Exponential Synchronization for Fractional-order Time-delayed Memristive Neural Networks

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VOLUME 3 , ISSUE 3 (Oct 2018) > List of articles

Exponential Synchronization for Fractional-order Time-delayed Memristive Neural Networks

Ding Dawei * / Zhang Yaqin * / Wang Nian *

Keywords : Exponential Synchronization, Memristor-based Neural Networks, Fractional-order, Linear Error Feedback Control, Time-varying Delays

Citation Information : International Journal of Advanced Network, Monitoring and Controls. Volume 3, Issue 3, Pages 1-15, DOI: https://doi.org/10.21307/ijanmc-2019-001

License : (CC-BY-NC-ND 4.0)

Published Online: 01-October-2019

ARTICLE

ABSTRACT

Considering the fact that the exponential synchronization of neural networks has been widely used in theoretical research and practical application of many scientific fields, and there are a few researches about the exponential synchronization of fractional-order memristor-based neural networks (FMNN). This paper concentrates on the FMNN with time-varying delays and investigates its exponential synchronization. A simple linear error feedback controller is applied to compel the response system to synchronize with the drive system. Combining the theories of differential inclusions and set valued maps, a new sufficient condition concerning exponential synchronization is obtained based on comparison principle rather than the traditional Lyapunov theory. The obtained results extend exponential synchronization of integer-order system to fractional-order memristor-based neural networks with time-varying delays. Finally, some numerical examples are used to demonstrate the effectiveness and correctness of the main results.

I. INTRODUCTION

Chua already supposed the existence of memristor in 1971 [1], however, the practical device of memristor in electronics is obtained in [2] until 2008. In addition to the existing three kinds of circuit elements, memristor is regarded as the fourth basic circuit element and is defined by a nonlinear charge-flux characteristic. As everyone knows, resistors can be used to work as connection weights so that it can emulate the synapses in artificial neural networks. However, in the neural networks of biological individual, long-term memories is essential in the synapses among neurons, but for the general resistors, it is impossible to have the function of memory. Recently, due to the memory characteristics of memristor, memristor can replace the resistor to develop a new neural networks that is memristor-based neural networks (MNN) [3-6].

In recent years, more and more attentions have been put on the dynamical analysis of memristor-based neural networks, such as the investigation of stability [7-10], periodicity [11-13], system synchronization [14-22], passivity analysis [23], dissipativity [24-25] and attractivity [26]. Particularly, the stability and synchronization of MNN has been widely studied in [27-30]. In fact, synchronization means the dynamics of nodes share the common time-spatial property. Therefore we can understand an unknown dynamical system by achieving the synchronization with the well-known dynamical systems [18]. Moreover, in the transmission of digital signals, communication will become security, reliable and secrecy by achieving synchronization between the various systems. Therefore, the synchronization of MNN is still worth further research.

Moreover, the fractional-order models can better describe the memory and genetic properties of various materials and process, so the fractional-order models have received a lot of research attentions than integer-order models. In recent years, with the improvement of fractional-order differential calculus and fractional-order differential equations, it is easy to model and analyze practical problems [31, 32]. Therefore, there have been a lot of researches about the dynamical analysis and synchronization of fractional-order memristor-based neural networks (FMNN) [34-39]. Finite-time synchronization, hybrid projective synchronization and adaptive synchronization of FMNN have all been researched [34-36]. However, there are only a very few research results on exponential synchronization of FMNN. In fact, the exponential synchronization of neural networks has been widely used in the theoretical research and practical application of many scientific fields, for example, associative memory, ecological system, combinatorial optimization, military field, artificial intelligence system and so on [40-43]. So the exponential synchronization of FMNN is still worth further studying as it is a significant academic problem.

On the other hand, the stability and synchronization of FMNN without time delay have been deeply studied such as in [33]. However, in hardware implementation of neural networks, time delay is unavoidable owing to the finite switching speeds of the amplifiers. And it will cause instability, oscillation and chaos phenomena of systems. So the investigation for stability and synchronization of FMNN cannot be independent on the time delay.

Motivated by the above discussion, this paper studies the exponential synchronization of FMNN with time-varying delays. The main contributions of this paper can be listed as follow. (1) This is the first attempt to achieve exponential synchronization of FMNN with time-varying delays by employing a simple linear error feedback controller. (2) The sufficient condition for exponential synchronization of FMNN with time delays is obtained based on comparison principle instead of the traditional Lyapunov theory. (3) Some previous research results of exponential synchronization for integer-order memristor-based system are the special cases of our results. Furthermore, some numerical examples are given to demonstrate the effectiveness and correctness of the main results.

The rest of this paper is organized as follows. Preliminaries including the introduction of Caputo fractional-order derivative, model description, assumptions, definitions and lemmas are presented in Section 2. Section 3 introduces the sufficient condition for exponential synchronization of the FMNN. In Section4, the numerical simulations are presented. Section5 gives the conclusion of this paper.

II. PRELIMINARIES

Compared to the integer-order derivatives, we know the distinct advantage of Caputo derivative is that it only requires initial conditions from the Laplace transform of fractional derivative, and it can represent well-understood features of physical situations and making it more applicable to real world problems [36]. So in the rest of this paper, we apply the Caputo fractional-order derivative for the fractional-order memristor-based neural networks (FMNN) and investigate the exponential synchronization of FMNN.

A. The Caputo fractional-order derivative

Definition1 [32] The Caputo fractional-order derivative is defined as follows:

(1)
Dtqf(t)=1Γ(mq)t0tf(m)(τ)(tτ)qm+1dτ, 10.21307_ijanmc-2019-001-eqn1.jpg
where q is the order of fractional derivative, m is the first integer larger than q, m − 1 ≤ q < m,Γ(⋅) is the Gamma function,
(2)
T(x)=0tx1etdt. 10.21307_ijanmc-2019-001-eqn2.jpg

Particularly, when 0<q<1,

(3)
Dtqf(t)=1Γ(1q)t0tf'(τ)(tt0)qdτ. 10.21307_ijanmc-2019-001-eqn3.jpg

B. Model description

In this paper, referring to some relevant works on FMNN [35,36], we consider a class of FMNN with time-varying delays described by the following equation,

