A novel and effective approach to synchronization analysis of neural networks is
investigated by using the nonlinear operator named the generalized Dahlquist constant and the general
intermittent control. The proposed approach offers a design procedure for synchronization of a large
class of neural networks. The numerical simulations whose theoretical results are applied to typical neural
networks with and without delayed item demonstrate the effectiveness and feasibility of the proposed
technique.

1. Introduction

Since its introduction by Pecora and Carrol in 1990, synchronization of chaotic systems [1–10] is of great practical significance and has received great interest in recent years. In the above literature, the approach applied to stability analysis is basically Lyapunov's method. As we all know, the construction of a proper Lyapunov function becomes usually very skillful, and Lyapunov's method does not specifically describe the convergence rate near the equilibrium of the system. Hence, there is little compatibility among all of the stability criteria obtained so far.

The concept named the generalized Dahlquist constant [11] has been applied to the investigation of impulsive synchronization [12, 13] analysis.

Intermittent control [14–18] has been used for a variety of purposes in engineering fields such as manufacturing, transportation, air-quality control, and communication. Synchronization using an intermittent control method has been discussed [15–18]. Compared with continuous control methods [2–10], intermittent control is more efficient when the system output is measured intermittently rather than continuously. Our interest focuses on the class of intermittent control with time duration, wherein the control is activated in certain nonzero time intervals and is off in other time intervals. A special case of such a control law is of the form U(t)={-k(y(t)-x(t)),(nT≤t<nT+δ),0,(nT+δ≤t<(n+1)T),

where k denotes the control strength, δ>0 denotes the switching width, and T denotes the control period. In this paper, based on the generalized Dahlquist constant and the Gronwall inequality, a general intermittent controller U(t)={-k(y(t)-x(t)),(h(n)T≤t<h(n)T+δ),0,(h(n)T+δ≤t<h(n+1)T),

is designed, where h(n) is a strictly monotone increasing function on n with h(0)=0 then sufficient yet generic criteria for synchronization of typical neural networks with and without delayed item are obtained.

This paper is organized as follows. In Section 2, some necessary background materials are presented, and a simple configuration of coupled neural networks is formulated. Section 3 deals with synchronization. The theoretical results are applied to typical chaotic neural networks, and numerical simulations are shown in this section. Finally, some concluding remarks are given in Section 4.

2. Formulations

Let X be a Banach space endowed with the Euclidean norm ∥·∥, that is, ∥x∥=xTx=〈x,x〉, where 〈·,·〉 is inner product, and, let Ω be a open subset of X. We consider the following system:
dx(t)dt=F(x(t))+G(x(t-τ)),
where F,G are nonlinear operators defined on Ω, x(t),x(t-τ)∈Ω, τ is a time-delayed positive constant, and F(0)=0,G(0)=0.

Definition 1.

System (3) is called to be exponentially stable on a neighborhood Ω of the equilibrium point if there exist constants μ>0,M>0, such that
‖x(t)‖≤Me-μt‖x0‖(t≥0),
where x(t) is any solution of (3) initiated from x(t0)=x0.

Definition 2 (see [<xref ref-type="bibr" rid="B11">11</xref>]).

Suppose Ω is an open subset of Banach space X, and F:Ω→X is an operator.

The constant
α(F)=supx,y∈Ω,x≠y1‖x-y‖limr→+∞f(r)
is called to be the generalized Dahlquist constant of F on Ω, where f(r)=∥(F+rI)x-(F+rI)y∥-r∥x-y∥; here, denote by F+rI the operator mapping every point x∈Ω onto F(x)+rx.

For r≥0, f(r)=‖(F+rI)x-(F+rI)y‖-r‖x-y‖=(k(r))Tk(r)-r(x-y)T(x-y)=f1(r)-f2(r),

where k(r)=F(x)-F(y)+r(x-y),f1(r)=(k(r))Tk(r),f2(r)=r(x-y)T(x-y)df1(r)dr=(F(x)-F(y))T(x-y)+r‖x-y‖2f1(r)=f3(r)f1(r),df2(r)dr=(x-y)T(x-y),

where f3(r)=(F(x)-F(y))T(x-y)+r∥x-y∥2. According to the Cauchy-Bunie Khodorkovsky inequality, we obtain (f1(r)(x-y)T(x-y))2-(f3(r))2=‖F(x)-F(y)‖2‖x-y‖2-((F(x)-F(y))T(x-y))2=〈F(x)-F(y),F(x)-F(y)〉〈x-y,x-y〉-(〈F(x)-F(y),x-y〉)2≥0.

Therefore,
|f1(r)(x-y)T(x-y)|≥|f3(r)|≥f3(r).

That is
f1(r)(x-y)T(x-y)≥f3(r),df1(r)dr-df2(r)dr≤0,df(r)dr≤0.
So the function f(r),r≥0, is monotone decreasing; thus, the limit limr→+∞f(r)

exists.

