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忆阻器因具有非线性和记忆性[1-3],在人工神经网络[4-5]、保密通信[6-7]、计算机内存[8-9]、生物仿真[10-11]等领域有着广泛的应用前景,得到了各领域学者的关注,并取得了丰富的研究成果。特别在电路中,忆阻器作为非线性元件引入电路之后,电路的混沌振荡将变得更加容易实现,其动力学行为更为复杂,所以忆阻混沌电路及其动力学特性研究逐渐成为了学术研究的新热点[12],近年来获得了系列研究成果[13-18]。
隐藏吸引子是近几年新提出的一类吸引子。为了与传统的含有一个或者多个不稳定焦鞍点的混沌系统进行分类,文献[19]将传统的混沌系统定义为自激系统,以区别近年来发现的没有平衡点激发也能产生吸引子的隐藏混沌系统。隐藏吸引子存在于不与任何平衡点的小邻域相交的吸引盆中。为了更好地把握隐藏系统的潜在功能,对于存在隐藏吸引子的系统研究也成为了学术研究的新热点。隐藏吸引子最早是在经典Chua电路中发现的[20],而后学者们利用基于二极管对的非线性滤波电路[21]或者基于二极管桥的广义忆阻器[22]替换Chua电路中的蔡氏二极管,同样发现了这些新型Chua电路具有复杂的隐藏动力学行为。进一步,将实际物理电路中的隐藏动力学行为特点延伸至混沌系统中,通过改造Chua[23]、Lorenz[24-25]、Rabinorich[26]等系统以及忆阻自激振荡系统[27]也产生了具有多稳定性的隐藏混沌或超混沌吸引子。这些电路或系统中除了具有隐藏动力学行为,还存在吸引子的共存现象[27],即不同的初始条件会导致系统轨线分别演化到不同的吸引子上,从而产生共存吸引子即多稳定性[28]。多稳定性现象是非线性动力学系统中普遍存在的一种物理现象,而隐藏吸引子也是一种特殊的多稳定性现象[29]。另外,在文献[30]构建的简单四维系统中,隐藏吸引子有着分别从瞬态超混沌和瞬态周期过渡到稳态混沌的复杂瞬态行为。
已有文献通过在三阶混沌电路或三维混沌系统中引入忆阻[31],得到了一类忆阻混沌或超混沌电路和系统[13-18],但关于忆阻电路和系统产生隐藏超混沌吸引子并具有共存多吸引子的文献报道相对较少[27]。本文通过在经典Lü系统[32]的第二方程中添加一个线性反馈项和一个常数项,并在第一方程中引入一个忆阻,提出了一种新颖的、无任何平衡点的、基于忆阻的改进型Lü系统。改进后的忆阻Lü系统可以产生周期、准周期、混沌和超混沌等复杂隐藏动力学行为,且能在不同的初始条件下产生不同的周期极限环或弱混沌与周期极限环的隐藏多吸引子共存现象。
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从混沌系统的电路实现角度来看,在积分电路的输入项中,电阻表示线性增益,忆阻表示非线性增益,而直流分量,则代表常数项[33]。一种常用的忆阻模型可以描述为[33]:
$$\left\{ \begin{array}{l} i = (\alpha + \beta {\varphi ^2})u\\ \dot \varphi = u \end{array} \right.$$ (1) 式中,u是忆阻的输入电压变量;i是忆阻的输出电流变量;φ是忆阻的内部磁通变量;α+βφ2是描述忆导变化的非线性函数。
在经典Lü系统的第二方程中添加线性反馈项dx和常数项e,且在第一方程中引入一个式(1)描述的忆阻,即可构建出一种新颖的忆阻超混沌Lü系统,其数学模型表示为:
$$ \left\{ \begin{array}{l} \dot x = a(y - x) + g(\alpha + \beta {w^2})y\\ \dot y = by - xz - dx + e\\ \dot z = xy - cz\\ \dot w = y \end{array} \right. $$ (2) 式中,x、y、z和w为4个状态变量;a、b和c为经典Lü系统3个控制参数。为了研究该忆阻Lü系统的隐藏动力学特性,固定系统参数a=36,b=20,c=3,d=5,e=0.1,α=4,β=0.18,选择增益g为系统唯一的可调节参数。
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通过忆阻Lü系统的方程式(2)可以得到:
$$ \begin{array}{c} \nabla V = \frac{{\partial \dot x}}{{\partial x}} + \frac{{\partial \dot y}}{{\partial y}} + \frac{{\partial \dot z}}{{\partial z}} + \frac{{\partial \dot w}}{{\partial w}} = \\ - a + b - c = - 19 < 0 \end{array} $$ (3) 由式(3)可知,$\nabla V < 0$,说明系统是耗散的,同时也说明Lü系统的改进和忆阻的引入没有改变原经典Lü系统的耗散度。
