Neural Networks 55 (2014) 1–10

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Exponential synchronization of delayed memristor-based chaotic neural networks via periodically intermittent control Guodong Zhang, Yi Shen ∗ School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Wuhan 430074, China

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Article history: Received 25 December 2013 Received in revised form 10 March 2014 Accepted 20 March 2014 Keywords: Chaotic dynamical systems Exponential synchronization Memristor Time-varying delays Intermittent control

abstract This paper investigates the exponential synchronization of coupled memristor-based chaotic neural networks with both time-varying delays and general activation functions. And here, we adopt nonsmooth analysis and control theory to handle memristor-based chaotic neural networks with discontinuous righthand side. In particular, several new criteria ensuring exponential synchronization of two memristorbased chaotic neural networks are obtained via periodically intermittent control. In addition, the new proposed results here are very easy to verify and also complement, extend the earlier publications. Numerical simulations on the chaotic systems are presented to illustrate the effectiveness of the theoretical results. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction Memristor-based neural networks made of hybrid complementary metal–oxide–semiconductors have a very wide range of uses in bioinspired engineering (Cantley, Subramaniam, Stiegler, Chapman, & Vogel, 2012; Itoh & Chua, 2009; Kim, Sah, Yang, Roska, & Chua, 2012; Pershin & Di Ventra, 2010; Sharifiy & Banadaki, 2010). Memristor-based neural networks are well suited to characterize the nonvolatile feature of the memory cell because of hysteresis effects. The studies of memristor-based neural networks would benefit a number of important applications in neural learning circuits (Cantley et al., 2012; Itoh & Chua, 2009; Sharifiy & Banadaki, 2010), associative memories (Pershin & Di Ventra, 2010), new classes of artificial neural systems (Hu & Wang, 2010; Kim et al., 2012), and so on. The memristor-based neural networks are a class of statedependent nonlinear systems from a systems-theoretic point of view (Bao & Zeng, 2013; Chen, Zeng, & Jiang, 2014; Hu & Wang, 2010; Wen & Zeng, 2012; Wu, Wen, & Zeng, 2012; Wu & Zeng, 2012; Yang, Cao, & Yu, 2014; Zhang & Shen, 2013; Zhang, Shen, & Sun, 2012; Zhang, Shen, & Wang, 2013). Such a system can reveal coexisting solutions, jumped, transient chaos of rich and

∗ Corresponding author at: School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China. Tel.: +86 27 87543630; fax: +86 27 87543130. E-mail addresses: [email protected] (G. Zhang), [email protected] (Y. Shen). http://dx.doi.org/10.1016/j.neunet.2014.03.009 0893-6080/© 2014 Elsevier Ltd. All rights reserved.

complex nonlinear behaviors, whereas, in the past decades, statedependent nonlinear system has not received considerable attention. With the development and application of memristors, the studies of such state-dependent nonlinear system with its various generalizations may be an active area of research, to allow the memristors to be readily used in emerging technologies. As is well known, chaos synchronization in nonlinear science has been known for a rather long time, and its applications to diverse areas such as secure communications and biological and chemical reactions. Since then, many important and fundamental results have been reported on the synchronization and control of chaotic systems, e.g, see Cao and Wan (2014), Liu (2009) and Liu, Wang, and Liu (2008). And many control approaches have been proposed to stabilize chaotic systems such as adaptive control (Zhang, Xie, Wang, & Zheng, 2007), feedback control (Zhu, Zhang, Fei, Zhang, & Li, 2009), impulsive control (Guan & Zhang, 2008; Sheng & Yang, 2008; Sun, Chen, Lu, & Chen, 2012), and intermittent control (Cai, Liu, Xu, & Shen, 2009; Hu, Yu, Jiang, & Teng, 2010a; Huang & Li, 2010; Huang, Li, & Liu, 2008; Huang, Li, Yu, & Chen, 2009; Yu, Hu, Jiang, & Teng, 2011). Comparing with continuous control of chaos, the discontinuous control method, such as impulsive control and intermittent control, have received much interest because they are practical and easily implemented in engineering such as transportation and communication (Hu et al., 2010a; Huang, Li, Yu et al., 2009). The intermittent control is different from the impulsive control since impulsive control is activated only at some isolated instants, while intermittent control has a nonzero control width. In this scheme,

2

G. Zhang, Y. Shen / Neural Networks 55 (2014) 1–10

the synchronization signals are used in the slave system at periodic time intervals (control width) when the slave system tracks the orbit of the driving system (Hu et al., 2010a; Huang et al., 2008; Huang, Li, Yu et al., 2009). Moreover, to use intermittent control proves to be more cost effective than using control at all times (Amritkar & Gupte, 1993; Hu et al., 2010a; Huang, Li, Yu et al., 2009). After several decades, numerous studies with respect to intermittent control have been carried out (Hu, Yu, Jiang, & Teng, 2010b; Huang, Li, & Han, 2009; Li & Cao, 2014; Li, Feng, & Liao, 2007; Li, Liao, & Huang, 2007; Xia & Cao, 2009; Yang & Cao, 2009). However, on the synchronization of memristor-based chaotic neural networks via intermittent control, few results are found in the literature. On the other hand, the memristor-based chaotic neural networks with discontinuous right-hand side, this problem brings challenges to investigate the exponential synchronization of the systems via intermittent control. Recently, Bao, Liu, & Xu (2010a, 2010b) show that the memristor-based chaotic system is more safe in secure communications. Therefore, using intermittent control to get synchronization of memristor-based chaotic neural networks is also more safe in secure communications, because intermittent control is the discontinuous control method, which can increase the difficulty when capturing the information by sending a periodic signal. Motivated by the above discussions, in this paper, we will derive several new criteria ensuring exponential synchronization of memristor-based chaotic neural networks with both timevarying delays and general activation functions via periodically intermittent control. The main advantages of this paper lie in the following aspects. Firstly, the dynamic analysis here adopts nonsmooth analysis and control theory to handle memristor based neural networks with discontinuous right-hand side. Secondly, periodically intermittent controller technique, which is totally different from the techniques employed in Bao and Zeng (2013), Chen et al. (2014), Hu and Wang (2010), Wen and Zeng (2012), Wu et al. (2012), Wu and Zeng (2012), Yang et al. (2014), Zhang et al. (2012), Zhang et al. (2013) and Zhang and Shen (2013) is to study the addressed neural networks in the paper. Thirdly, as the generalization of the obtained results, exponential synchronization of addressed neural networks under various feedback functions are discussed in detail. Lastly, some new criteria are derived to ensure synchronization of the neural networks, and the new proposed results here are very easy to verify and they achieve a valuable improvement, and also complement, and extend the earlier publications. The organization of this paper is as follows. Some preliminaries are introduced in Section 2. In Section 3, some new criteria for the exponential synchronization are derived by using nonsmooth analysis and control theory. And then, numerical simulations are given to demonstrate the effectiveness of the proposed approach in Section 4. Finally, our conclusion is given in Section 5.

