G S, Stewart D R, Williams R S 2008 Nature 453 80Google Scholar

    [3]

    Wen S P, Wei H Q, Yan Z, Guo Z Y, Yang, Y, Huang T W, Chen Y R 2019 IEEE Trans. Netw. Sci. Eng. 7 1431Google Scholar

    [4]

    Liu S J, Wang Y Z, Fardad M, Varshney P K 2018 IEEE Circ. Syst. Mag. 18 29Google Scholar

    [5]

    Yao P, Wu H Q, Gao B, Tang J S, Zhang Q T, Zhang W Q, Yang J J, Qian H 2020 Nature 577 641Google Scholar

    [6]

    包涵, 包伯成, 林毅, 王将, 武花干 2016 物理学报 65 180501Google Scholar

    Bao H, Bao B C, Lin Y, Wang J, Wu H G 2016 Acta Phys. Sin. 65 180501Google Scholar

    [7]

    郑广超, 刘崇新, 王琰 2018 物理学报 67 050502Google Scholar

    Zheng G C, Liu C X, Wang Y 2018 Acta Phys. Sin. 67 050502Google Scholar

    [8]

    Li C B, Wang R, Ma X, Jiang Y C, Liu Z H 2021 Chin. Phys. B 30 201Google Scholar

    [9]

    秦铭宏, 赖强, 吴永红 2022 物理学报 71 160502Google Scholar

    Qing M H, Lai Q, Wu Y H 2022 Acta Phys. Sin. 71 160502Google Scholar

    [10]

    Chen M, Ren X, Wu H G, Xu Q, Bao B C 2019 Front. Inform. Tech. El. 20 1706Google Scholar

    [11]

    Wu H G, Ye Y, Chen M, Xu Q, Bao B C 2019 IEEE Access 7 145022Google Scholar

    [12]

    Wang N, Zhang G S, Kuznetsov N V, Bao H 2021 Commun. Nonlinear Sci. Numer. Simul. 92 105494Google Scholar

    [13]

    Wu H G, Bao B C, Liu Z, Xu Q, Jiang P 2016 Nonlinear Dyn. 83 893Google Scholar

    [14]

    俞亚娟, 王在华 2015 物理学报 64 238401Google Scholar

    Yu Y J, Wang Z H 2015 Acta Phys. Sin. 64 238401Google Scholar

    [15]

    Ramakrishnan B, Durdu A, Rajagopal K, Akgul A 2020 AEU-Int. J. Electron. Commun. 123 153319Google Scholar

    [16]

    Kengne J, Tabekoueng Z N, Tamba V K, Negou A N 2015 Chaos 25 103126Google Scholar

    [17]

    Hu W P, Wang Z, Zhao Y P, Deng Z C 2020 Appl. Math. Lett. 103 106207Google Scholar

    [18]

    Kengne L K, Kengne J, Telem N A K, Pone J R M 2021 J. Circuit. Syst. Comp. 30 2150077Google Scholar

    [19]

    Kengne J, Mogue R L T, Fozin T F, Telem A N K 2019 Chaos Solitons Fractals 121 63Google Scholar

    [20]

    Cao H, Seoane J M, Sanjuán M A F 2007 Chaos Solitons Fractals 34 197Google Scholar

    [21]

    Kengne L K, Pone J R M, Tagne H T K, Kengne J 2020 AEU-Int. J. Electron. Commun. 118 153146Google Scholar

    [22]

    Wu H, Zhou J, Chen M, Xu Q, Bao B C 2022 Chaos, Solitons Fractals 154 111624Google Scholar

    [23]