令$ \dot x = \dot y = \dot z = \dot w = 0 $,显然式(2)不存在任何数学解,即所提出的忆阻Lü系统不存在任何平衡点,完全不同于具有3个平衡点的经典Lü系统。根据隐藏吸引子的定义[19-29],无平衡点的忆阻Lü系统所产生的周期极限环、准周期极限环、混沌吸引子和超混沌吸引子均是隐藏的。
系统(2)的雅克比矩阵为:
$$ \mathit{\boldsymbol{J}} = \left[ {\begin{array}{*{20}{c}} { - a}&{a + g(\alpha + \beta {w^2})}&0&{2g\beta yw}\\ { - d - z}&b&{ - x}&0\\ y&x&{ - c}&0\\ 0&1&0&0 \end{array}} \right] $$ (4) 相应的特征方程为:
$$ P(\lambda ) = \det (\mathit{\pmb{1}} \lambda - \mathit{\boldsymbol{J}}) = 0 $$ (5) 基于式(4),利用Wolf算法可计算得到忆阻Lü系统的李雅普诺夫指数。
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以忆阻的增益g作为调节参数,设置初始条件为(1, 0, 1, 0),随忆阻的增益g在区域[6, 16变化的李雅普诺夫指数谱如图 1a所示,与之对应的状态变量z的分岔图如图 1b所示。
当忆阻增益g=6时,LE1 =0,其余3个李雅普诺夫指数LE2、LE3、LE4 < 0,忆阻Lü系统处于周期态。在6 < g < 7.69内,忆阻Lü系统处于周期态或弱混沌态。当g =7.69且继续增大时,LE1、LE2 =0,LE3、LE4 < 0,忆阻Lü系统从周期态过渡到准周期态。随着忆阻增益的继续增大,在8.3 < g < 9.02内,LE1 > 0、LE2=0和LE3、LE4 < 0,忆阻Lü系统处于相对稳定的混沌参数区间。当忆阻增益继续增大时,LE1、LE2开始出现抖动,直到g =10.09时结束,忆阻Lü系统开始由混沌状态转移到超混沌状态。在10.09 < g < 14.55内,LE1 > 0、LE2 > 0、LE3=0和LE4 < 0,忆阻Lü系统处于超混沌状态。自g=14.55且逐步增大至15时,忆阻Lü系统交替处于超混沌状态或混沌状态。
在g > 15之后,LE1=0,其余3个李雅普诺夫指数LE2、LE3、LE4 < 0,忆阻Lü系统迅速地收敛到一个点吸引子。在各参数区间选取典型的参数值,忆阻Lü系统在x-z平面上典型的相轨图如图 2所示。
此外,图 3给出了典型超混沌吸引子在4个平面上的相轨图。
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采用电阻、电容、运算放大器和模拟乘法器等分立元器件构建的模拟电子电路实现本文提出的忆阻超混沌Lü系统。
在电路制作中,选用型号OP07CP作为运算放大器,型号AD633JN作为模拟乘积器,两者的工作电压动态范围分别为±16 V和±10 V。考虑到实际电路的输出信号幅度不应大于电路器件的饱和电压,也不应过小而导致信号失真。根据前述数值仿真结果,可知超混沌吸引子和共存多吸引子的x、y、z、w状态变量的动态范围分别在±15、±5、0~12、±4 V,因此,需要对系统(2)的状态变量进行线性缩放如下:
$$ (x, y, z, w, e) \to (2x, 2y, 2z, w, 0.5e) $$ (6) 选择时间尺度因子为k=1/(RC)。系统(2)经过时间尺度因子转换并进行状态变量线性缩放后的电路方程为:
$$ \left\{ \begin{array}{l} {{\dot u}_x} = k[a({u_y} - {u_x}) + g(\alpha + \beta u_w^2){u_y}]\\ {{\dot u}_y} = k(b{u_y} - 2{u_x}{u_z} - d{u_x} + 0.5e)\\ {{\dot u}_z} = k(2{u_x}{u_y} - c{u_z})\\ {{\dot u}_w} = k(2{u_y}) \end{array} \right. $$ (7) 式中,ux、uy、uz和uw表示电路中4个电容两端的电压,分别对应于x、y、z和w状态变量。
基于式(7)的电路方程,可设计并实现的电路原理图如图 8所示。选择时间尺度因子k中的R=36 kΩ和C=100 nF。与系统(2)比较,可得图 8中电容值均为C1=C2=C3=C4=C =100 nF,电阻值分别为R1=R2=R/a = 1 kΩ,R3=R/c=12 kΩ,R4=R5=R/20=1.8 kΩ,R6= R/b=1.8 kΩ,R7=R/d=7.2 kΩ,R8=R9=R=36 kΩ,R10=R/2=18 kΩ和R13=2R=72 kΩ。此外,电阻R11和R12是联动可调的,其参数值分别为:R11=R/gα,R12=R/100gβ。实现图 8的硬件实验电路图如图 9所示。