where

dxi (t ) dt

= −xi (t ) +

n 

aij (xj (t ))fj (xj (t ))

j =1

+

n 

bij (xj (t − τj (t )))gj (xj (t − τj (t ))) + Ii ,

j =1

t ≥ 0, i ∈ N ,

× sgnij , Wij Ci

× sgnij

where sgnij = 1, if i ̸= j holds, otherwise, −1. Mij and Wij denote the memductances of memristors Rij and  Rij , respectively. In addition, Rij represents the memristor between the neuron activation functions fj (xj (t )) and xi (t ),  Rij represents the memristor between the neuron activation functions gj (xj (t − τj (t ))) and xi (t ). And aij (xj (t )), bij (xj (t − τj (t ))) are memristors synaptic connection weights, denote the strengths of the jth unit on the ith unit at time t and time t −τj (t ), respectively. fj , gj : R → R denotes the neuron activation functions, τj (t ) corresponds to the transmission delays and satisfies 0 ≤ τj (t ) ≤ τ , τ˙j (t ) ≤ σ0 < 1 (τ > 0, σ0 is a constant), Ii is an external constant input, i, j ∈ N , N = 1, 2, . . . , n. As we know, capacitor Ci is changeless, memductances Mij and Wij respond to changes in pinched hysteresis loops. Thus, aij (xj (t )), bij (xj (t − τj (t ))) will change, as pinched hysteresis loops change (Hu & Wang, 2010; Wen & Zeng, 2012; Wu et al., 2012; Wu & Zeng, 2012; Zhang et al., 2012, 2013). According to the feature of the memristor and the current–voltage characteristic, then

|xj (t )| ≤ Tj , aij , |xj (t )| > Tj ,  ∗ bij , |xj (t − τj (t ))| ≤ Tj , bij (xj (t − τj (t ))) = b∗∗ |xj (t − τj (t ))| > Tj , ij , aij (xj (t )) =



a∗ij , ∗∗

∗ ∗∗ in which switching jumps Tj > 0, a∗ij , a∗∗ ij , bij , bij , i, j ∈ N, are all constant numbers. Throughout this paper, we consider system (1) as the drive system and corresponding response system is as follows:

dyi (t ) dt

= −yi (t ) +

n 

aij (yj (t ))fj (yj (t ))

j=1

+

n 

bij (yj (t − τj (t )))gj (yj (t − τj (t )))

j =1

+ Ii + ui (t ),

t ≥ 0, i ∈ N ,

(2)

where i, j ∈ N,

|yj (t )| ≤ Tj , aij , |yj (t )| > Tj ,  ∗ bij , |yj (t − τj (t ))| ≤ Tj , bij (yj (t − τj (t ))) = ∗∗ bij , |yj (t − τj (t ))| > Tj , aij (yj (t )) =

a∗ij ,



∗∗

and ui (t ) is a periodically intermittent controller which is defined by

ui (t ) =

 n    ωij (yj (t ) − xj (t )),

mT ≤ t ≤ mT + δ,

j =1

 

0,

(3)

mT + δ < t ≤ (m + 1)T ,

where m = 0, 1, 2, . . . , and ωij are constants for all i, j ∈ N, which denote the control gains, T denotes the control period and 0 < δ < T is called the control width. In this paper, solutions of all systems considered in the following are intended in Filippov’s sense (Filippov, 1988). We den fine ∥φ∥ = sup−τ ≤t ≤0 [ i=1 |φi (t )|p ]1/p , where p is a constant and p ≥ 1, for ∀ φ = (φ1 (t ), φ2 (t ), . . . , φn (t )) ∈ C([−τ , 0], Rn ), co{ξ , ξ i } denotes the convex hull of {ξ , ξ i }. aij = min{a∗ij , a∗∗ ij }, i

(1)

Ci

bij (xj (t − τj (t ))) =

2. Preliminaries In this paper, based on the previous works (Chen et al., 2014; Hu et al., 2010b; Wu et al., 2012; Zhang & Shen, 2013), we consider a class of memristor-based neural networks with time-varying delays as follows:

Mij

aij (xj (t )) =

i

∗ ∗∗ ∗ ∗∗ aij = max{a∗ij , a∗∗ ij }, bij = min{bij , bij }, bij = max{bij , bij }. For

G. Zhang, Y. Shen / Neural Networks 55 (2014) 1–10

a continuous function k(t ) : R → R, D+ k(t ) is called the upper right dini derivative and defined as D+ k(t ) = limh→0+ 1h (k(t + h) − k(t )). The initial conditions of systems (1) and (2) are assumed to be xi (s) = ϕi (s), yi (s) = ψi (s), respectively, s ∈ [−τ , 0], ϕi (s), ψi (s) ∈ C([−τ , 0], R), i ∈ N. Now, we do following assumption for system (1):

(H1 ) For j ∈ N , ∀s1 , s2 ∈ R, the neuron activation functions fj , gj are bounded, fj (±Tj ) = gj (±Tj ) = 0 and satisfy σj− ≤

fj (s1 ) − fj (s2 ) s1 − s2

≤ σj+ ,

ρj− ≤

gj (s1 ) − gj (s2 ) s1 − s2

≤ ρj+ ,

where s1 ̸= s2 , and σj− , σj+ , ρj− , ρj+ are constants. Let Rn be the space of n-dimensional real column vectors. For any h = (h1 , h2 , . . . , hn )T ∈ Rn , the norms are defined by ∥h∥p =  ( ni=1 |hi |p )1/p , where p ≥ 1 is a positive integer. Through the theories of differential inclusions and set-valued maps (Aubin & Cellina, 1984; Clarke, Ledyaev, Stem, & Wolenski, 1998; Filippov, 1988), from (1) and (2), it follows that dxi (t ) dt