    Yang N N, Xu C, Wu C J, Jia R, Liu C X 2018 Nonlinear Dyn. 97 33Google Scholar

  • [1] 郭慧朦, 梁燕, 董玉姣, 王光义. 蔡氏结型忆阻器的简化及其神经元电路的硬件实现. 物理学报, 2023, 72(7): 070501. doi: 10.7498/aps.72.20222013
    [2] 徐威, 王钰琪, 李岳峰, 高斐, 张缪城, 连晓娟, 万相, 肖建, 童祎. 新型忆阻器神经形态电路的设计及其在条件反射行为中的应用. 物理学报, 2019, 68(23): 238501. doi: 10.7498/aps.68.20191023
    [3] 王伟, 曾以成, 孙睿婷. 含三个忆阻器的六阶混沌电路研究. 物理学报, 2017, 66(4): 040502. doi: 10.7498/aps.66.040502
    [4] 阮静雅, 孙克辉, 牟俊. 基于忆阻器反馈的Lorenz超混沌系统及其电路实现. 物理学报, 2016, 65(19): 190502. doi: 10.7498/aps.65.190502
    [5] 俞亚娟, 王在华. 一个分数阶忆阻器模型及其简单串联电路的特性. 物理学报, 2015, 64(23): 238401. doi: 10.7498/aps.64.238401
    [6] 贺少波, 孙克辉, 王会海. 分数阶混沌系统的Adomian分解法求解及其复杂性分析. 物理学报, 2014, 63(3): 030502. doi: 10.7498/aps.63.030502
    [7] 杨芳艳, 冷家丽, 李清都. 基于Chua电路的四维超混沌忆阻电路. 物理学报, 2014, 63(8): 080502. doi: 10.7498/aps.63.080502
    [8] 李志军, 曾以成, 李志斌. 改进型细胞神经网络实现的忆阻器混沌电路. 物理学报, 2014, 63(1): 010502. doi: 10.7498/aps.63.010502
    [9] 洪庆辉, 李志军, 曾金芳, 曾以成. 基于电流反馈运算放大器的忆阻混沌电路设计与仿真. 物理学报, 2014, 63(18): 180502. doi: 10.7498/aps.63.180502
    [10] 梁燕, 于东升, 陈昊. 基于模拟电路的新型忆感器等效模型. 物理学报, 2013, 62(15): 158501. doi: 10.7498/aps.62.158501
    [11] 洪庆辉, 曾以成, 李志军. 含磁控和荷控两种忆阻器的混沌电路设计与仿真. 物理学报, 2013, 62(23): 230502. doi: 10.7498/aps.62.230502
    [12] 许碧荣. 一种最简的并行忆阻器混沌系统. 物理学报, 2013, 62(19): 190506. doi: 10.7498/aps.62.190506
    [13] 周小勇. 一种具有恒Lyapunov指数谱的混沌系统及其电路仿真. 物理学报, 2011, 60(10): 100503. doi: 10.7498/aps.60.100503
    [14] 包伯成, 胡文, 许建平, 刘中, 邹凌. 忆阻混沌电路的分析与实现. 物理学报, 2011, 60(12): 120502. doi: 10.7498/aps.60.120502
    [15] 孙克辉, 杨静利, 丁家峰, 盛利元. 单参数Lorenz混沌系统的电路设计与实现. 物理学报, 2010, 59(12): 8385-8392. doi: 10.7498/aps.59.8385
    [16] 闵富红, 余杨, 葛曹君. 超混沌分数阶Lü系统电路实验与追踪控制. 物理学报, 2009, 58(3): 1456-1461. doi: 10.7498/aps.58.1456
    [17] 刘扬正, 姜长生, 李心朝, 孙 晗. 复杂超混沌Lü系统的电路实验. 物理学报, 2008, 57(11): 6808-6814. doi: 10.7498/aps.57.6808
    [18] 禹思敏, 禹之鼎. 一个新的五阶超混沌电路及其研究. 物理学报, 2008, 57(11): 6859-6867. doi: 10.7498/aps.57.6859
    [19] 陈向荣, 刘崇新, 王发强, 李永勋. 分数阶Liu混沌系统及其电路实验的研究与控制. 物理学报, 2008, 57(3): 1416-1422. doi: 10.7498/aps.57.1416
    [20] 陈菊芳, 程 丽, 刘 颖, 彭建华. 延迟变量反馈法控制离散混沌系统的电路实验. 物理学报, 2003, 52(1): 18-24. doi: 10.7498/aps.52.18
计量
  • 文章访问数:  2533
  • PDF下载量:  76
  • 被引次数: 0
出版历程
  • 收稿日期:  2023-07-26
  • 修回日期:  2023-09-12
  • 上网日期:  2023-10-09
  • 刊出日期:  2024-01-05

/

返回文章
返回