采用Agilent DSO7054B数字示波器捕捉图 9硬件实验电路测试结果,其中4个节点电压测量点被标记为ux,uy,uz,uw。在典型电路参数下,当g=13.4时,忆阻超混沌Lü系统硬件电路输出的隐藏超混沌吸引子在ux-uz、ux-uy、ux-uw和uw-uz平面上的相轨图分别如图 10a~图 10d所示。进一步地,当g=5.96时,联动可调电阻的参数值分别固定为:R11=1 510 Ω,R12=335.5 Ω,从数字示波器上可捕捉到的3个共存隐藏极限环在ux-uz和ux-uw平面上的相轨图分别如图 11a和图 11b所示。
当g=6.936时,联动可调电阻的参数值分别固定为:R11=1 297.5 Ω,R12=288.3 Ω时,从数字示波器上可捕捉到的共存隐藏弱混沌吸引子与隐藏周期极限环在ux-uz和ux-uw平面上的相轨图分别如图 12a和图 12b所示。这里,不同颜色标注的实验波形是通过不断接通和断开图 9硬件电路的供电电源随机捕捉到的。分别与图 3、6和7的数值仿真结果相比较,可见电路实验结果与数值仿真结果完全一致,从而由硬件电路验证了忆阻超混沌Lü系统的隐藏超混沌吸引子和共存多吸引子的存在性。
Hidden Dynamical Characteristics in Memristor-Based Hyperchaotic Lü System
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摘要: 通过改进经典Lü系统并引入忆阻元件,提出了一种新颖的基于忆阻的改进型Lü系统。该忆阻系统的最大特征是不存在任何平衡点,因此形成的动力学行为都是隐藏的。采用理论分析、李雅普诺夫指数和分岔图等非线性系统分析,研究了该忆阻系统随忆阻增益变化的周期、准周期、混沌和超混沌等复杂的隐藏动力学行为。此外,在初始条件不同时,该忆阻系统存在3个不同极限环以及混沌吸引子和周期极限环的共存多吸引子现象。制作硬件电路,验证了理论分析和数值仿真结果,表明了该忆阻超混沌Lü系统有着十分丰富而复杂的隐藏动力学特性。Abstract: By improving the classical Lü system and introducing a generalized memristor, a novel memristor-based modified Lü system is proposed. The most important feature of this memristive system is that there does not exist any equilibrium point, thereby leading to that the forming dynamical behaviors are all hidden. By utilizing theoretical analyses and nonlinear system analysis methods of Lyapunov exponent spectrum and bifurcation diagram, the complex hidden dynamical behaviors, such as period, quasi-period, chaos, hyperchaos, and so on, with the variation of memristor gain for the memristive system are studied. In addition, when different initial conditions are used, the memristive system exhibits coexisting multiple attractors' phenomena of three different limit cycles as well as chaotic attractor and limit cycle. The hardware circuit is made and the experimental results verify the theoretical analysis and numerical simulations, and demonstrate that the proposed memristive Lü system has very abundant and complex hidden dynamical characteristics.
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