+

∈ −xi (t ) +

n 

co[aij (xj (t ))]fj (xj (t ))

j =1 n 

co[bij (xj (t − τj (t )))]gj (xj (t − τj (t ))) + Ii ,

j =1

for a.e. t ≥ 0, i ∈ N ,

(4)

and

 n  dyi (t )   ∈ − yi (t ) + co[aij (yj (t ))]fj (yj (t ))   dt   j =1   n      + co[bij (yj (t − τj (t )))]gj (yj (t − τj (t ))) + Ii    j=1    n     + ωij (yj (t ) − xj (t )), mT ≤ t ≤ mT + δ, j=1    n     dyi (t ) ∈ −yi (t ) + co[aij (yj (t ))]fj (yj (t ))    dt  j =1   n      + co[bij (yj (t − τj (t )))]gj (yj (t − τj (t ))) + Ii ,     j=1  mT + δ < t ≤ (m + 1)T ,

(5)

dxi (t ) dt

( −xi (t ) +

n  j =1

co[aij (xj (t ))]fj (xj (t ))

for t ≥ 0, i = 1, 2, . . . , n

(8)

has nonempty compact convex values. Furthermore, it is uppersemicontinuous. Now, define the synchronization error e(t ) as follows: e(t ) = (e1 (t ), e2 (t ), . . . , en (t ))T , where ei (t ) = yi (t ) − xi (t ). And by the theories of set-valued maps and differential inclusions (Aubin & Cellina, 1984; Clarke et al., 1998; Filippov, 1988) and associated with the systems (4) and (5), then we can get the following synchronization error system:

 n  dei (t )   ∈ −ei (t ) + {co[aij (yj (t ))]fj (yj (t ))   dt  j =1     − co[aij (xj (t ))]fj (xj (t ))}   n      + {co[bij (yj (t − τj (t )))]gj (yj (t − τj (t )))     j =1    − co[bij (xj (t − τj (t )))]gj (xj (t − τj (t )))}    n     ωij ei (t ), mT ≤ t ≤ mT + δ, + j =1    n   dei (t )   ∈ − e ( t ) + {co[aij (yj (t ))]fj (yj (t ))  i   dt  j =1    − co[aij (xj (t ))]fj (xj (t ))}    n      + {co[bij (yj (t − τj (t )))]gj (yj (t − τj (t )))     j =1    − co[bij (xj (t − τj (t )))]gj (xj (t − τj (t )))},  mT + δ < t ≤ (m + 1)T .

(9)

Definition 1. A function (in Filippov’s sense) x(t ) = (x1 (t ), x2 (t ), . . . , xn (t ))T is a solution of system (1) with the initial conditions ϕ(s) = (ϕ1 (s), ϕ2 (s), . . . , ϕn (s))T ∈ C([−τ , 0], Rn ), if x(t ) is

1/p |yi (t ) − xi (t )|p

≤ β e−εt ∥ψ − ϕ∥,

for ∀ t ≥ 0,

(10)

i =1

where the constant ε is said to be the degree of exponential synchronization. (6)

Lemma 1. If assumption (H1 ) holds, then there is at least a local solution x(t ) of system (1) with initial condition ϕ(s) = (ϕ1 (s), ϕ2 (s), . . . , ϕn (s))T ∈ C((−τ , 0], Rn ), and the local solution x(t ) can be extended to the interval [0, +∞) in the sense of Filippov. In the following section, the paper aims to find some sufficient synchronization criteria for the purpose of exponentially synchronizing the unidirectional coupled identical systems (1) and (2).

(7)

It is obvious that the set-valued map

co[bij (xj (t − τj (t )))]gj (xj (t − τj (t ))) + Ii ,

j =1

 n 

 ∗ aij , |yj (t )| < Tj , ∗ ∗∗ co[aij (yj (t ))] = co{aij , aij }, |yj (t )| = Tj , a∗∗ , |y (t )| > T , j j ij  ∗ bij , |yj (t − τj (t ))| < Tj , }, |yj (t − τj (t ))| = Tj , = co{b∗ij , b∗∗ b∗∗ , |yij (t − τ (t ))| > T . j j j ij



Definition 2. Systems (1) and (2) are said to be exponentially synchronized if there exist constants β ≥ 1 and ε > 0 such that

and

co[bij (yj (t − τj (t )))]

+

an absolutely continuous function on any compact interval of [0, +∞) and satisfies the differential inclusion (4).

where

 ∗ aij , |xj (t )| < Tj , ∗ ∗∗ co[aij (xj (t ))] = co{aij , aij }, |xj (t )| = Tj , a∗∗ , |x (t )| > T , j j ij  ∗ b , | x ( t − τj (t ))| < Tj ,  ij j ∗ ∗∗ co[bij (xj (t − τj (t )))] = co{bij , bij }, |xj (t − τj (t ))| = Tj , b∗∗ , |x (t − τ (t ))| > T , j j j ij

3

n

3. Main results For convenience, the following denotations are introduced. Let

λi = p − (p − 1)

n 

pαij p−1

Aij

pβij

σj p−1 − (p − 1)

−

j =1

µi

pγij p−1

Bij

pξij

ρjp−1

j =1

j =1 n  µj

n 

p(1−αji ) p(1−βji ) Aji i

σ

,

(11)

4

G. Zhang, Y. Shen / Neural Networks 55 (2014) 1–10

κi = p − (p − 1)

pα ij p−1

n 

Aij

pβ ij

σj p−1 − (p − 1)

j =1

−

n  µj

µi

j =1

p(1−α ji )

p(1−β ji )

σi

n  µj j =1

µi

p(1−γji )

Bji

≤ Aij |fj (yj (t )) − fj (xj (t ))|

pξ ij

ρjp−1

n 

,

pζij

|ωij | p−1 +

j=1,j̸=i

ηi =

pγ ij p−1

Bij

≤ Aij σj |yj (t ) − xj (t )| = Aij σj |ej (t )|.

