Simulink modeling and dynamic characteristics of discrete memristor chaotic system

Fu Long-Xiang; He Shao-Bo; Wang Hui-Hai; Sun Ke-Hui

doi:10.7498/aps.71.20211549

Abstract

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1. 引　言
2. 离散忆阻建模与Simulink仿真
2.1 离散忆阻数学模型
2.2 Simulink仿真实现
3. 离散忆阻混沌系统及其Simulink仿真
3.1 离散忆阻混沌系统
3.2 离散忆阻混沌系统Simulink仿真
4. 动力学特性分析
4.1 平衡点及其稳定性分析
4.2 分岔图与Lyapunov指数谱分析
4.3 复杂度分析
4.4 多稳态分析
5. 结　论

References

Abstract

In the last two years, the discrete memristor has been proposed, and it is in the early stages of research. Now, it is particularly important to use various simulation softwares to expand the applications of the discrete memristor model. Based on the difference operator, in this paper, a discrete memristor model with quadratic nonlinearity is constructed. The addition, subtraction, multiplication and division of the discrete memristor mathematical model are clarified, and the charge q is obtained by combining the discrete-time summation module, thereby realizing the Simulink simulation of the discrete memristor. The simulation results show that the designed memristor meets the three fingerprints of memristor, indicating that the designed discrete memristor belongs to generalized memristor.Using memristors to construct chaotic systems is one of the current research hotspots, but most of the literature is about the introduction of continuous memristors into continuous chaotic systems. In this paper, the obtained discrete memristor is introduced into a three-dimensional chaotic map which is mentioned in a Sprott’s book titled as Chaos and Time-Series Analysis, and a new four-dimensional memristor chaotic map is designed. Meanwhile, the Simulink model of the chaotic map is established. It is found that attractors with different sizes and shapes can be observed by changing the parameters in the Simulink model, indicating that the changes of system parameters and memristor parameters can change the dynamic behavior of the system. The analyses of equilibria and equilibrium stability show that the four-dimensional chaotic map has infinite equilibrium points. The Lyapunov exponent spectra and bifurcation diagrams of the circuit imply that the map can transform between weak chaotic state, chaotic state, and hyperchaotic state. Meanwhile, the multistability and coexisting attractors are analyzed under different initial conditions. Moreover, by comparing the results of measuring the complexity, it is found that the chaotic map with discrete memristor has richer dynamical behaviors and higher complexity than the original map.From the perspective of system modeling, in this paper the discrete memristor modeling and discrete memristor map designing are discussed based on the Matlab/Simulink. It further verifies the realizability and lays a foundation for the future applications of discrete memristor.

Keywords:

PACS:

05.45.-a

(Nonlinear dynamics and chaos)

Authors and contacts

School of Physics and Electronics, Central South University, Changsha 410083, China

Corresponding author: He Shao-Bo, heshaobo@csu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 61901530, 62071496, 62061008) and the Natural Science Foundation of Hunan Province, China (Grant No. 2020JJ5767).

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1. 引　言

1971年, 蔡少棠教授^[1]在研究基本电学物理量即电压V、电流I、电荷q、磁通φ之间的相互关系时, 基于电路完备性考虑, 预测存在除电阻、电感和电容外的第4种无源非线性基本电路元件, 用以描述电荷与磁通之间的关系, 称其为忆阻器. 2008年, 惠普(Hewlett-Packard, HP)实验室^[2]利用TiO₂和金属铂首次实现了世界上纳米级忆阻器, 从此掀起了忆阻器研究热潮. 近年来, 忆阻器在许多应用领域都有深入研究, 如图像处理^[3]、存储器^[4-6]、神经网络^[7-9]、神经动力学^[10,11]等. 但由于HP忆阻器纳米级工艺以及严格实验条件的限制, 迄今为止, 忆阻器尚未得到商用. 为了推进对忆阻器等记忆元件的研究, 建立合适的仿真模型至关重要. 近年来, 很多基于SPICE (simulation program with integrated circuit emphasis)的各类忆阻仿真模型相继被提出^[12-14]. 同时, Simulink作为Matlab中一种可视化建模工具, 能对大多数非线性系统进行分析, 具有流程清晰、易观察、效率高等诸多优点, 便于更好地研究与分析非线性系统. 因此, 采用Matlab/Simulink仿真工具对忆阻器进行建模值得进一步研究^[15-17].

忆阻器是一种非线性元件, 其在混沌系统方面的应用也是忆阻器的研究热点之一. 在王春华等^[18]对忆阻器相关应用总结中提到, 可将忆阻器应用到混沌电路分为4种方式, 即用忆阻器替换蔡氏电路中非线性元件^[19-21]、新增忆阻器^[22-24]、忆阻器替换非Chua系统中线性/非线性项的方法^[25,26]、利用不同数学模型的忆阻器进行电路设计^[27,28]. 本文将利用新增忆阻的方式, 将离散忆阻引入到三维混沌映射中, 构建一种新型四维忆阻混沌映射, 并研究离散忆阻混沌系统建模方法.

本文内容安排如下. 第2节, 基于离散忆阻数学模型建立离散忆阻Simulink仿真模型; 第3节, 将离散忆阻模型加入到三维Lorenz混沌映射中, 设计一新型混沌映射, 并构建其Simulink模型; 第4节, 对该离散忆阻器及混沌系统Simulink模型的应用进行分析, 最后, 对本文进行总结.

