Loading [MathJax]/jax/output/SVG/jax.js
Special Issues

Synchronization for a class of complex-valued memristor-based competitive neural networks(CMCNNs) with different time scales

  • In this paper, the synchronization problem of complex-valued memristive competitive neural networks(CMCNNs) with different time scales is investigated. Based on differential inclusions and inequality techniques, some novel sufficient conditions are derived to ensure synchronization of the drive-response systems by designing a proper controller. Finally, a numerical example is provided to illustrate the usefulness and feasibility of our results.

    Citation: Yong Zhao, Shanshan Ren. Synchronization for a class of complex-valued memristor-based competitive neural networks(CMCNNs) with different time scales[J]. Electronic Research Archive, 2021, 29(5): 3323-3340. doi: 10.3934/era.2021041

    Related Papers:

    [1] Yong Zhao, Shanshan Ren . Synchronization for a class of complex-valued memristor-based competitive neural networks(CMCNNs) with different time scales. Electronic Research Archive, 2021, 29(5): 3323-3340. doi: 10.3934/era.2021041
    [2] Jun Guo, Yanchao Shi, Shengye Wang . Synchronization analysis of delayed quaternion-valued memristor-based neural networks by a direct analytical approach. Electronic Research Archive, 2024, 32(5): 3377-3395. doi: 10.3934/era.2024156
    [3] Chao Yang, Juntao Wu, Zhengyang Qiao . An improved fixed-time stabilization problem of delayed coupled memristor-based neural networks with pinning control and indefinite derivative approach. Electronic Research Archive, 2023, 31(5): 2428-2446. doi: 10.3934/era.2023123
    [4] Jun Guo, Yanchao Shi, Weihua Luo, Yanzhao Cheng, Shengye Wang . Exponential projective synchronization analysis for quaternion-valued memristor-based neural networks with time delays. Electronic Research Archive, 2023, 31(9): 5609-5631. doi: 10.3934/era.2023285
    [5] Xiangwen Yin . A review of dynamics analysis of neural networks and applications in creation psychology. Electronic Research Archive, 2023, 31(5): 2595-2625. doi: 10.3934/era.2023132
    [6] Shuang Liu, Tianwei Xu, Qingyun Wang . Effect analysis of pinning and impulsive selection for finite-time synchronization of delayed complex-valued neural networks. Electronic Research Archive, 2025, 33(3): 1792-1811. doi: 10.3934/era.2025081
    [7] Huijun Xiong, Chao Yang, Wenhao Li . Fixed-time synchronization problem of coupled delayed discontinuous neural networks via indefinite derivative method. Electronic Research Archive, 2023, 31(3): 1625-1640. doi: 10.3934/era.2023084
    [8] Yu-Jing Shi, Yan Ma . Finite/fixed-time synchronization for complex networks via quantized adaptive control. Electronic Research Archive, 2021, 29(2): 2047-2061. doi: 10.3934/era.2020104
    [9] Yilin Li, Jianwen Feng, Jingyi Wang . Mean square synchronization for stochastic delayed neural networks via pinning impulsive control. Electronic Research Archive, 2022, 30(9): 3172-3192. doi: 10.3934/era.2022161
    [10] Wanshun Zhao, Kelin Li, Yanchao Shi . Exponential synchronization of neural networks with mixed delays under impulsive control. Electronic Research Archive, 2024, 32(9): 5287-5305. doi: 10.3934/era.2024244
  • In this paper, the synchronization problem of complex-valued memristive competitive neural networks(CMCNNs) with different time scales is investigated. Based on differential inclusions and inequality techniques, some novel sufficient conditions are derived to ensure synchronization of the drive-response systems by designing a proper controller. Finally, a numerical example is provided to illustrate the usefulness and feasibility of our results.



    Since it was first put forward by Chua[3], memristor has attracted increasing attention in recent years[19,16]. As the fourth fundamental circuit element except resistor, inductor, and capacitor in circuity, the prototype of practical memristor device was successfully developed by Hewlett-Packard Labs in 2008[17]. Moreover, memristor has many good properties, such as low power consumption, high density and good scalability. More importantly, the connection weight of the memristor is not fixed, it depends on the voltage applied in the corresponding state. Therefore, many researchers hope to use memristor as an artificial synapse to build a device similar to brain function. In view of its good charzcteristic, now memristor has been widely used to model memristor-based neural networks(MNNs)[24,6,15].

    As a collective dynamical behavior, synchronization extensively exists in life, society and neural systems. Synchronization plays important role in the activity of the brain and nervous system[14]. So synchronization of memristor-based neural networks is significant[22,1]. Liao et al.[22] discussed effects of initial conditions of memristor synapses on the synchronization of the coupled memristor neural circuits. Theoretical analysis and simulations show that the memristor synapse has played an important role in the synchronization of the coupled FitzHugh-Nagumo neural circuits. Ascoli et al.[1] found that the history of the memristor plays a critical role in the synchronous oscillations in the network and enhance synchronizaiton.

    In recent years, neurals networks have attracted a lot of researchers in different research areas[20,21,4,27,7,8]. It is worth mentioning that MCNNs with different time scales, which are extensions of conventional neural networks. It is a kind of unsupervised learning neural networks, which refers to the whole interconnection between input and output of the single layer neural networks[25]. MCNNs contain two types of state variables, including the aspects of long-term memory (LTM)and short-term memory(STM)[10], corresponding to the fast changes of the neural network states and the slow changes of the synapses by external stimuli, respectively. And up to now, various dynamical behaviors for competitive neural networks have been investigated[6,15,12,11,13] and have been successfully applied to control theory, signal processing, pattern recognition and optimization design and so on[25,11].

    We noticed that the results mentioned above have been achieved within the real number domain. However, as we all know, sometimes, it is unreasonable to deal with some problems only in the real number domain, such as symmetry detection, XOR problems, electromagnetic wave imaging and so on[23,18], which are more convenient and reasonable to deal with complex-value system. Thus, it is meaningful to study MCNNs as the generalization and extension for real-valued systems. In recent years, people have made a lot of achievements[9,26,2] in the field of complex number. Liu et al.[9] discussed global anti-synchronization of CMNNs with time delays by constructing an appropriate Lyapunov function. The proposed results of this paper are less conservative than existing literatures due to the characteristics of complex-valued memristive neural networks(CMNNs). Zhu et al.[26] investigated the synchronization of CMNNs with time delay by using the theory of the pinning control method, which control partial neurons instead of all neurons, and achieved new conclusions and progress. However, to the best of our knowledge, few scholars consider the synchronization problem of CMCNNs.