j =1

Aji

νi = p ωii + (p − 1)

n 

p(1−ξji )

ρi

(12) n  µj |ωji |p(1−ζij ) , µi j=1,j̸=i

,

(13)

(14)

where Aij = max{|a∗ij |, |a∗∗ ij |}, µi > 0, 0 ≤ αij ≤ 1, 0 ≤ βij ≤ 1, 0 ≤ γij ≤ 1, 0 ≤ ξij ≤ 1, 0 ≤ α ij ≤ 1, 0 ≤ β ij ≤ 1, 0 ≤

γ ij ≤ 1, 0 ≤ ξ ij ≤ 1, 0 ≤ ζ ij ≤ 1, σi = max{|σi− |, |σi+ |}, ρi = max{|ρi− |, |ρi+ |}, Bij = max{|b∗ij |, |b∗∗ ij |}, p > 1, i, j ∈ N. In the following, we will give an assumption: ηi 1−σ0

> 0, and there exist h¯ i > 0 such that κi + h¯ i − (H2 ) λi − νi − ηi > 0, where σ0 satisfies τ˙ij (t ) ≤ σ0 < 1, i, j ∈ N , t ≥ 0. 1−σ0 Now, under assumption (H2 ), we consider the functions

ii (˘εi ) = λi − νi − ε˘ i −

ηi 1 − σ0

eε˘ i τ ,

i ∈ N , ε˘ i ≥ 0.

By simple computing, we can easily get

i′i (˘εi ) = −1 −

ηi τ 1 − σ0

ii (0) = λi − νi −

eε˘ i τ < 0,

i ∈ N.

(15)

In a similar way, we get there exists a constant εˆ > 0 such that

ki (ˆε ) = κi + h¯ i −ˆε −

ηi eεˆ τ ≥ 0, 1 − σ0

i ∈ N.

(16)

Let ε = min{˘ε , εˆ }, then we can obtain ii (ε) ≥ 0, ki (ε) ≥ 0, i ∈ N. Now, we are in a position to present the following results: Theorem 1. Under assumptions (H1 ) and (H2 ) and τj (t ) satisfies τ˙j (t ) ≤ σ0 < 1, systems (1) and (2) are exponentially synchronized under controller (3) if the following condition is also satisfied:

(H3 ) ε −

(T −δ)h¯ T

≤ Aij σj |yj (t ) − xj (t )| = Aij σj |ej (t )|

> 0, where h¯ = max1≤i≤n {h¯ i }.

Proof. From (6), (7), (9) and under assumption (H1 ), we can get (1) For |yj (t )| ≤ Tj , |xj (t )| ≤ Tj

|co[aij (yj (t ))]fj (yj (t )) − co[aij (xj (t ))]fj (xj (t ))| × |a∗ij fj (yj (t )) − a∗ij fj (xj (t ))|

and if yj (t ) < −Tj , then

|co[aij (yj (t ))]fj (yj (t )) − co[aij (xj (t ))]fj (xj (t ))| = |co[aij (xj (t ))](fj (xj (t )) − fj (−Tj )) + co[aij (yj (t ))](fj (−Tj ) − fj (yj (t )))| ≤ |co[aij (xj (t ))](fj (xj (t )) − fj (−Tj ))| + |co[aij (yj (t ))](fj (−Tj ) − fj (yj (t )))| ≤ |Aij (fj (xj (t )) − fj (−Tj ))| + |Aij (fj (−Tj ) − fj (yj (t )))| (20)

|co[aij (yj (t ))]fj (yj (t )) − co[aij (xj (t ))]fj (xj (t ))| ≤ Aij σj |ej (t )|.

≤ Aij σj |yj (t ) − xj (t )| = Aij σj |ej (t )|. (2) For |yj (t )| > Tj , |xj (t )| > Tj

|co[aij (yj (t ))]fj (yj (t )) − co[aij (xj (t ))]fj (xj (t ))| ∗∗ × |a∗∗ ij fj (yj (t )) − aij fj (xj (t ))|

(21)

In a similar way, we get

|co[bij (yj (t − τj (t )))]gj (yj (t − τj (t ))) − co[bij (xj (t − τj (t )))]gj (xj (t − τj (t )))| ≤ Bij ρj |ej (t − τj (t ))|.

(22)

Now, we consider a Lyapunov functional as V (t , e) =

n 

Vi (t , ei (t )) +

i=1 n

×



ηi

i=1



exp{ετ } 1 − σ0

t t −τj (t )

Vi (s, ei (s))ds,

(23)

where Vi (t , ei (t )) = µi exp{ε t }|ei (t )|p , t ≥ 0, i ∈ N. We calculate the upper right derivation of V (t , e) along the solution of system (9). When mT ≤ t ≤ mT + δ , and from (21), (22), we obtain D+ V (t , e) =

n  

εVi (t , ei (t )) + pµi |ei (t )|p−1 sgn(ei (t ))

i=1

× D + ei ( t ) +

= |a∗ij (fj (yj (t )) − fj (xj (t )))| ≤ Aij |fj (yj (t )) − fj (xj (t ))|

= |a∗∗ ij (fj (yj (t )) − fj (xj (t )))|

(19)

So, from (17)–(20), we have

On the other hand, ii (˘εi ) is continuous on [0, +∞) and when ε˘ i → +∞, ii (˘εi ) < 0, then there exists a positive number ε˘ i∗ such that ii (˘εi∗ ) ≥ 0 and for ε˘ i ∈ (0, ε˘ i∗ ), ii (˘εi ) > 0. Denoting ε˘ = min1≤i≤n {˘εi∗ }, then we have

ηi eε˘ τ ≥ 0, 1 − σ0

|co[aij (yj (t ))]fj (yj (t )) − co[aij (yj (t ))]fj (xj (t ))| = |co[aij (yj (t ))](fj (yj (t )) − fj (Tj )) + co[aij (xj (t ))](fj (Tj ) − fj (xj (t )))| ≤ |co[aij (yj (t ))](fj (yj (t )) − fj (Tj ))| + |co[aij (xj (t ))](fj (Tj ) − fj (xj (t )))| ≤ |Aij (fj (yj (t )) − fj (Tj ))| + |Aij (fj (Tj ) − fj (xj (t )))|

≤ Aij σj |yj (t ) − xj (t )| = Aij σj |ej (t )|.