2. 离散忆阻建模与Simulink仿真

2.1 离散忆阻数学模型

类似于连续忆阻器模型, 离散忆阻器的定义为^[29]

$\left\{ \begin{aligned} & V\left( {{t_n}} \right) = M\left( {q\left( {{t_n}} \right)} \right)i\left( {{t_n}} \right), \hfill \\ & \Delta q\left( {{t_n}} \right) = ki\left( {{t_n}} \right), \end{aligned} \right.$

(1)

其中, n = 0, 1, 2, ···, $\Delta q\left( {{t_n}} \right) = q\left( {{t_{n + 1}}} \right) - q\left( {{t_n}} \right)$ 为前向差分. 因为

$\left\{ \begin{aligned} & q\left( {{t_1}} \right) - q\left( {{t_0}} \right) = ki\left( {{t_0}} \right), \\ & q\left( {{t_2}} \right) - q\left( {{t_1}} \right) = ki\left( {{t_1}} \right), \\ & \qquad\qquad \vdots \\ & q\left( {{t_{n + 1}}} \right) - q\left( {{t_n}} \right) = ki\left( {{t_n}} \right), \end{aligned} \right.$

(2)

等号两边相加, 则忆阻“内部电荷”计算式为

$q\left( {{t_{n + 1}}} \right) = q\left( {{t_0}} \right) + k\sum\limits_{j = 0}^n {i\left( {{t_j}} \right)} ,$

(3)

显然, q(t₀)为离散忆阻内部初始电荷. 在实际计算过程中, 该离散忆阻器为一离散累加数学模型. 一般地, 可以将该离散忆阻改写为迭代式, 即:

$\left\{ \begin{aligned} & V\left( {{t_n}} \right) = M\left( {{q_n}} \right)i\left( {{t_n}} \right), \hfill \\ & {q_{n + 1}} = {q_n} + k{i_n}, \end{aligned} \right.$

(4)

式中, i(t_n)为离散输入电流信号, V(t_n)为离散输出电压信号, q_n为忆阻内部电荷, M(q_n)为忆阻值. 其中M(·)函数将采用平方函数^[30], 因此, 该离散忆阻器模型可描述为

$\left\{ \begin{aligned} &{V_n} = M\left( {{q_n}} \right){i_n}, \hfill \\ &M\left( {{q_n}} \right) = \alpha + \beta q_n^2, \hfill \\ & {q_{n + 1}} = {q_n} + k{i_n}, \end{aligned} \right.$

(5)

式中, α和β为忆阻器参数, 令k = 1.

2.2 Simulink仿真实现

根据离散忆阻器的数学模型, 构建离散忆阻器的Simulink模型, 设计的模型如图1所示. 该模型将离散忆阻数学模型中的加减乘除分类整合, 且由离散时间求和模块结合前向差分算子实现电荷q的迭代, 其中可通过改变Constant与Gain1的取值来改变α与β的值, Scope1显示的是忆阻器输入电压v_n与流经其电流i_n之间的关系. 首先, 对该离散忆阻Simulink模型进行验证, 这里取α = 1, β = 0.000002. 发现在正弦电流的作用下, 忆阻器电流-电压特性曲线见图2(a), 具有迟滞环特性. 其次, 分别改变输入正弦信号 $I({t_n}) = {A_0}\sin \left( {2{\text{π}}w{t_n}} \right)$ 的幅值和频率以讨论输入信号幅值和频率对该忆阻模型的影响. 取不同的幅值A₀ = 1, 1.5, 2 A, 其中ω = 0.005 rad/s, 结果如图2(b)所示. 可见, 在输入信号频率保持不变的情况下, 该忆阻模型的滞回曲线随着输入信号幅值的增大而增大. 保持输入信号幅值A₀ = 1.5 A不变, 改变输入信号频率, 即ω分别取0.1, 0.01, 0.003 rad/s, 结果如图2(c)所示, 可见在输入信号幅值一定的情况下, 该忆阻模型的滞回曲线随着输入信号频率增大而减小, 最终退化为线性器件. 经以上分析, 根据忆阻器特性判定原则可知, 所构建的离散忆阻Simulink模型满足忆阻定义^[31], 可用于进一步的应用研究.

$图 1 离散忆阻Simulink模型\r\nFig. 1. Discrete memristor Simulink model.$

图 1 离散忆阻Simulink模型

Fig. 1. Discrete memristor Simulink model.

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$图 2 离散忆阻器的电流-电压特性曲线　(a)电流-电压特性曲线; (b) 幅值变化; (c) 频率变化\r\nFig. 2. Current-voltage characteristic curves of discrete memristor: (a) Current-voltage characteristic curve; (b) amplitude change; (c) frequency change.$

图 2 离散忆阻器的电流-电压特性曲线　(a)电流-电压特性曲线; (b) 幅值变化; (c) 频率变化

Fig. 2. Current-voltage characteristic curves of discrete memristor: (a) Current-voltage characteristic curve; (b) amplitude change; (c) frequency change.

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3. 离散忆阻混沌系统及其Simulink仿真

3.1 离散忆阻混沌系统

Sprott^[32]报道了一种三维混沌映射, 并称之为Lorenz混沌映射, 其动力学方程为

$\left\{ \begin{aligned} & {x_{n + 1}} = {x_n}{y_n} - {z_n}, \hfill \\ & {y_{n + 1}} = {x_n}, \hfill \\ & {z_{n + 1}} = {y_n}. \end{aligned} \right.$

(6)

采用结构化设计方法, 该映射可以被设计为图3所示的结构图. 可见, 该系统存在3个延时模块, 1个加法器和1个乘法器.

$图 3 三维Lorenz混沌映射(6)系统框图\r\nFig. 3. Structure diagram of the three-dimensional Lorenz chaotic map (6).$

图 3 三维Lorenz混沌映射(6)系统框图

Fig. 3. Structure diagram of the three-dimensional Lorenz chaotic map (6).

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引入离散忆阻, 得到一新型三维忆阻混沌映射, 其定义为

$\left\{ \begin{aligned} & {x_{n + 1}} = a\Bigg[ {\alpha + \beta {{\Bigg( {\sum\limits_{j = 0}^n {{x_n}} } \Bigg)}^2}} \Bigg]{x_n}{y_n} - {z_n}, \\ & {y_{n + 1}} = {x_n}, \hfill \\ & {z_{n + 1}} = {y_n}, \end{aligned} \right.$

(7)

其中, a为系统参数, 用于控制忆阻的输出值对系统动力学行为的影响. 根据上述系统方程(7), 绘制引入忆阻器后的混沌系统框图, 如图4所示. 可见, 离散忆阻增加在x_n与y_n的乘积项上.

$图 4 加入忆阻器后的三维混沌映射(7)系统框图\r\nFig. 4. Structure diagram of the three-dimensional chaotic map with memristor (7).$

图 4 加入忆阻器后的三维混沌映射(7)系统框图

Fig. 4. Structure diagram of the three-dimensional chaotic map with memristor (7).