(4)
Dqxi(t)=cixi(t)+j=1naij(xj(t))fj(xj(t))+j=1nbij(xj(tτj(t)))gj(xj(tτj(t)))+Iit0,iN.aij(xj(t))=MijCi×δij,bij(xj(tτj(t)))=WijCi×δij,δij={1,ij,1,i=j, 10.21307_ijanmc-2019-001-eqn4.jpg
where xi(t) is the state variable of the ith neuron (the voltage of capacitor Ci), q is the order of fractional derivative, ci > 0 is the self-regulating parameters of the neurons, 0 ≤ τjτ and (τ is a constant ) represents the transmission time-varying delay. fj,gj :RR are feedback functions without and with time-varying delay. aij(xj(t)) and bij(xj(t − τj(t))) are memristive connective weights, which denote the neuron interconnection matrix and the delayed neuron interconnection matrix, respectively. Wij and Mij denote the memductances of memristors Rij and Fij respectively. And Rij represents the memristor between the feedback function fi(xi(t)) and xi(t), Fij represents the memristor between the feedback function gi(xi(tτi(t))) and xi(t). Ii represents the external input. According to the feature of memristor, we denote
(5)
aij(xj(t))={aij,xj(t)0,aij,xj(t)>0,bij(xj(tτj(t)))={bij,xj(tτj(t))0.bij,xj(tτj(t))>0. 10.21307_ijanmc-2019-001-eqn5.jpg

C. Assumptions, Definitions and Lemmas

In the rest of paper, we first make following assumption for system (4).

Assumption1: For jN, ∀s1, s2R, the neuron activation functions fj, gj bounded, fi(0) = gj(0) = 0 and satisfy

(6)
0fj(s1)fj(s2)s1s2σj,0gj(s1)gj(s2)s1s2ρj, 10.21307_ijanmc-2019-001-eqn6.jpg
where s1s2 and σj, ρj are nonnegative constants.

We consider system (4) as drive system and corresponding response system is given as follows:3

(7)
Dqyi(t)=ciyi(t)+j=1naij(yj(t))fj(yj(t))+j=1nbij(yj(tτj(t)))gj(yj(tτj(t)))+Ii+ui,t0,iN, 10.21307_ijanmc-2019-001-eqn7.jpg

Where

(8)
aij(yj(t))={aij,yj(t)0,aij,yj(t)>0,bij(yj(tτj(t)))={bij,yj(tτj(t))0.bij,yj(tτj(t))>0. 10.21307_ijanmc-2019-001-eqn8.jpg
and ui(t) is a liner error feedback control function which defined by ui(t) = ωi(yi(t) − xi(t)), where ωi, iN are constants, which denotes the control gain. Next, we define the synchronization error e(t) as e(t) = (e1(t),e2(t),….,en(t))T, where ei(t) = yi(t) − xi(t). According to the system (4) and system (7), the synchronization error system can be described as follows:
(9)
Dqei(t)=ciei(t)+[j=1naij(yj(t))fj(yj(t))j=1naij(xj(t))fj(xj(t))]+[j=1nbij(yj(tτj(t)))gj(yj(tτj(t)))j=1nbij(xj(tτj(t)))gj(xj(tτj(t)))]+ui(t),t0,iN 10.21307_ijanmc-2019-001-eqn9.jpg
where aij(yj(t)), bij(yj(t − τj(t))), aij(xj(t)), bij(xj(t − τj(t))) are the same as those defined above, ui(t) = ωi(yi(t) − xi(t)) = ωiei(t), where ωi, iN are constants, which denotes the control again.

According to the theories of differential inclusions and set valued maps [40], if xi(t) and yi(t) are solutions of (4) and (7) respectively, system (4) and system (7) can be written as follow:

(10)
Dqxi(t)cixi(t)+j=1nco[aij(xj(t))]fj(xj(t))+j=1nco[bij(xj(tτj(t)))]gj(xj(tτj(t)))+Ii,t0,iN 10.21307_ijanmc-2019-001-eqn10.jpg

And

(11)
Dqyi(t)ciyi(t)+j=1nco[aij(yj(t))]fj(yj(t))+j=1nco[bij(yj(tτj(t)))]gj(yj(tτj(t)))+Ii+ui,t0,iN, 10.21307_ijanmc-2019-001-eqn11.jpg

Where

(12)
co[aij(xj(t))]={aij,xj(t)<0,co{aij,aij},xj(t)=0aij,xj(t)>0,,co[aij(yj(t))]={aij,yj(t)<0,co{aij,aij},yj(t)=0aij,yj(t)>0, 10.21307_ijanmc-2019-001-eqn12.jpg

And

(13)
co[bij(xj(tτ(j)))]={bij,xj(tτ(j))<0,co{bij,bij},xj(tτ(j))=0,bij,xj(tτ(j))>0,co[bij(yj(tτ(j)))]={bij,yj(tτ(j))<0,co{bij,bij},yj(tτ(j))=0,bij,yj(tτ(j))>0, 10.21307_ijanmc-2019-001-eqn13.jpg

where co{u, v} denotes the closure of convex hull generated by real numbers u and v or real matrices u and v. Then the synchronization error system can be described as follows:

(14)
Dqei(t)ciei(t)+{j=1nco[aij(yj(t))]fj(yj(t))j=1nco[aij(xj(t))]fj(xj(t))}+{j=1nco[bij(yj(tτj(t)))]gj(yj(tτj(t)))j=1nco[bij(xj(tτj(t)))]gj(xj(tτj(t)))}+ωiei,t0,iN. 10.21307_ijanmc-2019-001-eqn14.jpg

Definition2 [8] For ∀t ≥ 0, the exponential synchronization of system (4) and system (7) can be transformed to the exponential stability of the error system (9) (error approaches to zero). The error system (9) is said to be exponentially stable, if there exist constant Qi > 0, Pi > 0, such that the solution e(t) = (e1(t),e2(t),…,en(t))T of error system (9) with initial condition e(s) = ϕ(s) ∈ ([t0τ, t0],Rn) satisfies

|ei(t)|Qimax1in{supt0τst0|ϕi(s)|}exp{Pi(tt0)},tt0>0, 10.21307_ijanmc-2019-001-eqn15.jpg
i = 1,2,…,n, where Pi is called the estimated rate of exponential convergence.