ηi > 0. 1 − σ0

ii (˘ε ) = λi − νi − ε˘ −

(18)

(3) For |xj (t )| ≤ Tj < |yj (t )|, or |yj (t )| ≤ Tj < |xj (t )|, here we only consider |xj (t )| ≤ Tj < |yj (t )|, because the other case is similarly discussed. Then, we have yj (t ) < −Tj ≤ xj (t ) or xj (t ) ≤ Tj < yj (t ). Here, we only consider Tj < yj (t ) or yj (t ) < −Tj , because other cases can be similarly discussed as above. If Tj < yj (t ), then

ηi exp{ετ } 1 − σ0

× [Vi (t , ei (t )) − Vi (t − τj (t ), ei (t − τj (t )))]

(17)

≤



n   ε Vi (t , ei (t )) + µi exp{ε t } [p(−1 + ωii ) i=1

× |ei (t )|p +

n  j =1

pσj Aij |ei (t )|p−1 |ej (t )|

G. Zhang, Y. Shen / Neural Networks 55 (2014) 1–10



+

p |ωij | |ei (t )|p−1 |ej (t )| +



j=1,j̸=i

n

pρj Bij

p

n 

+

+ (p − 1) n 

+

pAij σj |ei (t )|

p−1

 p

αij p−1 Aij

≤ (p − 1)

|ej (t )|

σj

βij p−1

p−1

ρ

(1−αij )

|ei (t )|

(Aij

(1−βij )

σj

|ej (t )|)

n 

pαij

pβij

≤

σj p−1 |ei (t )|p

p−1

Aij

n 

p(1−αij )

Aij

p(1−βij )

σj

|ej (t )|p ,

(25)

×

=

p |ωij |

n 

pαij p−1



Aij

≤ (p − 1)

+

+

pζij

+

|ωij | p−1 |ei (t )|p

+

|ωij |p(1−ζij ) |ej (t )|p ,

(26)

j=1,j̸=i

pBij ρj |ei (t )|p−1 |ej (t − τj (t ))|

j =1

=

n 

γij p−1



ρj

p Bij

ξij p−1

p−1 |ei (t )|

(1−γij ) (1−ξij )

(Bij

ρj

=−

≤ (p − 1)

pγij p−1

Bij

ρj

pξij p−1

n 

p(1−γij )

Bij

p(1−ξij )

ρj

(27)

n  µi

µj

n  

|ωij |

pζij p−1

 Vi (t , ei (t ))

p(1−αij )

Aij

p(1−βij )

σj

Vj (t , ej (t ))

p(1−γij )

Bij

p(1−ξij )

ρj

 ηi exp{ετ } Vi (t , ei (t )) λi − νi − ε − 1 − σ0 (28)

V (t , e) ≤ V (mT , e),

t ∈ [mT , mT + δ].

(29)

Similarly, for t ∈ (mT + δ, (m + 1)T ], we can obtain

 ε Vi (t , ei (t )) + µi exp{ε t }

i=1

D+ V (t , e) ≤ −

n  

κi + h¯ i −ε −

i=1

 ×

n 

which implies that

|ej (t − τj (t ))|p .

Now, from (25)–(27), we obtain n 

ρj



≤ 0,

j =1

D+ V (t , e) ≤

µj

i =1

|ei (t )|p

j =1

+

+ Bij

pξij p−1

n  µi |ωij |p(1−ζij ) Vj (t , ej (t )) µ j j=1,j̸=i

j =1

|ej (t − τj (t ))|)

j =1 n 

pγij p−1

× eετ Vj (t − τj (t ), ej (t − τj (t )))  Vi (t , ei (t )) + ηi eετ 1 − σ0  − Vi (t − τj (t ), ei (t − τj (t )))

and n 

n  µi j =1

j=1,j̸=i n 

σj

pβij p−1

j=1,j̸=i

j=1,j̸=i n 

1 − σ0

ε − p(1 − ωii ) + (p − 1)

+ (p − 1)

p−1 |ei (t )| (|ωij |(1−ζij ) |ej (t )|)

ζij p−1

Vi (t , ei (t ))

j =1

j=1,j̸=i





i =1

p |ωij | |ei (t )|p−1 |ej (t )| n 

|ej (t − τj (t ))|



j =1 n 

 p

 − Vi (t − τj (t ), ei (t − τj (t )))

j =1 n



pξij

ρjp−1 |ei (t )|p

p(1−γij ) p(1−ξij ) Bij j

+ ηi exp{ετ }

j =1

+

pγij p−1

Bij

j =1

j =1

=

n  j =1

ai ≥ 0, i = 1, 2, . . . , p,

we can get

n 

|ωij |p(1−ζij ) |ej (t )|p

j=1,j̸=i

Furthermore, it follows from (H1 ) and the fact p

|ωij | p−1 |ei (t )|p

j=1,j̸=i

j=1

p a1 a2 · · · ap ≤ a1 + a2 + · · · + app ,

pζij



+ (p − 1)

 × |ei (t )|p−1 |ej (t − τj (t ))| + ηi exp{ετ }   Vi (t , ei (t )) × −Vi (t − τj (t ), ei (t − τj (t ))) . (24) 1 − σ0

n 

5

n

n

p(−1 + ωii )|ei (t )|p + (p − 1)

+

n 

 ηi exp{ετ } Vi (t , ei (t )) 1 − σ0

h¯ i Vi (t , ei (t ))

i=1

×

n 

pαij p−1

Aij

pβij

≤ h¯ V (t , e),

σj p−1 |ei (t )|p

j=1

+

n  j =1

p(1−αij ) p(1−βij ) Aij j

σ

h¯ = max h¯ i ,

(30)

1≤i≤n

for t ∈ (mT + δ, (m + 1)T ], which leads to

|ej (t )|

p

V (t , e) ≤ V (mT + δ, e) exp{h¯ (t − mT − δ)}.