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结合离散忆阻数学模型, 令:

${w_{n + 1}} = \sum\limits_{j = 0}^n {{x_j}} = \sum\limits_{j = 0}^{n - 1} {{x_j}} + {x_n} = {w_n} + {x_n} ,$

(8)

进而可以得到系统的四维表达式为

$\left\{ \begin{aligned} & {x_{n + 1}} = a\left( {\alpha + \beta w_n^2} \right){x_n}{y_n} - {z_n}, \hfill \\ & {y_{n + 1}} = {x_n}, \hfill \\ & {z_{n + 1}} = {y_n}, \hfill \\ & {w_{n + 1}} = {w_n} + {x_n}. \end{aligned} \right.$

(9)

根据(9)式绘制设计的忆阻混沌系统框图, 如图5所示, 其中阴影部分描述的是离散忆阻的结构.

$图 5 四维忆阻混沌映射(9)系统框图\r\nFig. 5. Structure diagram of the four-dimensional memristor chaotic map (9).$

图 5 四维忆阻混沌映射(9)系统框图

Fig. 5. Structure diagram of the four-dimensional memristor chaotic map (9).

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3.2 离散忆阻混沌系统Simulink仿真

根据四维忆阻混沌系统, 建立该系统Simulink仿真模型, 如图6所示, 其中忆阻模块采用图1所示的模型. 模型中增益Gain2表示系统参数a的值, Scope2显示的是系统x_n, y_n, z_n时间序列, 并可改变3个延时器以及忆阻模块中离散时间求和模块中的初值, 从而分别改变系统初始状态x₀, y₀, z₀, w₀值.

$图 6 四维忆阻混沌映射(9) Simulink模型\r\nFig. 6. Simulink model of thefour-dimensional memristor chaotic map (9).$

图 6 四维忆阻混沌映射(9) Simulink模型

Fig. 6. Simulink model of thefour-dimensional memristor chaotic map (9).

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设置该模型系统参数a = 0.9, 初始状态为x₀ = 0.5, y₀ = 0.5, z₀ = 0.5, w₀ = 0.4, 忆阻参数α = 1, β取值分别为–0.1, –0.02, –0.000002. 将示波器中的图像保存到WorkSpace, 再利用Matlab绘制出图像, 得到x-y平面的吸引子相图, 如图7(a)—(c)所示. 可见, 随着β的取值的不同, 吸引子大小形状产生较大变化. 设置系统初始状态为x₀ = 0.5, y₀ = 0.5, z₀ = 0.5, w₀ = 0.4, 忆阻参数α = 1, β = –0.001, 改变系统参数a, 分别取值为0.25, 0.5, 0.9, 得到的x_n-y_n平面吸引子相图, 如图7(d)—(f)所示, 表明参数a也能够改变吸引子大小与形状. 绘制图7中各个状态下对应的Lyapunov指数, 如图8所示. 可见, 系统参数以及忆阻参数的变动能够改变系统的动力学行为.

$图 7 四维忆阻混沌映射(9)的Simulink仿真吸引子图　(a) β = –0.1, 超混沌状态; (b) β = –0.02, 超混沌状态; (c) β = –0.000002, 混沌状态; (d) a = 0.25, 混沌状态; (e) a = 0.5, 混沌状态; (f) a = 0.9, 混沌状态\r\nFig. 7. Simulink simulation results of the four-dimensional memristor chaotic map (9): (a) β = –0.1, hyperchaotic; (b) β = –0.02, hyperchaotic; (c) β = –0.000002, chaos; (d) a = 0.25, chaos; (e) a = 0.5, chaos; (f) a = 0.9, chaos.$

图 7 四维忆阻混沌映射(9)的Simulink仿真吸引子图　(a) β = –0.1, 超混沌状态; (b) β = –0.02, 超混沌状态; (c) β = –0.000002, 混沌状态; (d) a = 0.25, 混沌状态; (e) a = 0.5, 混沌状态; (f) a = 0.9, 混沌状态

Fig. 7. Simulink simulation results of the four-dimensional memristor chaotic map (9): (a) β = –0.1, hyperchaotic; (b) β = –0.02, hyperchaotic; (c) β = –0.000002, chaos; (d) a = 0.25, chaos; (e) a = 0.5, chaos; (f) a = 0.9, chaos.

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$图 8 四维忆阻混沌映射(9)的Simulink仿真吸引子图对应Lyapunov指数谱　(a) β = –0.1, 超混沌状态; (b) β = –0.02, 超混沌状态; (c) β = –0.000002, 混沌状态; (d) a = 0.25, 混沌状态; (e) a = 0.5, 混沌状态; (f) a = 0.9, 混沌状态\r\nFig. 8. Simulink simulation attractor diagram of the four-dimensional memristor chaotic map (9) corresponds to Lyapunov Exponent spectra: (a) β = –0.1, hyperchaotic; (b) β = –0.02, hyperchaotic; (c) β = –0.000002, chaos; (d) a = 0.25, chaos; (e) a = 0.5, chaos; (f) a = 0.9, chaos.$

图 8 四维忆阻混沌映射(9)的Simulink仿真吸引子图对应Lyapunov指数谱　(a) β = –0.1, 超混沌状态; (b) β = –0.02, 超混沌状态; (c) β = –0.000002, 混沌状态; (d) a = 0.25, 混沌状态; (e) a = 0.5, 混沌状态; (f) a = 0.9, 混沌状态

Fig. 8. Simulink simulation attractor diagram of the four-dimensional memristor chaotic map (9) corresponds to Lyapunov Exponent spectra: (a) β = –0.1, hyperchaotic; (b) β = –0.02, hyperchaotic; (c) β = –0.000002, chaos; (d) a = 0.25, chaos; (e) a = 0.5, chaos; (f) a = 0.9, chaos.

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4. 动力学特性分析

4.1 平衡点及其稳定性分析

令(9)式左右两边相等, 即可求出系统平衡点为

$\left\{ \begin{aligned} & {x^{\left( {\text{e}} \right)}} = {y^{\left( {\text{e}} \right)}} = {z^{\left( {\text{e}} \right)}} = 0, \hfill \\ & {w^{\left( {\text{e}} \right)}} = C, \end{aligned} \right.$

(10)

其中, C可以为任意值, 也就是说w^(e)可以取任意值, 即系统具有无限多的平衡点.