    Based on the above analysis, this paper aims to investigate the synchronization problem of CMCNNs. By designing a proper controller, we achieve asymptotically stable of the error system such that achieve synchronization of the drive-response system. The contributions of this article can be summarized as follows.

    (1) Different from the neural networks discussed earlier, the systems considered in this article are discussed based on complex-valued, which are an extension of the general real-valued networks.

    (2) Different from asymptotic or exponential synchronization, it is shown that both the STM and LTM play a regulatory role in the systems so that the systems can show better performance.

    (3) In this paper, the sufficiency of the synchronization of CMCNNs is derived by constructing a proper controller and use some inequality techniques.

    The rest of the paper is organized as follows. In Section 2, some useful assumptions, definitions and lemmas needed in the paper are presented. In Section 3, a controller is designed to investigate the synchronization of CMCNNs by constructing a proper Lyapunov functional. In Section 4, a numerical example is given to illustrate the effectiveness of the obtained results. Finally, some conclusions are drawn.

    In this paper, the solutions of all the systems considered below are intended in Filippov's sense[5]. Rn and Cn denote the ndimensional Euclidean space and complex space, respectively. co[a,b] represents closure of the convex hull of Rn generated by real numbers a and b. Now, the model of CMCNNs with different time scales to be introduced as follows:

    {STM:ε˙zk(t)=zk(t)+nl=1akl(zk(t))fl(zl(t))                      +nl=1bkl(zk(t))fl(zl(tτ(t)))+Hkmk(t),l=1,2,,n,LTM:˙mk(t)=mk(t)+fk(zk(t)), k=1,2,,n, (1)

    where z(t)=(z1(t),z2(t),...,zn(t))TCn. Let zk(t)=xk(t)+iyk(t), xk(t) and yk(t) are real part and imaginary part of zk(t), i is used to denote imaginary unit and satisfies i=1. akl(zk(t)) and bkl(zk(t)) are complex-valued memristive connection weights; f(z(.))=(f1(z1(.)),f2(z2(.)),...,fn(zn(.)))T stands for the complex-valued activation function with f(0)=0; τ(t) is used to express the variable time delay, which satisfies 0τ(t)τ,˙τ(t)γ<1, where τ and γ are positive constants.

    The initial conditions of system (1) are assumed to be

    z(s)=ϕ(s),τs0

    where ϕ(s)=(ϕ1(s),ϕ2(s),...,ϕn(s))TC([τ,0],Cn). The memristive connection weights of (1) satisfy the following conditions:

    akl(zk(t))={ˆakl, |zk(t)|>Tk,ˇakl, |zk(t)|Tk,bkl(zk(t))={ˆbkl, |zk(t)|>Tk,ˇbkl, |zk(t)|Tk,

    for t>0, where the switching jumps Tk>0,ˆakl,ˇakl,ˆbkl and ˇbkl are the complex-valued connected weights.

    Remark 1. From above analysis, we know that the connection weights akl(zk(t)) and bkl(zk(t)) in system (1) are complex-valued and discontinuous due to the characteristics of state-dependent switched nonlinear dynamical system. Therefore, we will study the characteristics of solutions for differential equations with discontinuous right-hand sides by using the theory of Filippov in this paper.

    In this paper, consider system (1) as drive system and corresponding response system can be described as follows;

    {STM:ε˙˜zk(t)=˜zk(t)+nl=1akl(˜zk(t))fl(˜zl(t))                      +nl=1bkl(˜zk(t))fl(˜zl(tτ(t)))+Hk˜mk(t)                      +uk(t),l=1,2,,n,LTM:˙˜mk(t)=˜mk(t)+fk(˜zk(t)), k=1,2,,n, (2)

    with initial condition ˜z(s)=φ(s)C([τ,0],Cn). uk(t) is the control input to be designed.

    Before proceeding further, uk(t),akl(z(t)),bkl(z(t)) and activation function fk(zk(t)) can be separated into real and imaginary parts as

    uk(t)=uRk(t)+iuIk(t);
    akl(z(t))=aRkl(z(t))+iaIkl(z(t));
    bkl(z(t))=bRkl(z(t)+ibIkl(z(t));
    fk(zk(t))=fRk(xk(t),yk(t))+ifIk(xk(t),yk(t)),

    where fRk(xk(t),yk(t)),fIk(xk(t),yk(t)):R2R, and they are odd functions. And the following assumption need to be introduced.

    Assumption 1. Suppose that fRk(xk(t),yk(t)) and fIk(xk(t),yk(t)) satisfy the following conditions.

    (1)The partial derivatives of fRk(.,.) and fIk(.,.) with respect to x,y:fRk/x, fRk/y,fIk/x and fIk/y are exist and continuous.

    (2) The partial derivatives fRk/x,fRk/y,fIk/x and fIk/y are bounded, namely, there exist positive constants λRRk,λRIk,λIRk, and λIIk such that

    |fRk/x|λRRk,|fRk/y|λRIk,
    |fIk/x|λIRk,|fIk/y|λIIk.

    Then we have

    |fRk(˜xk(t),˜yk(t))fRk(xk(t),yk(t))|λRRk|˜x(t)x(t)|+λRIk|˜y(t)y(t)|,
    |fIk(˜xk(t),˜yk(t))fIk(xk(t),yk(t))|λIRk|˜x(t)x(t)|+λIIk|˜y(t)y(t)|.

    Under Assumption 1, separating system (1) into real and imaginary parts as follows

    {STM:ε˙xRk(t)=xRk(t)+nl=1aRkl(xk(t))fRl(xl(t),yl(t))    nl=1aIkl(xk(t))fIl(xl(t),yl(t))+nl=1bRkl(xk(t))fRl(xl(tτ(t)),yl(tτ(t)))                      nl=1bIkl(xk(t))fIl(xl(tτ(t)),yl(tτ(t)))+HRkmRk(t)                      HIkmIk(t),l=1,2,,n,LTM:˙mRk(t)=mRk(t)+fRk(xk(t),yk(t)),k=1,2,,n, (3)
    {STM:ε˙yIk(t)=yIk(t)+nl=1aRkl(yk(t))fIl(xl(t),yl(t))    +nl=1aIkl(yk(t))fRl(xl(t),yl(t))+nl=1bRkl(yk(t))fIl(xl(tτ(t)),yl(tτ(t)))                      +nl=1bIkl(yk(t))fRl(xl(tτ(t)),yl(tτ(t)))+HRkmIk(t)                      +HIkmRk(t),l=1,2,,n,LTM:˙mIk(t)=mIk(t)+fIk(xk(t),yk(t)),k=1,2,,n, (4)

    with initial conditions x(s)=ϕR(s)C([τ,0],Rn) and y(s)=ϕI(s)C([τ,0],Rn), respectively.