Lemma1 [14] Under the assumption1, the following estimation can be obtained:

(i) co[aij(yj(t))]fj(yj(t)) − co[aij(xj(t))]fj(xj(t)) ≤ AijFj(ej(t)),

(ii) co[bij(yj(t − τj(t)))]gj(yj(t − τj(t))) − co[bij(xj(t − τj(t)))]gj(xj(t − τj(t))) ≤ BijGj(ej(t − τj(t))),

where

Aij=max{|aij|,|aij|},Bij=max{|bij|,|bij|},i,jN, 10.21307_ijanmc-2019-001-eqn16.jpg
Fj(ej(t))=fj(yj(t))fj(xj(t)),Gj(ej(tτj(t)))=gj(yj(tτj(t)))gj(xj(tτj(t))),jN. 10.21307_ijanmc-2019-001-eqn17.jpg

Proof: If yi(t) = 0, xi(t) = 0, iN we can easily have part(i) hold. From (9) and (10), we can get

  • (1) For yi(t) < 0, xi(t) < 0, then

    co[aij(yj(t))]fj(yj(t))co[aij(xj(t))]fj(xj(t))=aijfj(yj(t))aijfj(xj(t)=aijFj(ej(t))AijFj(ej(t)). 10.21307_ijanmc-2019-001-eqn18.jpg

  • (2) For yi(t) > 0, xi(t) > 0, then

    co[aij(yj(t))]fj(yj(t))co[aij(xj(t))]fj(xj(t))=aijfj(yj(t))aijfj(xj(t)=aijFj(ej(t))AijFj(ej(t)). 10.21307_ijanmc-2019-001-eqn19.jpg

  • (3) For xi(t) < 0 < yi(t) or yi(t) < 0 < xi(t), then

    co[aij(yj(t))]fj(yj(t))co[aij(xj(t))]fj(xj(t))=aijfj(yj(t))f(0))+aij(f(0)fj(xj(t)))Aij(fj(yj(t))f(0))+Aij(f(0)fj(xj(t)))=Aij(fj(yj(t))fj(xj(t)))=AijFj(ej(t)). 10.21307_ijanmc-2019-001-eqn20.jpg

Then complete the proof of part (i). In the similar way, part(ii) can be easily hold.

III. MAIN RESULTS

We present the exponential stability results for the synchronization error system of FMNN when the error system (9) is exponentially stable, the system (4) and system (7) will achieve the exponential synchronization.

tt0>0,i{1,2,,n} 10.21307_ijanmc-2019-001-eqn21.jpg

Theorem1 If there exist positive constant ε,η1,η2,…,ηn such that for any

(15)
(ci+ωi+ε)ηi+j=1nAijσjηj+j=1nBijρjηjexp{ετj(t)}<0, 10.21307_ijanmc-2019-001-eqn22.jpg
then the error system (9) is globally exponentially stable.

Proof: Consider Wi(t) = |ei(t)|/ηi,i = 1,2,…,n, according to the error system (9) or (14) and lemma1, we can get the following inequality

(16)
Dqei(t)ciei(t)+j=1nAijFj(ej(t))+j=1nBijGj(ej(tτj(t)))+ωiei(t)=(ci+ωi)ei(t)+j=1nAijFj(ej(t))+j=1nBijGj(ej(tτj(t))). 10.21307_ijanmc-2019-001-eqn23.jpg

Evaluating the fractional order derivative of Wi(t) along the trajectory of error system, then

(17)
DqWi(t)(ci+ωi)|ei(t)|/ηi+1/ηi[j=1nAijσj|ej(t)|+j=1nBijρj|ej(tτj(t))|]=(ci+ωi)Wi(t)+1/ηi[j=1nAijσjηjWj(t)+j=1nBijρjηjWj(tτj(t)))]. 10.21307_ijanmc-2019-001-eqn24.jpg

Define W˜i(t)=Wi(t)W¯(t0)exp{ε(tt0)},tt0>0,i=1,2,,n, where

W¯(t0)=max1in{supt0τst0|ei(s)|/ηi}. 10.21307_ijanmc-2019-001-eqn25.jpg

We will prove that W˜i(t)0,i=1,2,,n, for any tt0>0. Otherwise, since W˜i(t)0,i=1,2,,n for t ∈ [t0τ, t0], there must exist t1t0 and some ς such that DqW˜ς(t1)0 and W˜ς(t1)=0. Then

(18)
DqW˜ς(t1)(cς+ως)Wς(t1)+1/ης[j=1nAςjσjηjWj(t1)+j=1nBςjρjηjWj(tτj(t1))]+εW¯(t0)exp{ε(t1t0)}=(cς+ως)W¯(t0)exp{ε(t1t0)}+1/ης[j=1nAςjσjηjW¯(t0)exp{ε(t1t0)}+j=1nBςjρjηjW¯(t0)exp{ε(t1τj(t1)t0)}]+εW¯(t0)exp{ε(t1t0)}=(cς+ως+ε)W¯(t0)exp{ε(t1t0)}+1/ης[j=1nAςjσjηjW¯(t0)exp{ε(t1t0)}+j=1nBςjρjηjW¯(t0)exp{ε(t1τj(t1)t0)}]=(cς+ως+ε)W¯(t0)exp{ε(t1t0)}+1/ης[j=1nAςjσjηjW¯(t0)+j=1nBςjρjηjW¯(t0)exp{ετj(t1)}]exp{ε(t1t0)}. 10.21307_ijanmc-2019-001-eqn26.jpg

Moreover, from inequality(15), we have

(cς+ως+ε)+1/ηi[j=1nAijσjηj+j=1nBijσjηjexp{ετj(t)}]<0,tt0>0,i=1,2,,n, 10.21307_ijanmc-2019-001-eqn27.jpg

Therefore

(19)
(ci+ωi+ε)W¯(t0)exp{ε(t1t0)}+1/ηi[j=1nAijσjηj+j=1nBijρjηjexp{ετj(t)}]W¯(t0)exp{ε(t1t0)}<0,tt0>0,i=1,2,,n, 10.21307_ijanmc-2019-001-eqn28.jpg
so it is easy to find that DqW˜ς(t1)<0, which contradicts DqW˜ς(t1)0. That shows W˜i(t)=Wi(t)W¯(t0)exp{ε(tt0)}0,tt0>0,i=1,2,,n. Thus
|ei(t)|/ηimax1in{supt0τst0|ei(s)|/ηi}exp{ε(tt0)},tt0>0,i=1,2,,n. 10.21307_ijanmc-2019-001-eqn29.jpg

It shows

(20)
|ei(t)|ηimax1in{supt0τst0|ei(s)|/ηi}exp{ε(tt0)},tt0>0,i=1,2,,n. 10.21307_ijanmc-2019-001-eqn30.jpg

This completes the proof.