对系统进行线性化分析, 得其Jacobian矩阵为

$\begin{split} {\boldsymbol{J}} = \left[ {\begin{array}{*{20}{c}} {a\left( {\alpha + \beta {w^2}} \right)y}&{a\left( {\alpha + \beta {w^2}} \right)x}&{ - 1}&{2a\beta xyw} \\ 1&0&0&0 \\ 0&1&0&0 \\ 1&0&0&1 \end{array}} \right] ,\end{split}$

(11)

代入平衡点后矩阵为

${{\boldsymbol{J}}^{\left( {\text{e}} \right)}} = \left[ {\begin{array}{*{20}{c}} 0&0&{ - 1}&0 \\ 1&0&0&0 \\ 0&1&0&0 \\ 1&0&0&1 \end{array}} \right] ,$

(12)

经计算得其特征值为λ_i= [1, –1, 0.5+0.866i, 0.5–0.866i]. 可见, 系统存在不稳定平衡点, 可产生混沌, 即验证了前述Simulink仿真结果.

4.2 分岔图与Lyapunov指数谱分析

系统(9)具有无限多不稳定的平衡点, 表明此离散忆阻混沌系统具有丰富的动力学行为, 为了进一步研究系统的复杂动力学特性, 分析了当a在[0, 1]范围变化时, Lyapunov指数谱和系统的分岔图.

Lyapunov指数是相空间中邻近轨迹发散或收敛的平均指数率, 出现正的Lyapunov指数时, 表明系统是混沌的. 初始状态设置为x₀ = 0.5, y₀ = 0.5, z₀ = 0.5, w₀ = 0.4, 忆阻参数α = 1, β = –0.001, 迭代次数为N = 10000, 通过Matlab程序仿真绘制x_n随参数a变化的分岔图和Lyapunov指数谱, 如图9所示. 可见, Lyapunov指数谱与分岔图吻合, 两图都表明了系统能产生复杂的动力学行为. 当a∈[0, 0.26)时, Lyapunov指数有两个大于零的值, 但其值较小, 且分岔图形状与a∈[0.26, 1]时有明显差别. 因此, 进一步分析a的取值与该区间x_n序列值及对应的Lyapunov指数, 即a = 0.1, 忆阻参数和初值保持不变, 绘制x_n序列值及对应Lyapunov指数如图10所示. 通过x_n序列发现此时系统处于非周期状态, 存在混沌成分, 且Lyapunov指数值很小, 处于弱混沌状态. 当a = 0.9时, Lyapunov指数中有两个正数, 且值相对较大, 结合分岔图可以发现该系统处于超混沌状态. 通过分岔图还可以发现在超混沌区间中出现了周期窗口, 表明系统的动力学行为对参数a的取值比较敏感.

$图 9 四维忆阻混沌映射(9)分岔图与Lyapunov指数谱　(a) 分岔图; (b) Lyapunov指数谱\r\nFig. 9. Bifurcation and Lyapunov exponent spectra of the four-dimensional memristor chaotic map (9): (a) Bifurcation diagram; (b) Lyapunov exponent (LE) spectra.$

图 9 四维忆阻混沌映射(9)分岔图与Lyapunov指数谱　(a) 分岔图; (b) Lyapunov指数谱

Fig. 9. Bifurcation and Lyapunov exponent spectra of the four-dimensional memristor chaotic map (9): (a) Bifurcation diagram; (b) Lyapunov exponent (LE) spectra.

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$图 10 弱混沌系统动力学分析　(a) xn序列图; (b) Lyapunov指数谱\r\nFig. 10. Dynamic analysis of weakly chaotic system: (a) xn sequence diagram; (b) Lyapunov exponent spectra.$

图 10 弱混沌系统动力学分析　(a) x_n序列图; (b) Lyapunov指数谱

Fig. 10. Dynamic analysis of weakly chaotic system: (a) x_nsequence diagram; (b) Lyapunov exponent spectra.

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初始状态设置为x₀ = 0.5, y₀ = 0.5, z₀ = 0.5, w₀ = 0.4, N = 10000, 忆阻参数α = 1, β = –0.001, 系统参数a分别取0.25, 0.5, 绘制四维忆阻混沌映射(9)随初值z₀变化的Lyapunov指数谱, 结果如图11(a)和图11(b)所示, 原三维Lorenz混沌映射(6)的Lyapunov指数谱, 如图11(c)所示, 对比可见, 加入忆阻器后系统动力学行为变得更加复杂了, 由混沌状态进入了超混沌状态.

$图 11 随初值z0变化的Lyapunov指数谱　(a)四维忆阻混沌映射(9), a = 0.25; (b) 四维忆阻混沌映射(9), a = 0.5; (c) 三维Lorenz混沌映射(6)\r\nFig. 11. Lyapunov exponent spectra with initial value z0: (a) Four-dimensional memristor chaotic map (9), a = 0.5; (b) four-dimensional memristor chaotic map (9), a = 0.5; (c) three-dimensional Lorenz chaotic map (6).$

图 11 随初值z₀变化的Lyapunov指数谱　(a)四维忆阻混沌映射(9), a = 0.25; (b) 四维忆阻混沌映射(9), a = 0.5; (c) 三维Lorenz混沌映射(6)

Fig. 11. Lyapunov exponent spectra with initial value z₀: (a) Four-dimensional memristor chaotic map (9), a = 0.5; (b) four-dimensional memristor chaotic map (9), a = 0.5; (c) three-dimensional Lorenz chaotic map (6).

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4.3 复杂度分析

复杂度是衡量一个混沌序列与随机序列接近程度的指标, 复杂度越大意味着混沌序列越随机^[33]. 绘制当初始状态设置为x₀ = 0.5, y₀ = 0.5, z₀ = 0.5, w₀ = 0.4, 忆阻参数α = 1, β = –0.001, 迭代次数N = 10000, a在[0, 1]之间的系统样本熵复杂度^[34], 如图12所示. 可见系统样本熵复杂度呈上升趋势, 且在a = 0.26后, 系统进入高复杂度状态, 系统中的混沌序列越来越接近随机序列, 再次表明a值约为0.26, 系统处于混沌状态. 同时, 可以发现该混沌系统高复杂度区域较宽, 且与对应的分岔图和Lyapunov指数谱匹配.