(31)

6

G. Zhang, Y. Shen / Neural Networks 55 (2014) 1–10

Corollary 1. Under assumption (H1 ) and τj (t ) satisfies τ˙j (t ) ≤ σ0 < 1, systems (1) and (2) are exponentially synchronized under

Combining (29) and (31), we summarize that:

(S1 ) For t ∈ [0, δ], it follows from (29) that V (t , e) ≤ V (0, e). (S2 ) For t ∈ (δ, T ], from (31), we have

controller (3) if the following conditions are also satisfied: η

(i) νi < 0, λi − νi − 1−σi > 0, i ∈ N. 0 (T −δ)ν (ii) ε − T > 0, where ν = max1≤i≤n {|νi |}.

V (t , e) ≤ V (δ, e) exp{h¯ (t − δ)} ≤ V (0, e) exp{h¯ (t − δ)}. (S3 ) For t ∈ [T , T + δ], we get V (t , e) ≤ V (T , e) ≤ V (0, e) exp{h¯ (T − δ)}.

Proof. Let α ij = αij , β ij = βij , γ ij = γij , ξ ij = ξij for all i, j ∈ N in (12), then λi = κi . Under condition (i), select h¯ i = −νi , then, Corollary 1 can be immediately derived from Theorem 1.

(S4 ) For t ∈ (T + δ, 2T ], we obtain V (t , e) ≤ V (T + δ, e) exp{h¯ (t − T − δ)}

In Theorem 1, the results need p > 1. Now, we give the following Theorem 2 for p ≥ 1. Let

≤ V (0, e) exp{h¯ (t − 2δ)}. Repeating the above procedure, we obtain that for t ∈ [mT , mT + δ], V (t , e) ≤ V (mT , e) ≤ V (0, e) exp{mh¯ (T − δ)}.

λi = p − (p − 1)

Moreover, for t ∈ (mT + δ, (m + 1)T ],

−

V (t , e) ≤ V (mT + δ, e) exp{h¯ (t − mT − δ)} If t ∈ [mT , mT + δ], we have m ≤ t /T , then it follows from (32) that V (t , e) ≤ V (0, e) exp



h¯ (T − δ) T

t .

V (t , e) ≤ V (0, e) exp

h¯ (T − δ) T

t .



h¯ (T − δ) T



t .

(34)

(37) n 

|ωij | +

n  µj |ωji |, µi j=1,j̸=i

Bji ρi .

(38)

(39)

Then as proof of Theorem 1, we have following results. Theorem 2. Under assumption (H1 ) and τj (t ) satisfies τ˙j (t ) ≤ σ0 < 1, systems (1) and (2) are exponentially synchronized under

(iii) ε −

(T −δ)h¯ T

> 0, where h¯ = max1≤i≤n {h¯ i }.

Combining Corollary 1 and Theorem 2, we can easily obtain the following results. Corollary 2. Under assumption (H1 ) and τj (t ) satisfies τ˙j (t ) ≤ σ0 < 1, systems (1) and (2) are exponentially synchronized under controller (3) if the following condition are also satisfied: η

(i) ν i < 0, λi − ν i − 1−σi > 0, i ∈ N 0 (T −δ)ν (ii) ε − T > 0, where ν = max1≤i≤n {|ν i |}.

T

t ≥ 0,

(i) λi − ν i − 1−σi > 0, i ∈ N. 0 η (ii) There exist h¯ i > 0, such that λi + h¯ i − 1−σi > 0. 0 Proof. Theorem 2 can be similarly proved as Theorem 1, so the process is omitted here.

    1 h¯ (T − δ) ∥e(t )∥pp ≤ V (0, e) exp − ε − t µ T   n  1  ηi exp{ετ } ≤ Vi (0, ei ) + max 1≤i≤n µ i=1 1 − σ0   0 × Vi (s, ei (s))ds −τ     h¯ (T − δ) × exp − ε − t T     h¯ (T − δ) ≤ M ∥ψ − ϕ∥pp exp − ε − t , (35)

Remark 1. In order to study the dynamical behaviors of memristorbased neural networks, the previous works do the following assumptions: co[aij , aij ]fj (yj (t )) − co[aij , aij ]fj (xj (t )) ⊆ co[aij , aij ]

where

µ0 ηi exp{ετ } 1 + max 1 ≤ i ≤ n µ 1 − σ0 µ0 = max {µi }. M =

Aji σi ,

η

It follows from (23) and (34) that



µi

Bij ρj

controller (3) if the following conditions are also satisfied:

Therefore, for any t ∈ [0, +∞), we always have V (t , e) ≤ V (0, e) exp

n  µj

n  j =1

j=1,j̸=i

j =1



µi

ν i = p ωii + (p − 1) ηi =



Similarly, if t ∈ (mT + δ, (m + 1)T ], we have m + 1 ≤ t /T , then it follows from (33) that



n  µj j=1

(33)

Aij σj − (p − 1)

j =1

(32)

≤ V (0, e) exp{h¯ (t − (m + 1)δ)}.

n 





≥ 1,

× (fj (yj (t )) − fj (xj (t ))), co[bij , bij ]gj (yj (t )) − co[bij , bij ]

µ = min {µi },

× gj (xj (t )) ⊆ co[bij , bij ](gj (yj (t )) − gj (xj (t ))),

1≤i≤n

1≤i≤n

h(T −δ) Let ϵ = 1p (ε − ¯ T ), under (H3 ), we know that ϵ > 0, it follows from (35) that

∥y(t ) − x(t )∥p ≤ M 1/p ∥ψ − ϕ∥p exp{−ϵ t },

t ≥ 0,

(36)

which implies that system (1) can be globally exponentially synchronized with system (2) under controller (3). This completes the proof of Theorem 1.

(40)

however, the above conditions are difficult to verify, and they may not hold in some special cases (Yang et al., 2014). So, basing on the conditions fj (±Tj ) = gj (±Tj ) = 0 in (H1 ), this paper does not need the above conditions (40). In fact, when y(t ) = x∗ is an equilibrium point of system (1), the stability of x∗ can be similarly studied as in this paper. And other dynamical behaviors of system (1) can be also similarly investigated. Because in the proof of these dynamical behaviors of system (1), Eqs. (21) and (22) hold under the conditions fj (±Tj ) = gj (±Tj ) = 0.