IV. NUMERICAL RESULTS

In this section, we will give two numerical examples to demonstrate our analysis on exponential synchronization of FMNN.

Example1 Consider two-dimension fractional-order memristor-based neural networks

(21)
{Dqx1(t)=c1x1(t)+a11(x1(t))f1(x1(t))+a12(x2(t))f2(x2(t))+b11(x1(tτ1(t)))g1(x1(tτ1(t)))+b12(x2(tτ2(t)))g2(x2(tτ2(t)))+I1Dqx2(t)=c2x2(t)+a21(x1(t))f1(x1(t))+a22(x2(t))f2(x2(t))+b21(x1(tτ1(t)))g1(x1(tτ1(t)))+b22(x2(tτ2(t)))g2(x2(tτ2(t)))+I2 10.21307_ijanmc-2019-001-eqn31.jpg
where c1 = c2 = 1, a11(x1(t)) = 1, a22(x2(t)) = 1.8,
a12(x2(t))={12,x2(t)0,14,x2(t)>0,a21(x1(t))={0.1,x1(t)0,0.05,x1(t)>0, 10.21307_ijanmc-2019-001-eqn32.jpg
b11(x1(tτ1(t)))={1.2,x1(tτ1(t))0,1.5,x1(tτ1(t))>0,b12(x2(tτ2(t)))={0.8,x2(tτ2(t))0,1.0,x2(tτ2(t))>0,b21(x1(tτ1(t)))={0.05,x1(tτ1(t))0,0.1,x1(tτ1(t))>0,b22(x2(tτ2(t)))={1.6,x2(tτ2(t))0,1.4,x2(tτ2(t))>0, 10.21307_ijanmc-2019-001-eqn33.jpg
where τj(t) = et/1 + et, I = (I1,I2)T = (0,0)T, q = 0.92 and take the activation function as fi(xi) = sin(xi), gi(xi) = 0.5(|xi + 1| − |xi − 1|), i, j = 1,2. The model (21) has chaotic attractors with initial values x(0) = (0.45,0.65)T which can be seen in Figure1.

We consider system (21) as the drive system and corresponding response system is defined as Eq.(7). And for the controller ui(t) = ωi(yi(t) − xi(t)), the parameter ωi is chosen as ω1 = −9.5, ω2 = −10.5. From Theorem1, when we take ε = 0.7, τj(t) = 1, η1 = η2 = ρ1 = ρ2 = σ1= σ2 = 0.1, we can easily know (ci+ωi+ε)ηi+j=1nAijσjηj+j=1nBijρjηjexp{ετj(t)}<0 ω1 = −9.5, ω2 = −10.5, we can get is true when ω1 < −1.703, ω2 < −0.232. So when

(c1+ω1+ε)η1+A11σ1η1+A12σ2η2+B11ρ1η1exp{ετ1(t)}+B12ρ2η2exp{ετ2(t)}=0.798<0,(c2+ω2+ε)η2+A21σ1η1+A22σ2η2+B21ρ1η1exp{ετ1(t)}+B22ρ2η2exp{ετ2(t)}=1.027<0. 10.21307_ijanmc-2019-001-eqn34.jpg

It satisfies the condition of Theorem 1, then the exponential synchronization of drive-response system is achieved.

When the response system with this controller, we get state trajectories of variable x1(t), y1(t) and x2(t), y2(t) are depicted in Figure2a and 2b. Moreover, Figure3a and 3b depict the synchronization error curves e1(t), e2(t) between the drive system and response system. These numerical simulations show the state trajectories of variable x1(t), y1(t) and x2(t), y2(t) are synchronous and synchronization error e1(t), e2(t) are converge to zero. These prove the correctness of the Theorem1.

Figure 1.

The chaotic attractors of fractional-order memristor-based neural networks(18)

10.21307_ijanmc-2019-001-f001.jpg
Figure 2.

Exponential synchronization of state variable with cntroller (a: x1(t),y1(t),b : x2(t),y2(t))

10.21307_ijanmc-2019-001-f002.jpg
Figure 3.

Synchronization error between the drive and response system (a : e1(t),b : e2(t))

10.21307_ijanmc-2019-001-f003.jpg

Example2 Consider three-dimension fractional-order memristor-based neural networks

(22)
{Dqx1(t)=c1x1(t)+a11(x1(t))f1(x1(t))+a12(x2(t))f2(x2(t))+a13(x3(t))f3(x3(t))+b11(x1(tτ1(t)))g1(x1(tτ1(t)))+b12(x2(tτ2(t)))g2(x2(tτ2(t)))+b13(x3(tτ3(t)))g3(x3(tτ3(t)))+I1Dqx2(t)=c2x2(t)+a21(x1(t))f1(x1(t))+a22(x2(t))f2(x2(t))+a23(x3(t))f3(x3(t))+b21(x1(tτ1(t)))g1(x1(tτ1(t)))+b22(x2(tτ2(t)))g2(x2(tτ2(t)))+b23(x3(tτ3(t)))g3(x3(tτ3(t)))+I2Dqx3(t)=c3x3(t)+a31(x1(t))f1(x1(t))+a32(x2(t))f2(x2(t))+a33(x3(t))f3(x3(t))+b31(x1(tτ1(t)))g1(x1(tτ1(t)))+b32(x2(tτ2(t)))g2(x2(tτ2(t)))+b33(x3(tτ3(t)))g3(x3(tτ3(t)))+I3 10.21307_ijanmc-2019-001-eqn35.jpg
where c1 = c2 = c3 = 1,
a11(x1(t))={1,x1(t)0,1x1(t)>0,a21(x1(t))={1,x1(t)0,1x1(t)>0,a31(x1(t))={1,x1(t)0,1x1(t)>0,a12(x2(t))={1,x2(t)0,1x2(t)>0,a22(x2(t))={1,x2(t)0,1x2(t)>0,a32(x2(t))={1,x2(t)0,1x2(t)>0,a13(x3(t))={1,x3(t)0,1x3(t)>0,a23(x3(t))={1,x3(t)0,1x3(t)>0,a33(x3(t))={1,x3(t)0,1x3(t)>0, 10.21307_ijanmc-2019-001-eqn36.jpg
b11(x1(tτ1(t)))={1,x1(tτ1(t))0,1x1(tτ1(t))>0,b12(x2(tτ2(t)))={1,x2(tτ2(t))0,1x2(tτ2(t))>0,b21(x1(tτ1(t)))={1,x1(tτ1(t))0,1x1(tτ1(t))>0,b22(x2(tτ2(t)))={1,x2(tτ2(t))0,1x2(tτ2(t))>0,b31(x2(tτ2(t)))={1,x2(tτ2(t))0,1x2(tτ2(t))>0,b32(x2(tτ2(t)))={1,x2(tτ2(t))0,1x2(tτ2(t))>0,b13(x3(tτ3(t)))={1,x3(tτ3(t))0,1x3(tτ3(t))>0,b23(x3(tτ3(t)))={1,x3(tτ3(t))0,1x3(tτ3(t))>0,b33(x3(tτ3(t)))={1,x3(tτ3(t))0,1x3(tτ3(t))>0. 10.21307_ijanmc-2019-001-eqn37.jpg