$图 12 四维忆阻混沌映射(9)样本熵复杂度\r\nFig. 12. Sample entropy (SampEn) complexity of the four-dimensional memristor chaotic map (9).$

图 12 四维忆阻混沌映射(9)样本熵复杂度

Fig. 12. Sample entropy (SampEn) complexity of the four-dimensional memristor chaotic map (9).

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4.4 多稳态分析

设系统初始状态为x₀ = 0.5, z₀ = 0.5, w₀ = 0.4, N = 10000, 系统参数a = 0.9, 忆阻参数α = 1, β =–0.001, 忆阻初值y₀分别取值为1, 0.5, 0.25, 绘制x_n-y_n平面的吸引子共存图, 如图13(a)所示. 取系统参数a = 0.9, 初始状态设置为x₀ = 0.5, y₀ = 0.5, w₀ = 0.4, N = 10000, 忆阻参数α = 1, β = –0.001. 忆阻初值z₀分别取值为1, 0.5, 0.25, 绘制x_n-y_n平面的吸引子共存图, 如图13(b)所示. 设置系统初始状态为x₀ = 0.5, y₀ = 0.5, z₀ = 0.5, N = 10000, 系统参数a = 0.9, 忆阻参数α = 1, β = –0.001, 忆阻初值w₀分别取值为1, 0.5, 0.25, 绘制x-y平面的吸引子共存图, 如图13(c)所示. 可见设计的离散忆阻系统存在吸引子共存现象. 设系统参数a = 0.9, 图14分别表示随初值y₀, z₀, w₀变化的分岔图. 可见, 随着系统初值的变化, 各分岔图都出现了较多的周期窗口且吸引子大小也在不断变化, 进一步表明系统存在无限多共存吸引子.

$图 13 xn-yn平面上共存吸引子　(a) 初值y0 = 1, 0.5, 0.25; (b) 初值z0 = 1, 0.5, 0.25; (c) 初值w0 = 1, 0.5, 0.25\r\nFig. 13. Coexisting attractors in the xn-yn plane: (a) y0 = 1, 0.5, 0.25; (b) z0 = 1, 0.5, 0.25; (c) w0 = 1, 0.5, 0.25$

图 13 x_n-y_n平面上共存吸引子　(a) 初值y₀ = 1, 0.5, 0.25; (b) 初值z₀ = 1, 0.5, 0.25; (c) 初值w₀ = 1, 0.5, 0.25

Fig. 13. Coexisting attractors in the x_n-y_n plane: (a) y₀ = 1, 0.5, 0.25; (b) z₀ = 1, 0.5, 0.25; (c) w₀ = 1, 0.5, 0.25

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$图 14 随初值变化的分岔图　(a) 初值y0变化, x0 = 0.5, z0 = 0.5, w0 = 0.4; (b)初值z0变化, x0 = 0.5, y0 = 0.5, w0 = 0.4; (c) 初值w0变化, x0 = 0.5, y0 = 0.5, z0 = 0.5\r\nFig. 14. Bifurcation diagrams with the initial values: (a) Initial value y0 changes, x0 = 0.5, z0= 0.5, w0 = 0.4; (b) initial value z0 changes, x0 = 0.5, y0 = 0.5, w0 = 0.4; (c) initial value w0 changes, x0 = 0.5, y0 = 0.5, z0 = 0.5.$

图 14 随初值变化的分岔图　(a) 初值y₀变化, x₀ = 0.5, z₀ = 0.5, w₀ = 0.4; (b)初值z₀变化, x₀ = 0.5, y₀ = 0.5, w₀ = 0.4; (c) 初值w₀变化, x₀ = 0.5, y₀ = 0.5, z₀ = 0.5

Fig. 14. Bifurcation diagrams with the initial values: (a) Initial value y₀ changes, x₀ = 0.5, z₀= 0.5, w₀ = 0.4; (b) initial value z₀ changes, x₀ = 0.5, y₀ = 0.5, w₀ = 0.4; (c) initial value w₀ changes, x₀ = 0.5, y₀ = 0.5, z₀ = 0.5.

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对于不同系统初值条件下, 共存的无限多吸引子的吸引盆可以描绘吸引子与初始条件的关系^[35]. 系统参数分别为a = 0.25, 0.5, 0.9, 初始值为x₀ = 0.5, y₀ = 0.5, N = 10000, 忆阻参数α = 1, β =–0.001, z₀-w₀初值平面上的吸引盆如图15, 图中不同颜色代表不同的吸引子类型. 可见, 总体上, a较大时吸引子类型变化更加复杂. 根据图15中不同a值吸引盆, 分析对应的随初值z₀变化的样本熵复杂度如图16所示, 发现a = 0.9时, 复杂度值总体上明显高于a = 0.25和a = 0.5时的值, 且对初值的变化更加敏感, 进一步验证了图15中的结果.

$图 15 初值平面上吸引盆　(a) a = 0.25; (b) a = 0.5; (c) a = 0.9\r\nFig. 15. Attraction basins in the planes constructed by two different initial conditions: (a) a = 0.25; (b) a = 0.5; (c) a = 0.9.$

图 15 初值平面上吸引盆　(a) a = 0.25; (b) a = 0.5; (c) a = 0.9

Fig. 15. Attraction basins in the planes constructed by two different initial conditions: (a) a = 0.25; (b) a = 0.5; (c) a = 0.9.

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原系统随初值z₀变化的复杂度如图17所示, 对比图16, 再次验证加入忆阻器后系统复杂度变大, 且高复杂度的区域更宽, 表明设计的忆阻混沌系统具有较原系统更高的复杂度.