    The memristive connection weights of (3) and (4) satisfy the following conditions:

    aRkl(xk(t))={ˆaRkl, |xk(t)|>Tk,ˇaRkl, |xk(t)|Tk,aIkl(yk(t))={ˆaIkl, |yk(t)|>Tk,ˇaIkl, |yk(t)|Tk,
    bRkl(xk(t))={ˆbRkl, |xk(t)|>Tk,ˇbRkl, |xk(t)|Tk,bIkl(yk(t))={ˆbIkl, |yk(t)|>Tk,ˇbIkl, |yk(t)|Tk,

    where the switching jumps Tk>0, connections weights ˆaRkl,ˇaRkl,ˆaIkl,ˇaIkl,ˆbRkl,ˇbRkl,ˆbIkl and ˇbIkl are constants, k,l=1,...,n.

    Remark 2. Here, we transform a complex-valued system into two equivalent real-valued system. Similarly, the inequalities satisfied by activation function in Assumption 1 are equivalent to the Lipschitz continuity condition in the complex domain. The purpose of this process is to facilitate our discussion using the relevant theorems in the field of real numbers.

    Because the memristor-based connection weights in (3) and (4) are discontinuous, then by differential inclusions feature for system with the discontinuous right-hand sides, (3) and (4) will be written as follows:

    {STM:ε˙xRk(t)xRk(t)+nl=1co[aRkl,a+Rkl]fRl(xl(t),yl(t))nl=1co[aIkl,a+Ikl]fIl(xl(t),yl(t))                      +nl=1co[bRkl,b+Rkl]fRl(xl(tτ(t)),yl(tτ(t)))                      nl=1co[bIkl,b+Ikl]fIl(xl(tτ(t)),yl(tτ(t)))+HRkmRk(t)                      HIkmIk(t)),l=1,2,,n,LTM:˙mRk(t)=mRk(t)+fRk(xk(t),yk(t)),k=1,2,,n, (5)
    {STM:ε˙yIk(t)yIk(t)+nl=1co[aRkl,a+Rkl]fIl(xl(t),yl(t))+nl=1co[aIkl,a+Ikl]fRl(xl(t),yl(t))                      +nl=1co[bRkl,b+Rkl]fIl(xl(tτ(t)),yl(tτ(t)))                      +nl=1co[bIkl,b+Ikl]fRl(xl(tτ(t)),yl(tτ(t)))+HRkmIk(t)                      +HIkmRk(t),l=1,2,,n,LTM:˙mIk(t)=mIk(t)+fIk(xk(t),yk(t)),k=1,2,,n, (6)

    where

    a+Rkl=max{ˆaRkl,ˇaRkl},aRkl=min{ˆaRkl,ˇaRkl},b+Rkl=max{ˆbRkl,ˇbRkl},bRkl=min{ˆbRkl,ˇbRkl},
    a+Ikl=max{ˆaRkl,ˇaRkl},aIkl=min{ˆaRkl,ˇaRkl},b+Ikl=max{ˆbRkl,ˇbRkl},bIkl=min{ˆbRkl,ˇbRkl}.

    Or equivalently, for k,l=1,...,n, then there exists ˊaRklco[aRkl,a+Rkl],ˊaIklco[aIkl,a+Ikl],ˊbRklco[bRkl,b+Rkl],ˊbIklco[bIkl,b+Ikl] such that

    {STM:ε˙xRk(t)=xRk(t)+nl=1ˊaRklfRl(xl(t),yl(t))                      nl=1ˊaIklfIl(xl(t),yl(t))+nl=1ˊbRklfRl(xl(tτ(t)),yl(tτ(t)))                      nl=1ˊbIklfIl(xl(tτ(t)),yl(tτ(t)))+HRkmRk(t)                      HIkmIk(t),l=1,2,,n,LTM:˙mRk(t)=mRk(t)+fRk(xk(t),yk(t)),k=1,2,,n, (7)
    {STM:ε˙yIk(t)=yIk(t)+nl=1ˊaRklfIl(xl(t),yl(t))                      +nl=1ˊaIklfRl(xl(t),yl(t))+nl=1ˊbRklfIl(xl(tτ(t)),yl(tτ(t)))                      +nl=1ˊbIklfRl(xl(tτ(t)),yl(tτ(t)))+HRkmIk(t)                      +HIkmRk(t),l=1,2,,n,LTM:˙mIk(t)=mIk(t)+fIk(xk(t),yk(t)),k=1,2,,n. (8)

    Similar to the system (1), separating system (2) into real and imaginary parts as follows

    {STM:ε˙˜xRk(t)=˜xRk(t)+nl=1aRkl(˜xk(t))fRl(˜xl(t),˜yl(t))    nl=1aIkl(˜xk(t))fIl(˜xl(t),˜yl(t))+nl=1bRkl(˜xk(t))fRl(˜xl(tτ(t)),˜yl(tτ(t)))                      nl=1bIkl(˜xk(t))fIl(˜xl(tτ(t)),˜yl(tτ(t)))+HRk˜mRk(t)                      HIk˜mIk(t)+uRk(t),l=1,2,,n,LTM:˙˜mRk(t)=˜mRk(t)+fRk(˜xk(t),˜yk(t)),k=1,2,,n, (9)
    {STM:ε˙˜yIk(t)=˜yIk(t)+nl=1aRkl(˜yk(t))fIl(˜xl(t),˜yl(t))    +nl=1aIkl(˜yk(t))fRl(˜xl(t),˜yl(t))+nl=1bRkl(˜yk(t))fIl(˜xl(tτ(t)),˜yl(tτ(t)))                      +nl=1bIkl(˜yk(t))fRl(˜xl(tτ(t)),˜yl(tτ(t)))+HRk˜mIk(t)                      +HIk˜mRk(t)+uIk(t),l=1,2,,n,LTM:˙˜mIk(t)=˜mIk(t)+fIk(˜xk(t),˜yk(t)),k=1,2,,n, (10)

    with initial conditions ˜x(s)=˜ϕR(s)C([τ,0],Rn) and ˜y(s)=˜ϕI(s)C([τ,0],Rn), respectively. By differential inclusions feature for system with the discontinuous right-hand sides, (9) and (10) will be written as follows:

    {STM:ε˙˜xRk(t)˜xRk(t)+nl=1co[aRkl,a+Rkl]fRl(˜xl(t),˜yl(t))nl=1co[aIkl,a+Ikl]fIl(˜xl(t),˜yl(t))                      +nl=1co[bRkl,b+Rkl]fRl(˜xl(tτ(t)),˜yl(tτ(t)))                      nl=1co[bIkl,b+Ikl]fIl(˜xl(tτ(t)),˜yl(tτ(t)))+HRk˜mRk(t)                      HIk˜mIk(t)+uRk(t),l=1,2,,n,LTM:˙˜mRk(t)=˜mRk(t)+fRk(˜xk(t),˜yk(t)),k=1,2,,n, (11)
    {STM:ε˙˜yIk(t)˜xIk(t)+nl=1co[aRkl,a+Rkl]fIl(˜xl(t),˜yl(t))+nl=1co[aIkl,a+Ikl]fRl(˜xl(t),˜yl(t))                      +nl=1co[bRkl,b+Rkl]fIl(˜xl(tτ(t)),˜yl(tτ(t)))                      +nl=1co[bIkl,b+Ikl]fRl(˜xl(tτ(t)),˜yl(tτ(t)))+HRk˜mIk(t)                      +HIk˜mRk(t)+uIk(t),l=1,2,,n,LTM:˙˜mIk(t)=˜mIk(t)+fIk(˜xk(t),˜yk(t)),k=1,2,,n. (12)

    Or equivalently, for k,l=1,...,n, then there exists ˊaRklco[aRkl,a+Rkl],ˊaIklco[aIkl,a+Ikl],ˊbRklco[bRkl,b+Rkl],ˊbIklco[bIkl,b+Ikl] such that

    {STM:ε˙˜xRk(t)=˜xRk(t)+nl=1ˊaRklfRl(˜xl(t),˜yl(t))                      nl=1ˊaIklfIl(˜xl(t),˜yl(t))+nl=1ˊbRklfRl(˜xl(tτ(t)),˜yl(tτ(t)))                      nl=1ˊbIklfIl(˜xl(tτ(t)),˜yl(tτ(t)))+HRk˜mRk(t)                      HIk˜mIk(t)+uRk(t),l=1,2,,n,LTM:˙˜mRk(t)=˜mRk(t)+fRk(˜xk(t),˜yk(t)),k=1,2,,n, (13)
    {STM:ε˙˜yIk(t)=˜xIk(t)+nl=1ˊaRklfIl(˜xl(t),˜yl(t))                      +nl=1ˊaIklfRl(˜xl(t),˜yl(t))+nl=1ˊbRklfIl(˜xl(tτ(t)),˜yl(tτ(t)))                      +nl=1ˊbIklfRl(˜xl(tτ(t)),˜yl(tτ(t)))+HRk˜mIk(t)                      +HIk˜mRk(t)+uIk(t),l=1,2,,n,LTM:˙˜mIk(t)=˜mIk(t)+fIk(˜xk(t),˜yk(t)),k=1,2,,n. (14)

    Let eRk(t)=˜xk(t)xk(t),eIk(t)=˜yk(t)yk(t),hRk(t)=˜mRk(t)mRk(t),hIk(t)=˜mIk(t)mIk(t)(k,l=1,2,...,n) and make the following assumption:

    Assumption 2.

    (1)co[aRkl,a+Rkl]fRl(˜xl(t),˜yl(t))co[aRkl,a+Rkl]fRl(xl(t),yl(t))co[aRkl,a+Rkl](fRl(˜xl(t),˜yl(t))fRl(xl(t),yl(t))),(2)co[aIkl,a+Ikl]fIl(˜xl(t),˜yl(t))co[aIkl,a+Ikl]fIl(xl(t),yl(t))co[aIkl,a+Ikl](fIl(˜xl(t),˜yl(t))fIl(xl(t),yl(t))),(3)co[bRkl,b+Rkl]fRl(˜xl(t),˜yl(t))co[bRkl,b+Rkl]fRl(xl(t),yl(t))co[bRkl,b+Rkl](fRl(˜xl(t),˜yl(t))fRl(xl(t),yl(t))),(4)co[bIkl,b+Ikl]fIl(˜xl(t),˜yl(t))co[bIkl,b+Ikl]fIl(xl(t),yl(t))co[bIkl,b+Ikl](fIl(˜xl(t),˜yl(t))fIl(xl(t),yl(t))).

    Then real and imaginary parts of the error system can be obtained by (5)-(6) and (11)-(12) as follows:

    {STM:ε˙eRk(t)eRk(t)+nl=1co[aRkl,a+Rkl]gRl(eRl(t),eIl(t))nl=1co[aIkl,a+Ikl]gIl(eRl(t),eIl(t))                      +nl=1co[bRkl,b+Rkl]gRl(eRl(tτ(t)),eIl(tτ(t)))                      nl=1co[bIkl,b+Ikl]gIl(eRl(tτ(t)),eIl(tτ(t)))+HRkhRk(t)                      HIkhIk(t)+uRk(t),l=1,2,,n,LTM:˙hRk(t)=hRk(t)+gRk(eRk(t),eIk(t)),k=1,2,,n, (15)
    {STM:ε˙eIk(t)eIk(t)+nl=1co[aRkl,a+Rkl]gIl(eRl(t),eIl(t))+nl=1co[aIkl,a+Ikl]gRl(eRl(t),eIl(t))                      +nl=1co[bRkl,b+Rkl]gIl(eRl(tτ(t)),eIl(tτ(t)))                      +nl=1co[bIkl,b+Ikl]gRl(eRl(tτ(t)),eIl(tτ(t)))+HRkhIk(t)                      +HIkhRk(t)+uIk(t),l=1,2,,n,LTM:˙hIk(t)=hIk(t)+gIk(eRk(t),eIk(t)),k=1,2,,n, (16)

    with initial conditions ψR(s)=φR(s)ϕR(s) and ψI(s)=φI(s)ϕI(s),respectively. Where

    gRl(eRl(t),eIl(t))=fRl(˜xl(t),˜yl(t))fRl(xl(t),yl(t));
    gIl(eRl(t),eIl(t))=fIl(˜xl(t),˜yl(t))fIl(xl(t),yl(t));
    gRl(eRl(tτ(t)),eIl(tτ(t)))
    =fRl(˜xl(tτ(t)),˜yl(tτ(t)))fRl(xl(tτ(t)),yl(tτ(t)));
    gIl(eRl(tτ(t)),eIl(tτ(t)))=fIl(˜xl(tτ(t)),˜yl(tτ(t)))fIl(xl(tτ(t)),yl(tτ(t))).