And τj(t) = et/1 + et, I = (I1,I2,I3)T = (0,0,0)T q = 0.92 and take the activation function as fi(xi) = gi(xi) = tanh(xi),i = 1,2,3. We consider system(22) as the drive system and the corresponding response system is defined in Eq.(7). And for the controller ui(t) = ωi(yi(t) − xi(t)), ωi is chosen as ω1 = −9.5, ω2 = −10.5, ω3 = −11. From Theorem1, we take ε = 0.7, τj(t) = 1 and choose η1 = η2 = 0.1 ρ1 = ρ2 = σ1 = σ2 = 0.1. According to

Aij=max{|aij|,|aij|},Bij=max{|bij|,|bij|} 10.21307_ijanmc-2019-001-eqn38.jpg
i,j = 1,2,3 Aij = Bij = 1, we can easily know
(ci+ωi+ε)ηi+j=1nAijσjηj+j=1nBijρjηjexp{ετj(t)}<0 10.21307_ijanmc-2019-001-eqn39.jpg
is true when ωi < −0.604. So when ω1 = −9.5, ω2 = −10.5, ω3 = −11 we can get
(c1+ω1+ε)η1+A11σ1η1+A12σ2η2+A13σ3η3+(B11ρ1η1+B12ρ2η2+B13ρ3η3)exp{ετj(t)}=0.89<0,(c2+ω2+ε)η2+A21σ1η1+A22σ2η2+A23σ3η3+(B21ρ1η1+B22ρ2η2+B23ρ3η3)exp{ετj(t)}=0.99<0,(c3+ω3+ε)η1+A31σ1η1+A32σ2η2+A33σ3η3+(B31ρ1η1+B32ρ2η2+B33ρ3η3)exp{ετj(t)}=1.04<0. 10.21307_ijanmc-2019-001-eqn40.jpg

It suggests the condition of Theorem 1 is satisfied, then drive-response system achieves the synchronization.

When the response system with this controller, we get state trajectories of variable x1(t), y1(t) and x2(t), y2(t) and x3(t), y3(t) are depicted in Figure 4a,4b,4c. Moreover, Figure 5a,5b,5c depict the synchronization error curves e1(t), e2(t), e3(t) between the drive system and response system. It’s easy to see that the state trajectories of variable x1(t), y1(t), x2(t), y2(t), and x3(t), y3(t) are synchronous and synchronization error e1(t), e2(t), e3(t) are converge to zero. So the Theorem1 is proved to be correct.

In addition, we choose ω1 = −9.5, ω2 = −10.5, ω3 = −11, according to the Theorem1, it needs the following inequalities to hold:

{τ<1εln(1173103ε)τ<1εln(1123103ε)τ<1εln(1023103ε) 10.21307_ijanmc-2019-001-eqn41.jpg

So, we just need τ<1εln(1023103ε) holds. We have the exponential convergence rate 0 < ε < 1, figure 6 depicts the relation of time-varying delay τ and exponential convergence rate ε.

Figure 4.

Synchronization of state variable with controller (a:x1(t),y1(t),b:x2(t),y2(t),c:x3(t),y3(t))

10.21307_ijanmc-2019-001-f004.jpg
Figure 5.

Synchronization error between the drive and response system (a:e1(t),b:e2(t),c:e3(t))

10.21307_ijanmc-2019-001-f005.jpg
Figure 6.

The relation of time-varying delay τ and exponential convergence rate ε.

10.21307_ijanmc-2019-001-f006.jpg

V. CONCLUSION

This paper achieves the exponential synchronization of a class of FMNN with time-varying delays by using linear error feedback controller. Based on comparison principle, the new theorem is derived to guarantee the exponential synchronization between the drive system and response system. The methods proposed for synchronization is effective and it is easy to achieve than other complex control methods. Moreover, it can be extended to investigate other dynamical behaviors of fractional-order memristive neural networks, such as realizing the lag synchronization or anti-synchronizaton of this system based on the suitable controller. These issues will be the topic of future research. Finally, numerical examples are given to illustrate the effectiveness of the proposed theory.