G. Zhang, Y. Shen / Neural Networks 55 (2014) 1–10

7

Now, in order to show that our activation functions are more general, we denote G , {Φ ∈ C(R, R)|sΦ (s) > 0, s ̸= 0, D+ Φ (s) ≥ 0, Φ (0) = 0, s ∈ R}. For i = 1, 2, . . . , n, three different types of activation functions are listed as follows: (1) Γ1 , {hi (·)|hi (·) ∈ C(R, R), ∃ki > 0, |hi (si )| ≤ ki , si ∈ R}, i.e., hi (·) is the bounded continuous function. (2) Γ2 , {hi (·)|hi (·) ∈ G, ∃ki > 0, si hi (si ) ≤ ki s2i , si ∈ R}, i.e., hi (·) is the Lurie-type function. (3) Γ3 , {hi (·)|hi (·) ∈ C(R, R), |hi (xi ) − hi (yi )| ≤ ki |xi − yi |, xi , yi ∈ R}, i.e., hi (·) is the Lipschitz-type function. Remark 2. In Hu et al. (2010b), Wu and Zeng (2012), Zhang and Shen (2013) and Zhang et al. (2012), the scholars studied the synchronization of neural networks with Lurie-type activation functions. That is, they need σi− ≥ 0, ρi− ≥ 0, σi+ > 0, ρi+ > 0 in assumption (H1 ) of this paper. However, the activation function discussed in this paper maybe not monotonous nondecreasing. What is more, the constants σi− , ρi− , σi+ , ρi+ in assumption (H1 ) of this paper are allowed to be positive, negative or zero. Obviously, the assumption (H1 ) of this paper is more weaker and general than Γ1 , Γ2 and Γ3 . Remark 3. Compared with the results on synchronization of the neural networks with continuous right-hand side, such as Cai et al. (2009), Hu et al. (2010a), Huang and Li (2010), Huang et al. (2008), Huang, Li, Yu et al. (2009) and Yu et al. (2011), our results on synchronization of the neural networks are with discontinuous right-hand side. So the results of this paper are less conservative and more general. Remark 4. In this paper, the intermittent control is generalized to study a more general neural network model and the traditional restrictions that the control width is greater than the time delay are also removed. And compared with other researches by using LMIs technique to get the conditions of exponential synchronization, such as Huang and Li (2010) and Liu (2009), the conditions in our paper can be directly derived from the parameters of the neural networks, are very easily verified. Remark 5. The exponential synchronization of conventional neural networks via periodically intermittent control is guaranteed in the existing works, such as Hu et al. (2010b), Huang, Li, Han (2009), Li and Cao (2014), Li, Feng et al. (2007), Li, Liao et al. (2007) and Xia and Cao (2009). However, the memristor-based neural network model is also a system family. The existing results do not yield any exponential synchronization via periodically intermittent control, whereas, our criteria guarantee the global exponential synchronization of memristor-based neural networks via periodically intermittent control with various activation functions. Therefore, our results achieve a valuable improvement, and they also complement and extend the earlier publications. Remark 6. In Theorem 1, because all values of the parameters αij , βij , γij , ξij , α ij , β ij , γ ij , ξ ij , ζ ij can select randomly in the interval [0, 1], therefore, we can properly choose these parameters to get the parameter h¯ based on a practical problem, so Eqs. (11)–(14) are more general and flexible. In fact, Corollary 1 shows us an example for the application of Theorem 1. Remark 7. Here, we take an example for the application of Corollary 2 to shed light on how to design a suitable intermittent controller in real application to realize exponential synchronization, for a given chaotic network satisfying (H1 ), Step (i) Compute the value of τ according to its definition, and figure out the values of λi , ηi according to (37) and (39), respectively; Step (ii) Select control strengths ωij such that ν i satisfy condition (i) of Corollary 2; Step (iii) By using matlab software compute the values of ε˘ i∗ , ε, ν ; Step (iv) Choose randomly a control period T , and take the control width δ satisfying (1 − νε )T < δ < T , and then according to the above chosen ωij , T , δ , we can design an intermittent controller.

Fig. 1. The chaotic attractor of memristor-based neural networks (41).

4. Numerical example Now, we perform some numerical simulations to illustrate our analysis. Example. Consider the following memristor-based neural networks dxi (t ) dt

= −xi (t ) +

2 

aij (xj (t ))fj (xj (t ))

j =1

+

2 

bij (xj (t − τj (t )))gj (xj (t − τj (t ))) + Ii ,

j =1

i, j = 1, 2

(41)

where a11 (x1 (t )) = 1, a12 (x2 (t )) =



7, 5,

a22 (x2 (t )) = 1.8, |x2 (t )| ≤ 1, |x2 (t )| > 1,

|x1 (t )| ≤ 1, |x1 (t )| > 1,  −1.5, |x1 (t − τ1 (t ))| ≤ 1, b11 (x1 (t − τ1 (t ))) = −1.2, |x1 (t − τ1 (t ))| > 1,  1.0, |x2 (t − τ2 (t ))| ≤ 1, b12 (x2 (t − τ2 (t ))) = 0.8, |x2 (t − τ2 (t ))| > 1,  0.8, |x1 (t − τ1 (t ))| ≤ 1, b21 (x1 (t − τ1 (t ))) = 1, |x1 (t − τ1 (t ))| > 1,  −1.4, |x2 (t − τ2 (t ))| ≤ 1, b22 (x2 (t − τ2 (t ))) = −1.6, |x2 (t − τ2 (t ))| > 1 a21 (x1 (t )) =



0.8, 1,

we consider system (41) as the drive system and corresponding et response system is defined as in (2). And τ1 (t ) = 1+ , τ2 (t ) = et 0.75 − 0.25 cos(t ), and take the activation function as f1 (x1 ) = g1 (x1 ) = sin(|x1 | − 1), f2 (x2 ) = g2 (x2 ) = tanh(|x2 | − 1), I1 = I2 = 0. The model (41) has chaotic attractors with the initial condition x1 (s) = 0.45, x2 (s) = 0.6, ∀s ∈ [−1, 0) can be seen in Fig. 1. And Figs. 2, 3 depict the state variables x1 (t ) and x2 (t ) of system (41), respectively. By simple computing, we get A11 = 1, A12 = 7, A21 = 1, A22 = 1.8, B11 = 1.5, B12 = B21 = 1, B22 = 1.6, ρ1− = σ1− = −1, ρi+ = σi+ = 1, τ = 1, τ˙j (t ) ≤ σ0 = 14 < 1, i, j = 1, 2. Now, let

8

G. Zhang, Y. Shen / Neural Networks 55 (2014) 1–10

Fig. 2. The state x1 (t ) of system (41).