$图 16 在不同参数a下随初值z0变化的样本熵复杂度　(a) a = 0.25; (b) a = 0.5; (c) a = 0.9\r\nFig. 16. Sample entropy complexity with initial value z0 under different system parameter a: (a) a = 0.25; (b) a = 0.5; (c) a = 0.9.$

图 16 在不同参数a下随初值z₀变化的样本熵复杂度　(a) a = 0.25; (b) a = 0.5; (c) a = 0.9

Fig. 16. Sample entropy complexity with initial value z₀ under different system parameter a: (a) a = 0.25; (b) a = 0.5; (c) a = 0.9.

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$图 17 三维Lorenz混沌映射(6)样本熵复杂度\r\nFig. 17. Sample entropy complexity of the three-dimensional Lorenz chaotic map (6).$

图 17 三维Lorenz混沌映射(6)样本熵复杂度

Fig. 17. Sample entropy complexity of the three-dimensional Lorenz chaotic map (6).

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5. 结　论

本文在离散忆阻数学模型基础上构建了离散忆阻Simulink模型. 在Simulink模型中借助离散时间求和模块实现电荷的迭代, 以体现离散忆阻器独特的记忆特性. 仿真结果发现, 设计的忆阻器满足忆阻定义, 表明本文从仿真的角度实现了忆阻器. 同时, 将此离散忆阻器引入到三维Lorenz混沌映射中, 实现了一种新型的四维忆阻混沌映射, 并建立了该新型四维忆阻混沌映的Simulink模型. 通过分析该离散混沌映射的平衡点及稳定性, 发现其具有无穷多平衡点. 通过分析该混沌映射分岔图、Lyapunov指数谱、系统复杂度、吸引子共存、吸引盆以及对应初值平面复杂度, 研究了其复杂的动力学特性. 数值仿真分析结果表明新型四维忆阻混沌映具有较原系统更丰富的动力学行为以及更高的复杂度. 本文的研究进一步验证了离散忆阻模型的可实现性及其潜在的应用价值, 为离散忆阻在非线性系统中的应用研究奠定了基础.

References

[1]	Chua L O 1971 IEEE Trans. Circuit. Theory 18 507 Google Scholar
[2]	Strukov D B, Snider G S, Stewart D R, Williams R S 2008 Nature 453 80 Google Scholar
[3]	Haj-Ali A, Ben-Hur R, Wald N, Ronen R, Kvatinsky S 2018 IEEE Micro. 38 13 Google Scholar
[4]	Zhang Y, Shen Y, Wang X P, Cao L 2015 IEEE Trans. Circuits Syst. I 62 1402 Google Scholar
[5]	Ho P W C, Almurib H A F, Kumar T N 2016 J. Semicond. 37 104002 Google Scholar
[6]	Teimoori M, Amirsoleimani A, Ahmadi A, Ahmadi M 2018 IEEE Trans. Very Large Scale Integr. (VLSI) Syst. 26 2608 Google Scholar
[7]	Wang C H, Xiong L, Sun J R, Yao W 2019 Nonlinear Dyn. 95 2893 Google Scholar
[8]	Duan S K, Hu X F, Dong Z K, Wang L, Mazumder P 2015 IEEE Trans. Neural. Netw. Learn Syst. 26 1202 Google Scholar
[9]	Marco M D, Forti M, Pancioni L, Innocenti G, Tesi A 2020 IEEE Trans. Syst. Man. Cybern. DOI: 10.1109/TCYB.2020. 2997686
[10]	Pham V T, Jafari S, Vaidyanathan S, Volos C, Wang X 2016 Sci. China:Technol. Sci. 59 358 Google Scholar
[11]	Xu Q, Song Z, Bao H, Chen M, Bao B C 2018 Int. J. Electron. Commun. 96 66 Google Scholar
[12]	Pershin Y V, Di Ventra M 2010 IEEE Trans. Circuits Syst. 57 1857 Google Scholar
[13]	Biolek D, Di Ventra M, Pershin Y V 2013 Radioengineering 22 945
[14]	Gergel-Hackett N, Wright A, Fullerton F A, Joe A 2021 J. Circuits Syst. Comput. 30 2120002 Google Scholar
[15]	段飞腾, 崔宝同 2015 固体电子学研究与进展 35 231 Duan F T, Cui B T 2015 Res. Prog. Solid State Elec. Tron. 35 231
[16]	胡柏林, 王丽丹, 黄艺文, 胡小方, 张宇阳, 段书凯 2011 西南大学学报 33 50 Google Scholar Hu B L, Wang L D, Huang Y W, Hu X F, Zhang Y Y, Duan S K 2011 J. Southwest Univ. 33 50 Google Scholar
[17]	王晓媛, 俞军, 王光义 2018 物理学报 67 098501 Google Scholar Wang X Y, Yu J, Wang G Y 2018 Acta Phys. Sin. 67 098501 Google Scholar
[18]	王春华, 蔺海荣, 孙晶茹, 周玲, 周超, 邓全利 2020 电子与信息学报 42 795 Google Scholar Wang C H, Lin H R, Sun J R, Zhou L, Zhou C, Deng Q L 2020 J. Electr. Inf Technol. 42 795 Google Scholar
[19]	Fitch A L, Yu D S, Iu H H C, Sreeram V 2012 Int. J. Bifurcat. Chaos 22 1250133 Google Scholar
[20]	Bao H, Jiang T, Chu K B, Chen M, Xu Q, Bao B C 2018 Complexity 2018 1 Google Scholar
[21]	Buscarino A, Fortuna L, Frasca M, Gambuzza L V 2012 Chaos 22 023136 Google Scholar
[22]	Li Q D, Zeng H Z, Li J 2015 Nonlinear Dyn. 79 2295 Google Scholar
[23]	Ma J, Chen Z Q, Wang Z L, Zhang Q 2015 Nonlinear Dyn. 81 1275 Google Scholar
[24]	Zhou L, Wang C H, Zhou L L 2017 Bifurcat. Chaos 27 1750027 Google Scholar
[25]	阮静雅, 孙克辉, 牟俊 2016 物理学报 65 190502 Google Scholar Ruan J Y, Sun K H, Mou J 2016 Acta Phys. Sin. 65 190502 Google Scholar
[26]	Bao B C, Jiang T, Xu Q, Chen M, Wu H G, Hu Y H 2016 Nonlinear Dyn. 86 1711 Google Scholar
[27]	Teng L, Iu H H C, Wang X Y, Wang X K 2014 Nonlinear Dyn. 77 231 Google Scholar
[28]	Cang S J, Wu A G, Wang Z G, Xue W, Chen Z Q 2016 Nonlinear Dyn. 83 1987 Google Scholar
[29]	He S B, Sun K H, Peng Y X, Wang L 2020 AIP Adv. 10 015332 Google Scholar
[30]	Bao B C, Liu Z, Xu J P 2010 Chin. Phys. B 19 030510 Google Scholar
[31]	Adhikari S P, Sah M P, Kim H, Chua L O 2013 Trans. Circuits Syst. I, Reg. 60 3008 Google Scholar
[32]	Sprott J C 2003 Chaos and Time-Series Analysis (Oxford: Oxford University Press) pp46–102
[33]	Peng Y X, Sun K H, He S B 2020 Chaos Solitons Fract. 137 109873 Google Scholar
[34]	Chen W T, Zhuang J, Yu W X, Wang Z Z 2009 Med. Eng. Phys. 31 61 Google Scholar
[35]	Yuan F, Wang G Y, Wang X W 2016 Chaos 26 507 Google Scholar