    Or equivalently, for k,l=1,...,n, then there exists aRklco[aRkl,a+Rkl],aIklco[aIkl,a+Ikl],bRklco[bRkl,b+Rkl],bIklco[bIkl,b+Ikl] such that

    {STM:ε˙eRk(t)=eRk(t)+nl=1aRklgRl(eRl(t),eIl(t))                      nl=1aIklgIl(eRl(t),eIl(t))+nl=1bRklgRl(eRl(tτ(t)),eIl(tτ(t)))                      nl=1bIklgIl(eRl(tτ(t)),eIl(tτ(t)))+HRkhRk(t)                      HIkhIk(t)+uRk(t),l=1,2,,n,LTM:˙hRk(t)=hRk(t)+gRk(eRk(t),eIk(t)),k=1,2,,n, (17)
    {STM:ε˙eIk(t)=eIk(t)+nl=1aRklgIl(eRl(t),eIl(t))                      +nl=1aIklgRl(eRl(t),eIl(t))+nl=1bRklgIl(eRl(tτ(t)),eIl(tτ(t)))                      +nl=1bIklgRl(eRl(tτ(t)),eIl(tτ(t)))+HRkhIk(t)                      +HIkhRk(t)+uIk(t),l=1,2,,n,LTM:˙hIk(t)=hIk(t)+gIk(eRk(t),eIk(t)),l=1,2,,n. (18)

    Lemma 2.1. For any vector x,yRn and a matrix M>0Rn×n, one has the inequality that

    2xTyxTMx+yTM1y. (19)

    In this paper, the control inputs in (17) and (18) are taken as follows:

    uR(t)=KReR(t),uI(t)=KIeI(t), (20)

    where

    eR(t)=(eR1(t),eR2(t),...,eRn(t))T,eI(t)=(eI1(t),eI2(t),...,eIn(t))T,
    KR=diag(kR1,kR2,...,kRn),KI=diag(kI1,kI2,...,kIn),

    and the controller gains to be determined.

    Definition 2.2. The system (17) and (18) are asymptotically stable for any given initial conditions they satisfy:

    limte(t)2=0,limth(t)2=0,

    where

    e(t)=(eR(t))T,eI(t))T)T,h(t)=(hR(t))T,hI(t))T)T,
    hR(t)=(hR1(t),hR2(t),...,hRn(t))T,hI(t)=(hI1(t),hI2(t),...,hIn(t))T.

    Then drive-response systems (1) and (2) are said to be synchronized.

    In the section, we present synchronization problems of CMCNNs.

    Theorem 3.1. Under Assumptions 1-2, the systems (1) and (2) can be asymptotically synchronized with control inputs (20), if there exist constants r1,r2,r3, r4,r5,r6,r7,r8,r9,r0,r1,r2,r5,r6, r9,r0>0, diagonal matrix P,Q>0 such that TR,TI>0, where

    TR=1ε[2I2KR2ARλRRARλRI(AR)T+2AIλIR+AIλII(AI)Tr1(BRλRR)(BRλRR)Tr1(BRλRI)(BRλRI)Tr2(BIλIR)(BIλIR)Tr2(BIλII)(BIλII)Tr3HR(HR)T+r4HI(HI)TλIRIλRRI]r9λRRIr0λIRIP,
    TI=1ε[2I2KIλRII+λIIIARλIR(AR)T2ARλIIAIλRR(AI)T2AIλRIr5(BRλIR)(BRλIR)Tr5(BRλII)(BRλII)Tr6(BIλRR)(BIλRR)Tr6(BIλRI)(BIλRI)Tr7HR(HR)Tr8HI(HI)T]r9λRIIr0λIIIQ.

    Proof. Consider the following Lyapunov function

    V(t,e(t),h(t))=(eR(t))TeR(t)+(eI(t))TeI(t)+(hR(t))ThR(t)+(hI(t))ThI(t)+ttτ(t)(eR(s))TPeR(s)ds+ttτ(t)eI(s))TQeI(s)ds (21)

    Then, calculating the time derivative of V(t,e(t)) along the trajectories of (17) and (18), that is

    ˙V(t,e(t),h(t))2(eR(t))T˙eR(t)+2(eI(t))T˙eI(t)+2(hR(t))T˙hR(t)+2(hI(t))T˙hI(t)+(eR(t))TPeR(t)(eR(tτ(t)))TPeR(tτ(t))(1γ)+(eI(t))TQeI(t)(eI(tτ(t)))TQeI(tτ(t))(1γ) (22)

    According to Assumption 1-2 and Lemma 2.1, we have

    2(eR(t))T˙eR(t)=2(eR(t))T1ε[eR(t)+ARgR(e(t))AIgI(e(t))+BRgR(e(tτ(t)))BIgI(e(tτ(t)))+HRhR(t)HIhI(t)+uR(t)] (23)
    2(eR(t))TARgR(e(t))2(eR(t))TAR(λRReR(t)+λRIeI(t))=2(eR(t))TARλRReR(t)+2(eR(t))TARλRIeI(t)2(eR(t))TARλRReR(t)+(eR(t))TARλRI(AR)TeR(t)   +(eI(t))TλRIeI(t) (24)
    2(eR(t))TAIgI(e(t))2(eR(t))TAI(λIReR(t)+λIIeI(t))=2(eR(t))TAIλIReR(t)+2(eR(t))TAIλIIeI(t)2(eR(t))TAIλIReR(t)+(eR(t))TAIλII(AI)TeR(t)   +(eI(t))TλIIeI(t) (25)
    2(eR(t))TBRgR(e(tτ(t)))(eR(t))T(BRλRR)M1(BRλRR)TeR(t)+(eR(tτ(t)))TM11eR(tτ(t))+(eR(t))T(BRλRI)M1(BRλRI)TeR(t)+(eI(tτ(t)))TM11eI(tτ(t)) (26)

    Choose

    M1=(r100r1),M1=(r100r1),

    So (26) can be simplified as

    2(eR(t))TBRgR(e(tτ(t)))r1(eR(t))T(BRλRR)(BRλRR)TeR(t)+1r1(eR(tτ(t)))TeR(tτ(t))+r1(eR(t))T(BRλRI)(BRλRI)TeR(t)+1r1(eI(tτ(t)))TeI(tτ(t))

    (Note: The following inequalities use the same method.)