References


  1. L.O. Chua, “Memristor-the missing circuit element,” IEEE Trans Circuit Theory, vol.18, pp.507–519, 1971.
    [CROSSREF]
  2. D.B. Strukov, G.S. Snider and D.R. Stewart, “The missing memristor found,” Nature, vol.453, pp.80-83, 2008.
    [CROSSREF]
  3. L.O. Chua, “Resistance switching memories are memristors,” Applied Physics A Materials Science and Processing, vol.102,pp.765-783, 2011.
    [CROSSREF]
  4. M.J. Sharifiy and Y.M. Banadaki, “General spice models for memristor and application to circuit simulation of memristor-based synapses and memory cells,” J Circuit Syst Comp, vol.19, pp.407-424, 2010.
    [CROSSREF]
  5. Y.V. Pershin, V.M. Di, “Experimental demonstration of associative memory with memristive neural networks,” Neural Networks, vol.23, pp.881-886, 2010.
    [CROSSREF]
  6. M. Itoh and L.O. Chua, “Memristor cellular automata and memristor discrete-time cellular neural networks,” Int J Bifur Chaos,vol.19, pp. 3605–3656, 2010.
  7. G.D. Zhang, Y. Shen and J. Sun, “Global exponential stability of a class of memristor-based recurrent neural networks with time-varying delays,” Neurocomputing, vol.97, pp.149–154, 2012.
    [CROSSREF]
  8. A.L. Wu and Z.G. Zeng, “New global exponential stability results for a memristive neural system with time-varying delays,” Neurocomputing, vol.144, pp.553–559, 2014.
    [CROSSREF]
  9. A.L. Wu and Z.G. Zeng. “Exponential stabilization of memristive neural networks with time delays,” IEEE Trans on Neural Netw and Learn Syst, vol.23, pp.1919–1929, 2012.
    [CROSSREF]
  10. A.L. Wu and Z.G. Zeng, “Lagrange stability of neural networks with memristive synapses and multiple delays,” Inf Sci, vol.280, pp.135–151, 2014.
    [CROSSREF]
  11. G.D. Zhang, Y. Shen, Q. Yin and J.W. Sun, “Global exponential periodicity and stability of a class of memristor-based recurrent neural networks with multiple delays,”Inf Sci, vol.232, pp.386–396, 2013.
    [CROSSREF]
  12. J.J. Chen, Z.G. Zeng and P. Jiang, “On the periodic dynamics of memristor-based neural networks with time-varying delays,” Inf Sci, vol.279, pp.358–373, 2014.
    [CROSSREF]
  13. L. Duan and L.H. Huang, “Existence and stability of periodic solution for mixed time-varying delayed neural networks with discontinuous activations,” Neurocomputing, vol.123, pp.255–265, 2014.
    [CROSSREF]
  14. G. Zhang, J. Hu and Y. Shen, “New results on synchronization control of delayed memristive neural networks,” Nonlinear Dynamics, vol.81, pp. 1167-1178, 2015.
    [CROSSREF]
  15. X. Yang, J.D. Cao and W. Yu, “Exponential synchronization of memristive Cohen–Grossberg neural networks with mixed delays,” Cogn Neurodyn, vol.8, pp. 239–249, 2014.
    [CROSSREF]
  16. G.D. Zhang and Y. Shen, “Exponential synchronization of delayed memristor-based chaotic neural networks via periodically intermittent control,” Neural Networks, vol.55, pp.1-10, 2014.
    [CROSSREF]
  17. A. Dai, W. Zhou and Y. Xu, “Adaptive exponential synchronization in mean square for Markovian jumping neutral-type coupled neural networks with time-varying delays by pinning control,” Neurocomputing, vol.173, pp.809-818, 2016.
    [CROSSREF]
  18. Z. Cai, L. Huang L and L. Zhang, “New conditions on synchronization of memristor-based neural networks via differential inclusions,” Neurocomputing, vol.186, pp.235-250, 2016.
    [CROSSREF]
  19. J.N. Li, W.D. Bao and S.B. Li, “Exponential synchronization of discrete-time mixed delay neural networks with actuator constraints and stochastic missing data,” Neurocomputing, vol.207, pp. 700-707, 2016.
    [CROSSREF]
  20. X. Han, H. Wu and B. Fang, “Adaptive Exponential Synchronization of Memristive Neural Networks with mixed time-varying delays,” Neurocomputing, vol.201, pp.40-50, 2016.
    [CROSSREF]
  21. W. Zhang, C. Li and T. Huang, “Stability and synchronization of memristor-based coupling neural networks with time-varying delays via intermittent control,” Neurocomputing, vol.173, pp. 1066-1072, 2016.
    [CROSSREF]
  22. A. Abdurahman, H. Jiang and Z. Teng, “Exponential lag synchronization for memristor-based neural networks with mixed time delays via hybrid switching control,” Journal of the Franklin Institute, vol.353, pp.2859-2880, 2016.
    [CROSSREF]
  23. X.D. Li, R. Rakkiyappan and G. Velmurugan, “Passivity analysis of memristor-based complex-valued neural networks with time-varying delays,” Neural Processing Letters, vol.42, pp.517-540, 2014.
  24. L. Duan and L.H. Huang, “Periodicity and dissipativity for memristor-based mixed time-varying delayed neural networks via differential inclusions,” Neural Networks, vol.57, pp.12-22, 2014.
    [CROSSREF]
  25. Z.Y. Guo, J. Wang J and Z. Yan, “Global exponential dissipativity and stabilization of memristor-based recurrent neural networks with time-varying delays,” Neural Networks, vol.48, pp.158-172, 2013.
    [CROSSREF]
  26. Z.Y. Guo, J. Wang and Z. Yan, “Attractivity analysis of memristor-based cellular neural networks with time-varying delays,” IEEE Trans on Neural Networks and Learning System, vol.25, pp.704-717, 2014.
    [CROSSREF]
  27. M. Jiang, S. Wang and J. Mei, “Finite-time synchronization control of a class of memristor-based recurrent neural networks,” Neural Networks, vol.63, pp.133-140, 2015.
    [CROSSREF]
  28. A. Chandrasekar, A. Rakkiyappan and J.D. Cao, “Synchronization of memristor-based recurrent neural networks with two delay components based on second-order reciprocally convex approach,” Neural Networks, vol.57, pp.79-93, 2014.
    [CROSSREF]
  29. R. Rakkiyappan, G. Velmurugan and J.D. Cao, “Finite-time stability analysis of fractional-order complex-valued memristor-based neural networks with time delays,”Nonlinear Dynamics, vol.78, pp.2823-2836, 2014.
    [CROSSREF]
  30. N. Li and J.D. Cao, “New synchronization criteria for memristor-based networks: Adaptive control and feedback control schemes,” Neural Networks, vol.61, pp.1-9, 2015.
    [CROSSREF]
  31. M.D. Paola, F.P. Pinnola and M. Zingales, “Fractional differential equations and related exact mechanical models,” Computers and Mathematics with Applications, vol.66, pp. 608-620, 2013.
    [CROSSREF]
  32. V. Lakshmikantham and A.S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis Theory Methods and Applications, vol.69, pp.2677-2682, 2008.
    [CROSSREF]
  33. H.B. Bao and J.D. Cao, “Projective synchronization of fractional-order memristor-based neural networks,” Neural Networks, vol.63, pp.1-9, 2015.
    [CROSSREF]
  34. G. Velmurugan, R. Rakkiyappan and J.D. Cao, “Finite-time synchronization of fractional-order memristor-based neural networks with time delays,” Nonlinear Dynamics, vol.73, pp.36-46, 2016.
  35. G. Velmurugan and R. Rakkiyappan, “Hybrid projective synchronization of fractional-order memristor-based neural networks with time delays,” Nonlinear Dynamics, vol.11, pp.1-14, 2015.
  36. Bao H, Ju HP, Cao JD. Adaptive synchronization of fractional-order memristor-based neural networks with time delay [J]. Nonlinear Dynamics, 2015, 82:1-12.
    [CROSSREF]
  37. L. Chen, R. Wu and J.D. Cao, “Stability and synchronization of memristor-based fractional-order delayed neural networks,” Neural Networks the Official Journal of the International Neural Network Society, vol.71, pp.37-44, 2015.
    [CROSSREF]
  38. Y. Gu, Y. Yu and H. Wang, “Synchronization for fractional-order time-delayed memristor-based neural networks with parameter uncertainty,” Journal of the Franklin Institute, vol.353, pp.3657-3684, 2016.
    [CROSSREF]
  39. A. Abdurahman, H. Jiang and Z. Teng, “Finite-time synchronization for memristor-based neural networks with time-varying delays,” Neural Networks, vol.69, pp.20-28, 2015.
    [CROSSREF]
  40. X. Yang, J. Cao and W. Yu, “Exponential synchronization of memristive Cohen-Grossberg neural networks with mixed delays,” Cognitive Neurodynamics, vol.8, pp.239-249, 2014.
    [CROSSREF]
  41. J. Perez-Cruz, E. Portilla-Flores and P. Niño-Suarez, “Design of a nonlinear controller and its intelligent optimization for exponential synchronization of a new chaotic system,” Optik -International Journal for Light and Electron Optics, vol.130, pp.201-212, 2016.
    [CROSSREF]
  42. Q. Gan, “Exponential synchronization of stochastic neural networks with leakage delay and reaction-diffusion terms via periodically intermittent control,” Neural Processing Letters, vol.22, pp.393-410, 2013.
    [CROSSREF]
  43. S. Tyagi, S. Abbas and M. Kirane, “Global asymptotic and exponential synchronization of ring neural network with reaction-diffusion term and unbounded delay,” Neural Computing and Applications, pp.1-15, 2016.
  44. J.P. Aubin and A. Cellina, “Differential inclusions set-valued maps and viability theory,” Acta Applicandae Mathematica, vol.6, pp.215-217, 1986.
    [CROSSREF]
XML PDF Share