图 1 离散忆阻Simulink模型

Figure 1. Discrete memristor Simulink model.

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图 2 离散忆阻器的电流-电压特性曲线　(a)电流-电压特性曲线; (b) 幅值变化; (c) 频率变化

Figure 2. Current-voltage characteristic curves of discrete memristor: (a) Current-voltage characteristic curve; (b) amplitude change; (c) frequency change.

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图 3 三维Lorenz混沌映射(6)系统框图

Figure 3. Structure diagram of the three-dimensional Lorenz chaotic map (6).

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图 4 加入忆阻器后的三维混沌映射(7)系统框图

Figure 4. Structure diagram of the three-dimensional chaotic map with memristor (7).

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图 5 四维忆阻混沌映射(9)系统框图

Figure 5. Structure diagram of the four-dimensional memristor chaotic map (9).

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图 6 四维忆阻混沌映射(9) Simulink模型

Figure 6. Simulink model of thefour-dimensional memristor chaotic map (9).

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图 7 四维忆阻混沌映射(9)的Simulink仿真吸引子图　(a) β = –0.1, 超混沌状态; (b) β = –0.02, 超混沌状态; (c) β = –0.000002, 混沌状态; (d) a = 0.25, 混沌状态; (e) a = 0.5, 混沌状态; (f) a = 0.9, 混沌状态

Figure 7. Simulink simulation results of the four-dimensional memristor chaotic map (9): (a) β = –0.1, hyperchaotic; (b) β = –0.02, hyperchaotic; (c) β = –0.000002, chaos; (d) a = 0.25, chaos; (e) a = 0.5, chaos; (f) a = 0.9, chaos.

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图 8 四维忆阻混沌映射(9)的Simulink仿真吸引子图对应Lyapunov指数谱　(a) β = –0.1, 超混沌状态; (b) β = –0.02, 超混沌状态; (c) β = –0.000002, 混沌状态; (d) a = 0.25, 混沌状态; (e) a = 0.5, 混沌状态; (f) a = 0.9, 混沌状态

Figure 8. Simulink simulation attractor diagram of the four-dimensional memristor chaotic map (9) corresponds to Lyapunov Exponent spectra: (a) β = –0.1, hyperchaotic; (b) β = –0.02, hyperchaotic; (c) β = –0.000002, chaos; (d) a = 0.25, chaos; (e) a = 0.5, chaos; (f) a = 0.9, chaos.

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图 9 四维忆阻混沌映射(9)分岔图与Lyapunov指数谱　(a) 分岔图; (b) Lyapunov指数谱

Figure 9. Bifurcation and Lyapunov exponent spectra of the four-dimensional memristor chaotic map (9): (a) Bifurcation diagram; (b) Lyapunov exponent (LE) spectra.

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图 10 弱混沌系统动力学分析　(a) x_n序列图; (b) Lyapunov指数谱

Figure 10. Dynamic analysis of weakly chaotic system: (a) x_nsequence diagram; (b) Lyapunov exponent spectra.

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图 11 随初值z₀变化的Lyapunov指数谱　(a)四维忆阻混沌映射(9), a = 0.25; (b) 四维忆阻混沌映射(9), a = 0.5; (c) 三维Lorenz混沌映射(6)

Figure 11. Lyapunov exponent spectra with initial value z₀: (a) Four-dimensional memristor chaotic map (9), a = 0.5; (b) four-dimensional memristor chaotic map (9), a = 0.5; (c) three-dimensional Lorenz chaotic map (6).

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图 12 四维忆阻混沌映射(9)样本熵复杂度

Figure 12. Sample entropy (SampEn) complexity of the four-dimensional memristor chaotic map (9).

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图 13 x_n-y_n平面上共存吸引子　(a) 初值y₀ = 1, 0.5, 0.25; (b) 初值z₀ = 1, 0.5, 0.25; (c) 初值w₀ = 1, 0.5, 0.25

Figure 13. Coexisting attractors in the x_n-y_n plane: (a) y₀ = 1, 0.5, 0.25; (b) z₀ = 1, 0.5, 0.25; (c) w₀ = 1, 0.5, 0.25

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图 14 随初值变化的分岔图　(a) 初值y₀变化, x₀ = 0.5, z₀ = 0.5, w₀ = 0.4; (b)初值z₀变化, x₀ = 0.5, y₀ = 0.5, w₀ = 0.4; (c) 初值w₀变化, x₀ = 0.5, y₀ = 0.5, z₀ = 0.5

Figure 14. Bifurcation diagrams with the initial values: (a) Initial value y₀ changes, x₀ = 0.5, z₀= 0.5, w₀ = 0.4; (b) initial value z₀ changes, x₀ = 0.5, y₀ = 0.5, w₀ = 0.4; (c) initial value w₀ changes, x₀ = 0.5, y₀ = 0.5, z₀ = 0.5.