Fig. 4. The synchronization error e1 (t ) without periodically intermittent control.

Fig. 3. The state x2 (t ) of system (41).

Fig. 5. The synchronization error e2 (t ) without periodically intermittent control.

p = µ1 = µ2 = 1, then, from (37) to (39), we can get

λ1 = 1 − A11 − A21 = −1, λ2 = 1 − A22 − A12 = −7.8, η1 = B11 + B21 = 2.5, η2 = B22 + B12 = 2.6, Now, we take the controller matrix W = (ωij )n×n in (3) which is given as follows: ω11 = −8, ω22 = −17, ω12 = 2, ω21 = 1. And

ν 1 = ω11 + |ω21 | = −7,

ν 2 = ω22 + |ω12 | = −15.

Then, we obtain ε = 0.6381, And now, we choose T = 5, from (ii) of Corollary 2, we get δ > 4.7873. Selecting δ = 4.85, all the conditions of Corollary 2 are satisfied. It follows from Corollary 2, that system (41) and the corresponding response system are exponentially synchronized. Figs. 4 and 5 depict the synchronization error of the state variables e1 (t ), e2 (t ) between the drive system and the response system without the periodically intermittent control (3), respectively. Choose randomly 100 initial conditions, Figs. 6 and 7 depict the synchronization error of the state variables e1 (t ), e2 (t ) between the drive system and the response system with the periodically intermittent control (3), respectively. From Figs. 4–7, we can see that the periodically intermittent control inputs contribute to the chaos synchronization. When the response system is with the periodically intermittent control (3), we get the state trajectories of variables x1 (t ), y1 (t ) and x2 (t ), y2 (t ) are shown in

Figs. 8 and 9, respectively. Some examples can also be given for other theorems, here they are omitted. Remark 8. Because aij (xj (t )) and bij (xj (t − τj (t ))), i, j = 1, 2, . . . , n are discontinuous, so the results obtained in Cai et al. (2009), Hu et al. (2010a, 2010b), Huang and Li (2010), Huang, Li, Han (2009), Huang et al. (2008), Huang, Li, Yu et al. (2009), Li and Cao (2014), Li, Feng et al. (2007), Li, Liao et al. (2007), Xia and Cao (2009) and Yu et al. (2011) about neural networks with continuous righthand side cannot be used here. In addition, in system (41), f1 (·) = g1 (·) ∈ Γ3 , we know σ1− = −1, ρ1− = −1, and f1 (·) = g1 (·) are not monotonous nondecreasing functions, so the results obtained in Hu et al. (2010b), Wu and Zeng (2012), Zhang and Shen (2013) and Zhang et al. (2012) cannot be used here, so the conditions in (H1 ) are more weaker and general in this paper. 5. Conclusion In this paper, we adopt nonsmooth analysis and control theory to handle memristor-based neural networks with discontinuous right-hand side. In particular, several new sufficient conditions ensuring exponential synchronization of memristor-based chaotic neural networks are obtained via periodically intermittent control. The model based on the memristor widens the application scope

G. Zhang, Y. Shen / Neural Networks 55 (2014) 1–10

9

Fig. 8. State trajectories of variables x1 (t ), y1 (t ) with periodically intermittent control (3), respectively.

Fig. 6. Choose randomly 100 initial conditions, the synchronization error e1 (t ) with periodically intermittent control (3), where T = 5, δ = 4.85.

Fig. 9. State trajectories of variables x2 (t ), y2 (t ) with periodically intermittent control (3), respectively.

Acknowledgments

Fig. 7. Choose randomly 100 initial conditions, the synchronization error e2 (t ) with periodically intermittent control (3), where T = 5, δ = 4.85.

for the design of neural networks. And the new proposed results here are different from the present works (Bao & Zeng, 2013; Chen et al., 2014; Hu & Wang, 2010; Wen & Zeng, 2012; Wu et al., 2012; Wu & Zeng, 2012; Yang et al., 2014; Zhang & Shen, 2013; Zhang et al., 2012, 2013), and our results achieve a valuable improvement, and they also complement and extend the earlier publications. As the generalization of the obtained results, exponential synchronization of the addressed neural networks under various feedback functions are discussed in detail. Numerical simulations are given to illustrate effectiveness of the proposed theories. In addition, the intermittent control method can be applied for dealing with antisynchronization of memristor-based chaotic neural networks. This issue will be tthe topic of future research.

The authors gratefully acknowledge anonymous referees’ comments and patient work. This work is supported by the National Science Foundation of China (Grant Nos. 11271146 and 61374150), Science and Technology Program of Wuhan under Grant No. 2013010501010117, Key Program of National Natural Science Foundation of China under Grant No. 61134012, and the prior developing Field for the Doctoral Program of Higher Education of China under Grant No. 20130142130012, the National Natural Science Foundation of China under Grant No. 61304067. References Amritkar, R. E., & Gupte, N. (1993). Synchronization of chaotic orbits: the effect of a finite time step. Physical Review E, 47, 3889–3895. Aubin, J. P., & Cellina, A. (1984). Differential inclusions. Germany, Berlin: SpringerVerlag. Bao, B. C., Liu, Z., & Xu, J. P. (2010a). Steady periodic memristor oscillator with transient chaotic behaviours. Electronics Letters, 46, 228–230. Bao, B. C., Liu, Z., & Xu, J. P. (2010b). Dynamical analysis of memristor chaotic oscillator. Acta Physica Sinica, 59, 3785–3793. Bao, G., & Zeng, Z. G. (2013). Multistability of periodic delayed recurrent neural network with memristors. Neural Computing and Applications, 23, 1963–1967.

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