3. Synchronization Analysis and ExamplesTheorem 3.

If the operator G in the system (3) satisfies
‖G(x)-G(y)‖≤l‖x-y‖
for any x,y∈Ω, where l is a positive constant, then two solutions, x(t) and y(t), respectively, initiated from x(t0)=x0∈Ω, y(t0)=y0∈Ω satisfy
‖x-y‖≤‖x0-y0‖exp{λ(t-t0)},∀t≥0,
where λ=α(F)+exp{-α(F)τ}l.

Proof.

Assume x(t) and y(t) are the solutions of (3), respectively, under the initial conditions x(t0)=x0∈Ω, y(t0)=y0∈Ω. We have
(ertx(t))t′=rertx(t)+ertF(x(t))+ertG(x(t-τ))=ert(F+rI)x(t)+ertG(x(t-τ))
for all t≥0 and r>0.

For all x0,y0∈Ω,t>s≥0,ert[x(t)-y(t)]=ers[x(s)-y(s)]+∫stk(r,u)du,

where k(r,u)=eru[(F+rI)x(u)-(F+rI)y(u)+(G(x(u-τ))-G(y(u-τ)))].

So ert‖x(t)-y(t)‖-ers‖x(s)-y(s)‖≤∫sth(r,u)du,

where h(r,u)=eru(∥(F+rI)x(u)-(F+rI)y(u)∥+∥G(x(u-τ))-G(y(u-τ))∥).

Then for all t≥0, we infer thatert(‖x(t)-y(t)‖)t′≤h(r,t).
Therefore, we obtain ert(‖x(t)-y(t)‖)t′≤h(r,t)-rert‖x(t)-y(t)‖.

Letting r→+∞, then
(‖x(t)-y(t)‖)t′≤α(F)‖x(t)-y(t)‖+‖G(x(t-τ))-G(y(t-τ))‖≤α(F)‖x(t)-y(t)‖+l‖x(t-τ)-y(t-τ)‖.
Integrating inequality (19) over [t0,t], we have ‖x(t)-y(t)‖≤eα(F)(t-t0)‖x0-y0‖+∫t0teα(F)(t-s)l‖x(s-τ)-y(s-τ)‖ds.

That is e-α(F)(t-t0)‖x(t)-y(t)‖≤‖x0-y0‖+∫t0te-α(F)(s-t0)l‖x(s-τ)-y(s-τ)‖ds,≤‖x0-y0‖+e-α(F)τl∫t0-τt-τe-α(F)(s-t0)‖x(s)-y(s)‖ds.

Using the Gronwall inequality [19, 20], we have e-α(F)(t-t0)‖x(t)-y(t)‖≤‖x0-y0‖exp{e-α(F)τl(t-t0)}.

Then ‖x(t)-y(t)‖≤‖x0-y0‖exp{(α(F)+e-α(F)τl)(t-t0)}.

Let system (3) be the drive system, and we consider the response system
dy(t)dt=F(y(t))+G(y(t-τ))+U(t),
where x,y∈Rn are the state variables, F(·),G(·) are nonlinear operators, U(t) is a feedback control term, and
U(t)={-k(y(t)-x(t)),(h(n)T≤t<h(n)T+δ),0,(h(n)T+δ≤t<h(n+1)T),
where k denotes the control strength, T is the control period, δ is called the control width, and h(n) is a strictly monotone increasing function on n with h(0)=0.

In this paper, our goal is to design suitable function, h(n) and suitable parameters, δ, T, and k such that system (24) synchronizes to system (3).

Subtract (3) from (24), the error system is obtained
de(t)dt={F(y(t))-F(x(t))+G(y(t-τ))-G(x(t-τ))-ke(t),(h(n)T≤t<h(n)T+δ),F(y(t))-F(x(t))+G(y(t-τ))-G(x(t-τ)),(h(n)T+δ≤t<h(n+1)T),
where e=y-x. Then we have the following result.

Theorem 4.

Suppose that the operator G in the systems (3) and (24) satisfies condition (12), and α(F) is defined as in Definition 2, and λ=α(F)+exp{-α(F)τ}l. Then the synchronization of (3) and (24), given in (26), is asymptotically stable if the parameters δ, T, and kare such that
inf((r+λ)δh-1(t-δ/T)t-λ)>0,
where r=k-λ>0, h-1(·) is inverse function of the function h(·).

Proof.

From Theorem 3, we can get the conclusion as follows:
‖e(t)‖≤‖e(h(n)T)‖exp{-r(t-h(n)T)}
for any h(n)T≤t<h(n)T+δ,
‖e(t)‖≤‖e(h(n)T+δ)‖exp{λ(t-h(n)T-δ)}
for any h(n)T+δ≤t<h(n+1)T.