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图 15 初值平面上吸引盆　(a) a = 0.25; (b) a = 0.5; (c) a = 0.9

Figure 15. Attraction basins in the planes constructed by two different initial conditions: (a) a = 0.25; (b) a = 0.5; (c) a = 0.9.

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图 16 在不同参数a下随初值z₀变化的样本熵复杂度　(a) a = 0.25; (b) a = 0.5; (c) a = 0.9

Figure 16. Sample entropy complexity with initial value z₀ under different system parameter a: (a) a = 0.25; (b) a = 0.5; (c) a = 0.9.

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图 17 三维Lorenz混沌映射(6)样本熵复杂度

Figure 17. Sample entropy complexity of the three-dimensional Lorenz chaotic map (6).

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[1]	Chua L O 1971 IEEE Trans. Circuit. Theory 18 507 Google Scholar
[2]	Strukov D B, Snider G S, Stewart D R, Williams R S 2008 Nature 453 80 Google Scholar
[3]	Haj-Ali A, Ben-Hur R, Wald N, Ronen R, Kvatinsky S 2018 IEEE Micro. 38 13 Google Scholar
[4]	Zhang Y, Shen Y, Wang X P, Cao L 2015 IEEE Trans. Circuits Syst. I 62 1402 Google Scholar
[5]	Ho P W C, Almurib H A F, Kumar T N 2016 J. Semicond. 37 104002 Google Scholar
[6]	Teimoori M, Amirsoleimani A, Ahmadi A, Ahmadi M 2018 IEEE Trans. Very Large Scale Integr. (VLSI) Syst. 26 2608 Google Scholar
[7]	Wang C H, Xiong L, Sun J R, Yao W 2019 Nonlinear Dyn. 95 2893 Google Scholar
[8]	Duan S K, Hu X F, Dong Z K, Wang L, Mazumder P 2015 IEEE Trans. Neural. Netw. Learn Syst. 26 1202 Google Scholar
[9]	Marco M D, Forti M, Pancioni L, Innocenti G, Tesi A 2020 IEEE Trans. Syst. Man. Cybern. DOI: 10.1109/TCYB.2020. 2997686
[10]	Pham V T, Jafari S, Vaidyanathan S, Volos C, Wang X 2016 Sci. China:Technol. Sci. 59 358 Google Scholar
[11]	Xu Q, Song Z, Bao H, Chen M, Bao B C 2018 Int. J. Electron. Commun. 96 66 Google Scholar
[12]	Pershin Y V, Di Ventra M 2010 IEEE Trans. Circuits Syst. 57 1857 Google Scholar
[13]	Biolek D, Di Ventra M, Pershin Y V 2013 Radioengineering 22 945
[14]	Gergel-Hackett N, Wright A, Fullerton F A, Joe A 2021 J. Circuits Syst. Comput. 30 2120002 Google Scholar
[15]	段飞腾, 崔宝同 2015 固体电子学研究与进展 35 231 Duan F T, Cui B T 2015 Res. Prog. Solid State Elec. Tron. 35 231
[16]	胡柏林, 王丽丹, 黄艺文, 胡小方, 张宇阳, 段书凯 2011 西南大学学报 33 50 Google Scholar Hu B L, Wang L D, Huang Y W, Hu X F, Zhang Y Y, Duan S K 2011 J. Southwest Univ. 33 50 Google Scholar
[17]	王晓媛, 俞军, 王光义 2018 物理学报 67 098501 Google Scholar Wang X Y, Yu J, Wang G Y 2018 Acta Phys. Sin. 67 098501 Google Scholar
[18]	王春华, 蔺海荣, 孙晶茹, 周玲, 周超, 邓全利 2020 电子与信息学报 42 795 Google Scholar Wang C H, Lin H R, Sun J R, Zhou L, Zhou C, Deng Q L 2020 J. Electr. Inf Technol. 42 795 Google Scholar
[19]	Fitch A L, Yu D S, Iu H H C, Sreeram V 2012 Int. J. Bifurcat. Chaos 22 1250133 Google Scholar
[20]	Bao H, Jiang T, Chu K B, Chen M, Xu Q, Bao B C 2018 Complexity 2018 1 Google Scholar
[21]	Buscarino A, Fortuna L, Frasca M, Gambuzza L V 2012 Chaos 22 023136 Google Scholar
[22]	Li Q D, Zeng H Z, Li J 2015 Nonlinear Dyn. 79 2295 Google Scholar
[23]	Ma J, Chen Z Q, Wang Z L, Zhang Q 2015 Nonlinear Dyn. 81 1275 Google Scholar
[24]	Zhou L, Wang C H, Zhou L L 2017 Bifurcat. Chaos 27 1750027 Google Scholar
[25]	阮静雅, 孙克辉, 牟俊 2016 物理学报 65 190502 Google Scholar Ruan J Y, Sun K H, Mou J 2016 Acta Phys. Sin. 65 190502 Google Scholar
[26]	Bao B C, Jiang T, Xu Q, Chen M, Wu H G, Hu Y H 2016 Nonlinear Dyn. 86 1711 Google Scholar
[27]	Teng L, Iu H H C, Wang X Y, Wang X K 2014 Nonlinear Dyn. 77 231 Google Scholar
[28]	Cang S J, Wu A G, Wang Z G, Xue W, Chen Z Q 2016 Nonlinear Dyn. 83 1987 Google Scholar
[29]	He S B, Sun K H, Peng Y X, Wang L 2020 AIP Adv. 10 015332 Google Scholar
[30]	Bao B C, Liu Z, Xu J P 2010 Chin. Phys. B 19 030510 Google Scholar
[31]	Adhikari S P, Sah M P, Kim H, Chua L O 2013 Trans. Circuits Syst. I, Reg. 60 3008 Google Scholar
[32]	Sprott J C 2003 Chaos and Time-Series Analysis (Oxford: Oxford University Press) pp46–102
[33]	Peng Y X, Sun K H, He S B 2020 Chaos Solitons Fract. 137 109873 Google Scholar
[34]	Chen W T, Zhuang J, Yu W X, Wang Z Z 2009 Med. Eng. Phys. 31 61 Google Scholar
[35]	Yuan F, Wang G Y, Wang X W 2016 Chaos 26 507 Google Scholar

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