    2(eR(t))TBIgI(e(tτ(t)))r2(eR(t))T(BIλIR)(BIλIR)TeR(t)+1r2(eR(tτ(t)))TeR(tτ(t))+r2(eR(t))T(BIλII)(BIλII)TeR(t)+1r2(eI(tτ(t)))TeI(tτ(t)) (27)
    2(eR(t))THRhR(t)r3(eR(t))THR(HR)TeR(t)+1r3(hR(t))ThR(t) (28)
    2(eR(t))THIhI(t)r4(eR(t))THI(HI)TeR(t)+1r4(hI(t))ThI(t) (29)

    Similarly,

    2(eI(t))T˙eI(t)=2(eI(t))T1ε[eI(t)+ARgI(e(t))+AIgR(e(t))+BRgI(e(tτ(t)))+BIgR(e(tτ(t)))+HRhI(t)+HIhR(t)+uI(t)] (30)
    2(eI(t))TARgI(e(t))2(eI(t))TAR(λIReR(t)+λIIeI(t))=2(eI(t))TARλIReR(t)+2(eI(t))TARλIIeI(t)(eI(t))TARλIR(AR)TeI(t)+(eR(t))TλIReR(t)   +2(eI(t))TARλIIeI(t) (31)
    2(eI(t))TAIgR(e(t))2(eI(t))TAI(λRReR(t)+λRIeI(t))=2(eI(t))TAIλRReR(t)+2(eI(t))TAIλRIeI(t)(eI(t))TAIλRR(AI)TeI(t)+(eR(t))TλRReR(t)   +2(eI(t))TAIλRIeI(t) (32)
    2(eI(t))TBRgI(e(tτ(t)))r5(eI(t))T(BRλIR)(BRλIR)TeI(t)+1r5(eR(tτ(t)))TeR(tτ(t))+r5(eI(t))T(BRλII)(BRλII)TeI(t)+1r5(eI(tτ(t)))TeI(tτ(t)) (33)
    2(eI(t))TBIgR(e(tτ(t)))r6(eI(t))T(BIλRR)(BIλRR)TeI(t)+1r6(eR(tτ(t)))TeR(tτ(t))+r6(eI(t))T(BIλRI)(BIλRI)TeI(t)+1r6(eI(tτ(t)))TeI(tτ(t)) (34)
    2(eI(t))THRhI(t)r7(eI(t))THR(HR)TeI(t)+1r7(hI(t))ThI(t) (35)
    2(eI(t))THIhR(t)r8(eI(t))THI(HI)TeI(t)+1r8(hR(t))ThR(t) (36)
    2(hR(t))T˙hR(t)=2(hR(t))T(hR(t)+gR(e(t))) (37)
    2(hR(t))TgR(e(t))2(hR(t))T(λRReR(t)+λRIeI(t))1r9(hR(t))ThR(t)+r9λRR(eR(t))TeR(t)+1r9(hR(t))ThR(t)+r9λRI(eI(t))TeI(t) (38)
    2(hI(t))T˙hI(t)=2(hI(t))T(hI(t)+gI(e(t))) (39)
    2(hI(t))TgI(e(t))2(hI(t))T(λIReR(t)+λIIeI(t))1r0(hI(t))ThI(t)+r0λIR(eR(t))TeR(t)+1r0(hI(t))ThI(t)+r0λII(eI(t))TeI(t) (40)

    Substituting(23)-(40) to (22) yield

    ˙V(t)(eR(t))T{1ε[2I2KR2ARλRRARλRI(AR)T+2AIλIR+AIλII(AI)Tr1(BRλRR)(BRλRR)Tr1(BRλRI)(BRλRI)Tr2(BIλIR)(BIλIR)Tr2(BIλII)(BIλII)Tr3HR(HR)T+r4HI(HI)TλIRIλRRI]r9λRRIr0λIRIP}eR(t)(eI(t))T{1ε[2I2KIλRII+λIIIARλIR(AR)T2ARλIIAIλRR(AI)T2AIλRIr5(BRλIR)(BRλIR)Tr5(BRλII)(BRλII)Tr6(BIλRR)(BIλRR)Tr6(BIλRI)(BIλRI)Tr7HR(HR)Tr8HI(HI)T]r9λRIIr0λIIIQ}eI(t)+(hR(t))T[2I+Iεr3+Iεr8+Ir9+Ir9]hR(t)+(hI(t))T[2I+Iεr4+Iεr7+Ir0+Ir0]hI(t)+(eR(tτ(t)))T[Iεr1Iεr2+Iεr5+Iεr6(1γ)P]eR(tτ(t))+(eI(tτ(t)))T[Iεr1Iεr2+Iεr5+Iεr6(1γ)Q]eI(tτ(t)) (41)

    where I is the identity matrix of appropriate dimension.

    It is easy to know that there are real numbers r3,r4,r7,r8,r9,r0,r9, and r0 such that

    1εr3+1r8+1r9+1r92<0,1εr4+1r7+1r0+1r02<0. (42)

    Letting

    Iεr1Iεr2+Iεr5+Iεr6=(1γ)P,Iεr1Iεr2+Iεr5+Iεr6=(1γ)Q,λR=min{λmin(TR),21εr3+1r8+1r9+1r9},λI=min{λmin(TI),21εr4+1r7+1r0+1r0}. (43)

    From (41)-(43), it can be seen that

    ˙V(t)λR(eR(t)2+hR(t)2)λI(eI(t)2+hI(t)2). (44)

    Moreover, in (44), the equality holds if and only if eR(t)2+hR(t)2=0,eI(t)2+hI(t)2=0, i.e., eR(t)2=0,eI(t)2=0,hR(t)2=0, and hI(t)2=0. It can be concluded from Lyapunov stability theory that

    limte(t)2=0,limth(t)2=0.

    According to Definition 2.2, the trivial solution of system (17) and (18) are asymptotically stable. We can conclude that the neural networks (1) and (2) can be synchronized with control inputs (20). The proof is complete.