FIGURES & TABLES

Figure 1.

The chaotic attractors of fractional-order memristor-based neural networks(18)

Full Size   |   Slide (.pptx)

Figure 2.

Exponential synchronization of state variable with cntroller (a: x1(t),y1(t),b : x2(t),y2(t))

Full Size   |   Slide (.pptx)

Figure 3.

Synchronization error between the drive and response system (a : e1(t),b : e2(t))

Full Size   |   Slide (.pptx)

Figure 4.

Synchronization of state variable with controller (a:x1(t),y1(t),b:x2(t),y2(t),c:x3(t),y3(t))

Full Size   |   Slide (.pptx)

Figure 5.

Synchronization error between the drive and response system (a:e1(t),b:e2(t),c:e3(t))

Full Size   |   Slide (.pptx)

Figure 6.

The relation of time-varying delay τ and exponential convergence rate ε.

Full Size   |   Slide (.pptx)

REFERENCES

  1. L.O. Chua, “Memristor-the missing circuit element,” IEEE Trans Circuit Theory, vol.18, pp.507–519, 1971.
    [CROSSREF]
  2. D.B. Strukov, G.S. Snider and D.R. Stewart, “The missing memristor found,” Nature, vol.453, pp.80-83, 2008.
    [CROSSREF]
  3. L.O. Chua, “Resistance switching memories are memristors,” Applied Physics A Materials Science and Processing, vol.102,pp.765-783, 2011.
    [CROSSREF]
  4. M.J. Sharifiy and Y.M. Banadaki, “General spice models for memristor and application to circuit simulation of memristor-based synapses and memory cells,” J Circuit Syst Comp, vol.19, pp.407-424, 2010.
    [CROSSREF]
  5. Y.V. Pershin, V.M. Di, “Experimental demonstration of associative memory with memristive neural networks,” Neural Networks, vol.23, pp.881-886, 2010.
    [CROSSREF]
  6. M. Itoh and L.O. Chua, “Memristor cellular automata and memristor discrete-time cellular neural networks,” Int J Bifur Chaos,vol.19, pp. 3605–3656, 2010.
  7. G.D. Zhang, Y. Shen and J. Sun, “Global exponential stability of a class of memristor-based recurrent neural networks with time-varying delays,” Neurocomputing, vol.97, pp.149–154, 2012.
    [CROSSREF]
  8. A.L. Wu and Z.G. Zeng, “New global exponential stability results for a memristive neural system with time-varying delays,” Neurocomputing, vol.144, pp.553–559, 2014.
    [CROSSREF]
  9. A.L. Wu and Z.G. Zeng. “Exponential stabilization of memristive neural networks with time delays,” IEEE Trans on Neural Netw and Learn Syst, vol.23, pp.1919–1929, 2012.
    [CROSSREF]
  10. A.L. Wu and Z.G. Zeng, “Lagrange stability of neural networks with memristive synapses and multiple delays,” Inf Sci, vol.280, pp.135–151, 2014.
    [CROSSREF]
  11. G.D. Zhang, Y. Shen, Q. Yin and J.W. Sun, “Global exponential periodicity and stability of a class of memristor-based recurrent neural networks with multiple delays,”Inf Sci, vol.232, pp.386–396, 2013.
    [CROSSREF]
  12. J.J. Chen, Z.G. Zeng and P. Jiang, “On the periodic dynamics of memristor-based neural networks with time-varying delays,” Inf Sci, vol.279, pp.358–373, 2014.
    [CROSSREF]
  13. L. Duan and L.H. Huang, “Existence and stability of periodic solution for mixed time-varying delayed neural networks with discontinuous activations,” Neurocomputing, vol.123, pp.255–265, 2014.
    [CROSSREF]
  14. G. Zhang, J. Hu and Y. Shen, “New results on synchronization control of delayed memristive neural networks,” Nonlinear Dynamics, vol.81, pp. 1167-1178, 2015.
    [CROSSREF]
  15. X. Yang, J.D. Cao and W. Yu, “Exponential synchronization of memristive Cohen–Grossberg neural networks with mixed delays,” Cogn Neurodyn, vol.8, pp. 239–249, 2014.
    [CROSSREF]
  16. G.D. Zhang and Y. Shen, “Exponential synchronization of delayed memristor-based chaotic neural networks via periodically intermittent control,” Neural Networks, vol.55, pp.1-10, 2014.
    [CROSSREF]
  17. A. Dai, W. Zhou and Y. Xu, “Adaptive exponential synchronization in mean square for Markovian jumping neutral-type coupled neural networks with time-varying delays by pinning control,” Neurocomputing, vol.173, pp.809-818, 2016.
    [CROSSREF]
  18. Z. Cai, L. Huang L and L. Zhang, “New conditions on synchronization of memristor-based neural networks via differential inclusions,” Neurocomputing, vol.186, pp.235-250, 2016.
    [CROSSREF]
  19. J.N. Li, W.D. Bao and S.B. Li, “Exponential synchronization of discrete-time mixed delay neural networks with actuator constraints and stochastic missing data,” Neurocomputing, vol.207, pp. 700-707, 2016.
    [CROSSREF]
  20. X. Han, H. Wu and B. Fang, “Adaptive Exponential Synchronization of Memristive Neural Networks with mixed time-varying delays,” Neurocomputing, vol.201, pp.40-50, 2016.
    [CROSSREF]
  21. W. Zhang, C. Li and T. Huang, “Stability and synchronization of memristor-based coupling neural networks with time-varying delays via intermittent control,” Neurocomputing, vol.173, pp. 1066-1072, 2016.
    [CROSSREF]