Consider conditions (28) and (29), and we can get the conclusion that
‖e(t)‖≤{‖e(0)‖exp{-rt+(r+λ)h(n)T-n(r+λ)δ},(h(n)T≤t<h(n)T+δ),‖e(0)‖exp{λt-(n+1)(r+λ)δ}(h(n)T+δ≤t<h(n+1)T),≤{‖e(0)‖exp{-((r+λ)δh-1(t-δ/T)t-λ)t},(h(n)nT≤t<h(n)T+δ),‖e(0)‖exp{-((r+λ)δh-1(t/T)t-λ)t},(h(n)T+δ≤t<h(n+1)T).
When t→∞, ∥e(t)∥→0 is obtained under condition (27) and (26) becomes asymptotically stable.

Corollary 5.

Letting G(x(t-τ))=0, λ=α(F) be defined as in Definition 2, and condition (27) is satisfied, then result similar to Theorem 4 is obtained.

Corollary 6.

Supposing that h(n)=pn,p>0, the operator G in the systems (3) and (24) satisfies condition (12), and α(F) is defined as in Definition 2, and λ=α(F)+exp{-α(F)τ}l then; the synchronization of (3) and (24), given in (26), is asymptotically stable if the parameters δ, T, and k are such that
(r+λ)δ1pT-λ>0,
where r=k-λ>0.

In the simulations of the following examples, we always choose T=5,k=10 and make use of the norm ∥x∥=xTx, where x∈Rn.

Example 7.

Consider a typical delayed Hopfield neural network [21–23] with two neurons:
ẋ(t)=-Cx(t)+Af(x(t))+Bf(x(t-τ)),
where x(t)=(x1(t),x2(t))T, f(x(t))=(tanh(x1(t)),tanh(x2(t)))T,τ=(1), and C=(1001), A=(2.0-0.1-5.03.0), with B=(-1.5-0.1-0.2-2.5).

It should be noted that the network is actually a chaotic delayed Hopfield neural network.

Equation (32) is considered as the drive system, and the response system is defined as follows: ẏ(t)=-Cy(t)+Af(y(t))+Bf(y(t-τ))+U(t),y(t0)=y0.

We calculate and get the value l<9.15,α(F)≤0.7993, where F(x(t))=-Cx(t)+Af(x(t)),G(x(t-τ))=Bf(x(t-τ)). Choose h(n)=n,δ=4, and it is easy to verify that condition (31) is satisfied. Let the initial condition be (x1x2y1y2)T=(34712.5)T. Then it can be clearly seen in Figure 1 that the drive system (32) synchronizes with the response system (33).

(a) Synchronization of x1(t) and y1(t). (b) Synchronization of x2(t) and y2(t).

Example 8.

Considering a typical delayed chaotic neural network (29) with two neurons [24, 25] as the drive system, (31) as the response system, where x(t)=(x1(t),x2(t))T, f(x(t))=(f1(x1(t)),f2(x2(t)))T,fi(xi(t))=0.5(|xi(t)+1|-|xi(t)-1|),i=1,2, τ=(1), C=(1001),A=(1+π/4200.11+π/4), with B=(-1.32π/40.10.1-1.32π/4).

It is easily seen that the operator f(x(t)) is differential on x in Example 7, but the operator f(x(t)) is not so in this example.

We calculate and get the value l<1.32π/2+0.2,α(F)≤1.0855, where F(x(t))=-Cx(t)+Af(x(t)),G(x(t-τ))=Bf(x(t-τ)). Choose h(n)=2n,δ=4, and it is easy to verify that condition (31) is satisfied. Let the initial condition be (x1x2y1y2)T=(341712.8)T. Then the synchronization property of this example can be clearly seen in Figure 2.

(a) Synchronization of x1(t) and y1(t). (b) Synchronization of x2(t) and y2(t).

Example 9.

Consider an autonomous Hopfield neural network with four neurons [26, 27]:
ẋ(t)=-Cx(t)+Af(x(t)),
where x(t)=(x1(t),x2(t),x3(t),x4(t))T,f(x(t))=(tanh(x1(t)),tanh(x2(t)),tanh(x3(t)),tanh(x4(t)))T, and
C=(1000010000100001),A=(0.85-2-0.50.51.81.150.60.31.11.212.50.050.1-0.4-1.51.45).

Das II et al. [26] have reported that the system (34) posseses a chaotic behavior.

Equation (34) is considered as the drive system, and the response system is defined as follows: ẏ(t)=-Cy(t)+Af(y(t))+U(t),y(t0)=y0.

We calculate and get the value α(F)≤1.4369, where F(x(t))=-Cx(t)+Af(x(t)) and choose h(n)=n/2,δ=2. It is easy to verify that condition (31) is satisfied. Let the initial condition be (x1x2x3x4y1y2y3y4)T=(2-1-21765.49)T. Then it can be clearly seen in Figure 3 that the drive system (34) synchronizes with the response system (36).