    Remark 3. From Theorem 3.1, we can see that the existence of the variable delay affects the value of the matrix P and Q and then affects the value of the matrix TR and TI. In the following, we will give two corollaries to explain how TR and TI will change when τ(t)=τ (τ is a constant) and τ(t)=0.

    Corollary 1. Under Assumptions 1-2, and the controllers (20) when τ(t)=τ>0, system (1) and (2) can be asymptotically synchronized, if there exist constants r1,r2,r3,r4,r5,r6,r7,r8,r9,r0,r1,r2,r5,r6, r9,r0>0, diagonal matrix P,Q>0 such that TR,TI>0, where

    TR=1ε[2I2KR2ARλRRARλRI(AR)T+2AIλIR+AIλII(AI)Tr1(BRλRR)(BRλRR)Tr1(BRλRI)(BRλRI)Tr2(BIλIR)(BIλIR)Tr2(BIλII)(BIλII)Tr3HR(HR)T+r4HI(HI)TλIRIλRRI]r9λRRIr0λIRIP,
    TI=1ε[2I2KIλRII+λIIIARλIR(AR)T2ARλIIAIλRR(AI)T2AIλRIr5(BRλIR)(BRλIR)Tr5(BRλII)(BRλII)Tr6(BIλRR)(BIλRR)Tr6(BIλRI)(BIλRI)Tr7HR(HR)Tr8HI(HI)T]r9λRIIr0λIIIQ.

    Proof. The proof process is similar to Theorem 3.1 by taking

    Iεr1Iεr2+Iεr5+Iεr6=P,
    Iεr1Iεr2+Iεr5+Iεr6=Q,

    and is omitted here.

    Corollary 2. Under Assumptions 1-2, and the controllers (20) when τ(t)=0, system (1) and (2) can be asymptotically synchronized, if there exist constants r3,r4,r7,r8,r9,r0, r9,r0>0, such that TR,TI>0, where

    TR=1ε[2I2KR2(AR+BR)λRR(AR+BR)λRI+2(AI+BI)λIR+(AI+BI)λIIr3HR(HR)T+r4HI(HI)T(AR+BR)λIR(AI+BI)λRR]r9λRRIr0λIRI,
    TI=1ε[2I2KI(AR+BR)λRI+(AI+BI)λII(AR+BR)λIR2(AR+BR)λII(AI+BI)λRR2(AI+BI)λRIr7HR(HR)Tr8HI(HI)T]r9λRIIr0λIII.

    Proof. The proof process is similar to Theorem 3.1 and is omitted here.

    In the section, a numerical example is given to demonstrate the validity of the above results. Consider the following memristor-based complex-valued competitive neural networks:

    {STM:ε˙zk(t)=zk(t)+2l=1akl(zk(t))fl(zl(t))                      +2l=1bkl(zk(t))fl(zl(tτ(t)))+Hkmk(t),l=1,2LTM:˙mk(t)=mk(t)+fk(zk(t)), k=1,2, (45)

    where

         aR11(x1(t))={1.5,|x1|<1,0.4,|x1|1,aR12(x1(t))={0.2,|x1|<1,1.0,|x1|1,
         aR21(x2(t))={0.7,|x2|<1,1.4,|x2|1,aR22(x2(t))={2.0,|x2|<1,0.8,|x2|1,
        aI11(y1(t))={1.5,|y1|<1,0.06,|y1|1,aI12(y1(t))={1.5,|y1|<1,0.28,|y1|1,
         aI21(y2(t))={0.6,|y2|<1,0.25,|y2|1,aI22(y2(t))={1.0,|y2|<1,1.6,|y2|1,
         bR11(x1(t))={0.2,|x1|<1,1.0,|x1|1,bR12(x1(t))={3.0,|x1|<1,0.5,|x1|1,
         bR21(x2(t))={1.6,|x2|<1,2.0,|x2|1,bR22(x2(t))={1.0,|x2|<1,1.25,|x2|1,
         bI11(y1(t))={1.2,|y1|<1,0.6,|y1|1,bI12(y1(t))={1.25,|y1|<1,0.13,|y1|1,
         bI21(y2(t))={1.4,|y2|<1,0.03,|y2|1,bI22(y2(t))={1.5,|y2|<1,0.7,|y2|1,

    with initial values z1(t)=0.80.5i,z2(t)=0.46+0.5i,m1=0.820.14i,m2=0.4+0.82i. Consider system (45) as drive system and corresponding response system can be described as follows;

    {STM:˙εk(t)=˜zk(t)+2l=1akl(˜zk(t))fl(˜zl(t))+2l=1bkl(˜zk(t))fl(˜zl(tτ(t)))+Hk˜mk(t),l=1,2LTM:˜mk(t)=˜mk(t)+fk(˜zk(t)),k=1,2 (46)

    with initial values ˜z1(t)=0.150.63i,˜z2(t)=0.36+0.35i,˜m1=0.63+0.2i,˜m2=0.460.9i. Now we denote eRk(t)=˜xkxk(t),eIk(t)=˜ykyk(t),˙hRk(t)=˜mRk(t)mRk(t) and ˙hIk(t)=˙˜mIk(t)mIk(t)(k=1,2),

    Let ε=0.9,τ(t)=[(et)/(1+et)] such that ˙τ(t)γ=0.25, the activation function is consider as f(z(.))=tanh(x(.))+itanh(y(.)). And choose λRR1=λIR1=λRI1=λII1=λRR2=λIR2=λRI2=λII2=1. Meanwhile, let HR1=0.5,HR2=1.0,HI1=1.5,HI2=1.8,KR=diag(7,7),KI=diag(11,11).

    When the controllers is the form of (20), then by simple computation we get

    AR=(1.50.20.72),AI=(1.51.50.61),
    BR=(0.231.61),BI=(1.21.251.41.5).

    And choose

    r1=r2=12,r3=2,r4=3,r5=r6=13,r7=r8=2,r9=3,r0=2,
    r1=r2=13,r5=r6=14,r9=3,r0=2.

    so from Theorem 3.1, we have

    TR=(5.854.537.647.72),TI=(4.473.963.074.41).

    Then, all the conditions of Theorem 3.1 are satisfied. Under the controllers (20), the synchronization errors of real parts and imaginary parts are depicted by Figure 1. Therefore, according to Theorem 3.1, the systems (45) and (46) are synchronized.