  22. A. Abdurahman, H. Jiang and Z. Teng, “Exponential lag synchronization for memristor-based neural networks with mixed time delays via hybrid switching control,” Journal of the Franklin Institute, vol.353, pp.2859-2880, 2016.
    [CROSSREF]
  23. X.D. Li, R. Rakkiyappan and G. Velmurugan, “Passivity analysis of memristor-based complex-valued neural networks with time-varying delays,” Neural Processing Letters, vol.42, pp.517-540, 2014.
  24. L. Duan and L.H. Huang, “Periodicity and dissipativity for memristor-based mixed time-varying delayed neural networks via differential inclusions,” Neural Networks, vol.57, pp.12-22, 2014.
    [CROSSREF]
  25. Z.Y. Guo, J. Wang J and Z. Yan, “Global exponential dissipativity and stabilization of memristor-based recurrent neural networks with time-varying delays,” Neural Networks, vol.48, pp.158-172, 2013.
    [CROSSREF]
  26. Z.Y. Guo, J. Wang and Z. Yan, “Attractivity analysis of memristor-based cellular neural networks with time-varying delays,” IEEE Trans on Neural Networks and Learning System, vol.25, pp.704-717, 2014.
    [CROSSREF]
  27. M. Jiang, S. Wang and J. Mei, “Finite-time synchronization control of a class of memristor-based recurrent neural networks,” Neural Networks, vol.63, pp.133-140, 2015.
    [CROSSREF]
  28. A. Chandrasekar, A. Rakkiyappan and J.D. Cao, “Synchronization of memristor-based recurrent neural networks with two delay components based on second-order reciprocally convex approach,” Neural Networks, vol.57, pp.79-93, 2014.
    [CROSSREF]
  29. R. Rakkiyappan, G. Velmurugan and J.D. Cao, “Finite-time stability analysis of fractional-order complex-valued memristor-based neural networks with time delays,”Nonlinear Dynamics, vol.78, pp.2823-2836, 2014.
    [CROSSREF]
  30. N. Li and J.D. Cao, “New synchronization criteria for memristor-based networks: Adaptive control and feedback control schemes,” Neural Networks, vol.61, pp.1-9, 2015.
    [CROSSREF]
  31. M.D. Paola, F.P. Pinnola and M. Zingales, “Fractional differential equations and related exact mechanical models,” Computers and Mathematics with Applications, vol.66, pp. 608-620, 2013.
    [CROSSREF]
  32. V. Lakshmikantham and A.S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis Theory Methods and Applications, vol.69, pp.2677-2682, 2008.
    [CROSSREF]
  33. H.B. Bao and J.D. Cao, “Projective synchronization of fractional-order memristor-based neural networks,” Neural Networks, vol.63, pp.1-9, 2015.
    [CROSSREF]
  34. G. Velmurugan, R. Rakkiyappan and J.D. Cao, “Finite-time synchronization of fractional-order memristor-based neural networks with time delays,” Nonlinear Dynamics, vol.73, pp.36-46, 2016.
  35. G. Velmurugan and R. Rakkiyappan, “Hybrid projective synchronization of fractional-order memristor-based neural networks with time delays,” Nonlinear Dynamics, vol.11, pp.1-14, 2015.
  36. Bao H, Ju HP, Cao JD. Adaptive synchronization of fractional-order memristor-based neural networks with time delay [J]. Nonlinear Dynamics, 2015, 82:1-12.
    [CROSSREF]
  37. L. Chen, R. Wu and J.D. Cao, “Stability and synchronization of memristor-based fractional-order delayed neural networks,” Neural Networks the Official Journal of the International Neural Network Society, vol.71, pp.37-44, 2015.
    [CROSSREF]
  38. Y. Gu, Y. Yu and H. Wang, “Synchronization for fractional-order time-delayed memristor-based neural networks with parameter uncertainty,” Journal of the Franklin Institute, vol.353, pp.3657-3684, 2016.
    [CROSSREF]
  39. A. Abdurahman, H. Jiang and Z. Teng, “Finite-time synchronization for memristor-based neural networks with time-varying delays,” Neural Networks, vol.69, pp.20-28, 2015.
    [CROSSREF]
  40. X. Yang, J. Cao and W. Yu, “Exponential synchronization of memristive Cohen-Grossberg neural networks with mixed delays,” Cognitive Neurodynamics, vol.8, pp.239-249, 2014.
    [CROSSREF]
  41. J. Perez-Cruz, E. Portilla-Flores and P. Niño-Suarez, “Design of a nonlinear controller and its intelligent optimization for exponential synchronization of a new chaotic system,” Optik -International Journal for Light and Electron Optics, vol.130, pp.201-212, 2016.
    [CROSSREF]
  42. Q. Gan, “Exponential synchronization of stochastic neural networks with leakage delay and reaction-diffusion terms via periodically intermittent control,” Neural Processing Letters, vol.22, pp.393-410, 2013.
    [CROSSREF]
  43. S. Tyagi, S. Abbas and M. Kirane, “Global asymptotic and exponential synchronization of ring neural network with reaction-diffusion term and unbounded delay,” Neural Computing and Applications, pp.1-15, 2016.
  44. J.P. Aubin and A. Cellina, “Differential inclusions set-valued maps and viability theory,” Acta Applicandae Mathematica, vol.6, pp.215-217, 1986.
    [CROSSREF]

EXTRA FILES

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