(a) Synchronization of x1(t) and y1(t). (b) Synchronization of x2(t) and y2(t). (c) Synchronization of x3(t) and y3(t). (d) Synchronization of x4(t) and y4(t).

Example 10.

Consider a typical hyperchaotic neural network (32) with two neurons [28] as the drive system, (33) as the response system, where x(t)=(x1(t),x2(t),x3(t),x4(t))T, f(x(t))=(0,0,0,|x4+1|-|x4-1|)T, and
C=(00110-2-10-141400-10000100),A=(000000000000000100).

We calculate and get the value α(F)≤14.8559, where F(x(t))=-Cx(t)+Af(x(t)) and choose h(n)=n2/(n+1),δ=2. It is easy to verify that the condition (27) is satisfied. Letting the initial condition be (x1x2x3x4y1y2y3y4)T=(2-1-21765.49)T. Then the synchronization property of this example can be clearly seen in Figure 4.

(a) Synchronization of x1(t) and y1(t). (b) Synchronization of x2(t) and y2(t). (c) Synchronization of x3(t) and y3(t). (d) Synchronization of x4(t) and y4(t).

4. Conclusion

Approaches for synchronization of two coupled neural networks which use the nonlinear operator named the generalized Dahlquist constant and the general intermittent control have been presented in this paper. Strong properties of global and asymptotic synchronization have been achieved in a finite number of steps. The techniques have been successfully applied to typical neural networks. Numerical simulations have verified the effectiveness of the method.

Acknowledgment

This work is supported by Research Fund Project of the Heze University under Grant: XY10KZ01.

DengW.LüJ.LiC.Stability of N-dimensional linear systems with multiple delays and application to synchronizationElabbasyE. M.AgizaH. N.El-DessokyM. M.Global synchronization criterion and adaptive synchronization for new chaotic systemZhangQ.ZhouJ.ZhangG.Stability concerning partial variables for a class of time-varying systems and its applications in chaos synchronizationProceedings of the 24th Chinese Control Conference2005South China University of Technology Press135139ZhangQ.JiaG.Chaos synchronization of Morse oscillator via backstepping designZhangQ. L.Synchronization of multi-chaotic systems via ring impulsive controlCaoJ.WangZ.SunY.Synchronization in an array of linearly stochastically coupled networks with time delaysCaoJ.LiL.Cluster synchronization in an array of hybrid coupled neural networks with delayLiL.CaoJ.Cluster synchronization in an array of coupled stochastic delayed neural networks via pinning controlKarimiH. R.MaassP.Delay-range-dependent exponential H∞ synchronization of a class of delayed neural networksCaoJ.HoD. W. C.YangY.Projective synchronization of a class of delayed chaotic systems via impulsive controlPengJ. G.XuZ. B.On asymptotic behaviours of nonlinear semigroup of Lipschitz operators with applicationsZhangQ. L.Generalized Dahlquist constant with application in impulsive synchronization analysisthe International Conference on Logistics Systems and Intelligent Management (ICLSIM '10)January 2010189619002-s2.0-7795296503110.1109/ICLSIM.2010.5461300JiaG.ZhangQ.Impulsive synchronization of hyperchaotic Chen systemProceeding of the 20th Chinese Control and Decision Conference2008123127HuangJ.LiC.HanQ.Stabilization of delayed chaotic neural networks by periodically intermittent controlYuJ.JiangH.TengZ.Synchronization of nonlinear systems with delay via periodically intermittent controlDongZ.WangY.BaiM.ZuoZ.Exponential synchronization of uncertain master-slave Lur'e systems via intermittent controlXiaW.CaoJ.Pinning synchronization of delayed dynamical networks via periodically intermittent controlHuangJ.LiC.HanQ.Stabilization of delayed chaotic neural networks by periodically intermittent controlShiB.ZhangD.GaiM.KuangJ.XiangL.ZhouJ.LiuZ. R.SunS.On the asymptotic behavior of Hopfield neural network with periodic inputsGopalsamyK.HeX. Z.Stability in asymmetric Hopfield nets with transmission delaysLuH.Chaotic attractors in delayed neural networksGilliM.Strange attractors in delayed cellular neural networksWangZ.-S.zhanshan_wang@163.comZhangH.-G.WangZ.-L.Global synchronization of a class of chaotic neural networksDasP. K.IISchieveW. C.ZengZ.Chaos in an effective four-neuron neural networkChengC.-J.LiaoT.-L.tlliao@mail.ncku.edu.twYanJ.-J.HwangC.-C.Synchronization of neural networks by decentralized feedback controlWuZ. Q.TanF. X.WangS. X.Synchronization of the hyper-chaotic system of cellular neural network based on passivity