    Figure 1.  Synchronization errors of eR1,eR2,hR1,hR2 of system (45) and (46) with the controllers (20).
    Figure 2.  Synchronization errors of eI1,eI2,hI1,hI2 of system (45) and (46) with the controllers (20).

    This paper is concerned with synchronization of CMCNNs with different time scales. Firstly, we improved the model: (1) improved the ordinary neural network model to CMCNNs with different time scales; (2) extended the a common real-valued system to a complex-valued system. Then, we achieved the synchronization problem of the drive and response systems by designing a proper controller. In theory, the control design is operable and can be easily realized. Moreover, our results are more general and extend the previously known results. Finally, the effectiveness of our results has been demonstrated by Section 4. In further research, the main results of this paper can be extended to no time-delay for the feedback controller. We will also explore more dynamical behaviors of CMCNNs, for example, finite-time synchronization, fixed-time synchronization and anti-synchronization.

    This work was supported by the National Natural Science Foundation of China (No.11972115, No.12062004) and Talent Special Projects of School-level Scientific Research Programs under Guangdong Polytechnic Normal University (No.2021SDKYA004).



    [1] Synchronization conditions in simple memristor neural networks. Journal of the Franklin Institute (2015) 352: 3196-3220.
    [2] Singularity formation and blowup of complex-valued solutions of the modified KdV equation. Discrete Cont. Dyn. (2013) 33: 4811-4840.
    [3] Memristor–The missing circuit element. IEEE Trans. Circ. Theor. (1971) 18: 507-519.
    [4] Improving control effects of absence seizures using single-pulse alternately resetting stimulation (SARS) of corticothalamic circuit. Appl. Math. Mech. -Engl. Ed. (2020) 41: 1287-1302.
    [5] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers, Dordrecht, 18 1988. doi: 10.1007/978-94-015-7793-9
    [6] Global exponential synchronization of memristive competitive neural networks with time-varying delay via nonlinear control. Neural Process. Lett. (2018) 49: 103-119.
    [7] Excitement and synchronization of small-world neuronal networks with short-term synaptic plasticity. Int. J. Neural Syst. (2011) 21: 415-425.
    [8] Global Hopf bifurcation analysis on a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete Cont. Dyn. Syst.-B (2011) 16: 457-474.
    [9] Global anti-synchronization of complex-valued memristive neural networks with time delays. IEEE T. Cybernetics (2019) 49: 1735-1747.
    [10] Singular perturbation analysis of competitive neural networks with different time scales. Neural. Comput. (1996) 8: 1731-1742.
    [11] Local and global stability analysis of an unsupervised competitive neural network. IEEE T. Neural Networ. (2008) 19: 346-351.
    [12] Local exponential stability of competitive neural networks with different time scales. Eng. Appl. Artif. Intell. (2004) 17: 227-232.
    [13] Anti-synchronization of a class of fuzzy memristive competitive neural networks with different time scales. Neural Process. Lett. (2020) 52: 647-661.
    [14] Spike synchronization and rate modulation differentially involved in motor cortical function. Science (1997) 278: 1950-1953.
    [15] Synchronization of memristive competitive neural networks with different time scales. Neural. Comput. and Applic. (2014) 25: 1163-1168.
    [16] Hybrid memristor chaotic system. J. Nanorlectron. Optoe. (2018) 13: 812-818.
    [17] Electronics: The fourth element. Nature (2008) 453: 42-43.
    [18] Complex-valued recurrent correlation neural networks. IEEE Trans. Neural. Netw. Learn. Syst. (2014) 25: 1600-1612.
    [19] Circuit elements with memory: Memristors, memcapacitors, and meminductors. P. IEEE (2009) 97: 1717-1724.
    [20] D. Wang, L. Huang and L. Tang, New results for global exponential synchronization in neural networks via functional differential inclusions, Chaos, 25 (2015), 083103, 11 pp. doi: 10.1063/1.4927737
    [21] A review of computational modeling and deep brain stimulation: Applications to Parkinson's disease. Appl. Math. Mech. -Engl. Ed. (2020) 41: 1747-1768.
    [22] Effects of initial conditions on the synchronization of the coupled memristor neural circuits. Nonlinear Dynamics (2019) 95: 1269-1282.
    [23] W. Zhang, C. Li and T. Huang,, Global robust stability of complex-valued recurrent neural networks with time-delays and uncertainties, Int. J. Biomath., 7 (2014), 1450016. doi: 10.1142/S1793524514500168
    [24] Input-to-state stability analysis for memristive Cohen-Grossberg-type neural networks with variable time delays. Chaos Soliton. Fract. (2018) 114: 364-369.
    [25] Exponential stability of a class of competitive neural networks with multi-proportional delays. Neural Process. Lett. (2016) 44: 651-663.
    [26] Synchronization of memristive complex-valued neural networks with time delays via pinning control method. IEEE T. Cybernetics (2020) 50: 3806-3815.
    [27] J. Zhuang, Y. Zhou and Y. Xia, Synchronization analysis of drive-response multi-layer dynamical networks with additive couplings and stochastic perturbations, Discrete Cont. Dyn.-S., 14 (2021), 1607–-1629. doi: 10.3934/dcdss.2020279
  • This article has been cited by:

    1. Shaohui Yan, Yuyan Zhang, Yu Ren, Xi Sun, Ertong Wang, Zhenlong Song, Four-dimensional Hindmarsh–Rose neuron model with hidden firing multistability based on two memristors, 2022, 97, 0031-8949, 125203, 10.1088/1402-4896/ac99ad
    2. Chenguang Xu, Minghui Jiang, Junhao Hu, Fixed-time synchronization of complex-valued memristive competitive neural networks based on two novel fixed-time stability theorems, 2023, 35, 0941-0643, 22605, 10.1007/s00521-023-08874-6
    3. Shaohui Yan, Jiawei Jiang, Yuyan Zhang, Bian Zheng, Hanbing Zhan, Defeng Jiang, Multiple firing patterns, energy conversion and hardware implementation within Hindmarsh-Rose-improved neuron model, 2024, 99, 0031-8949, 055265, 10.1088/1402-4896/ad3eec
    4. Jiapeng Han, Liqun Zhou, Fixed-time synchronization of proportional delay memristive complex-valued competitive neural networks, 2025, 08936080, 107411, 10.1016/j.neunet.2025.107411
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2407) PDF downloads(198) Cited by(4)

Figures and Tables

Figures(2)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog