# Synchronization between Bidirectional Coupled Nonautonomous Delayed Cohen-Grossberg Neural Networks

Authors: Qiming Liu; Rui Xu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Based on using suitable Lyapunov function and the properties ofM-matrix, sufficient conditions for complete synchronization of bidirectional coupled nonautonomous Cohen-Grossberg neural networks are obtained. The methods for discussing synchronization avoid complicated error system of Cohen-Grossberg neural networks. Two numerical examples are given to show the effectiveness of the proposed synchronization method.


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## Body

## 1. Introduction

It is well known that chaos synchronization has gained considerable attention due to the truth that many benefits of chaos synchronization exist in various engineering fields such as secure communication, image processing, and harmless oscillation generation. Since the pioneering works of Pecora and Carroll [1], chaos synchronization has been widely investigated, and many effective research methods such as feedback control, impulsive control, adaptive control, and important results have been presented in [2–10] and references cited therein. Synchronization of coupled chaotic systems also has received considerable attention [11–13].Since Cohen and Grossberg proposed the Cohen-Grossberg neural networks (CGNNs) in 1983 [14] and soon the networks have been the subject of active research due to their many applications in various engineering and scientific areas such as patter recognition, associative memory, and combination, and many dynamic analyses of equilibria or periodic solutions of the networks with delays are investigated [15–20]. Unfortunately, so far, only a few works have been done on synchronization of CGNNs, which remains challenging. In 2010,  Li et al. investigated the synchronization of discrete-time CGNNs with delays [6] and Chen discussed the synchronization of impulsive delayed CGNNs under noise perturbation [7], in 2010, Zhu and Cao discussed adaptive synchronization of delayed CGNNs [21] and in  2012, Gan also discussed adaptive synchronization of delayed CGNNs [22], and in 2011, Yu et al. discuss synchronization of delayed CGNNs via periodically intermittent control [23].Note that the previous methods in [6, 7, 21–23] have obtained error dynamical system first, and then discuss the error system, but, the error dynamical system of Cohen-Grossberg neural network which comparing the Hopfield neural networks becomes very complex because of existence of the amplification functions of the neural network; hence the results above turn to be quite complex, too. Comparing with foregoing works, the objective of this paper is to study the complete synchronization of nonautonomous CGNNs with mixed time delays by using suitable Lyapunov function and the properties of M-matrix, and our methods can avoid writing explicit complex error system which leads to complicated conditions for synchronization.The rest of this paper is organized as follows. The model, some definitions, and assumptions are presented in Section2. The sufficient conditions for complete synchronization of bidirectional coupled CGNNs are obtained in Section 3. Two examples are given in Section 4 to demonstrate the main results.


## 2. Model Description and Preliminaries

Consider the following bidirectional coupled nonautonomous CGNNs with time delays:(2.1)x˙i(t)=-ai(xi(t))[bi(t,xi(t))-∑j=1n‍uij(t)fj(xj(t))-∑j=1n‍vij(t)fj(xj(t-τij(t)))-ai(xi(t))-∑j=1n‍wij(t)∫0+∞‍kij(s)fj(xj(t-s))ds-Ii(t)]+∑j=1n‍αij(t)[yj(t)-xj(t)],(2.2)y˙i(t)=-ai(yi(t))[bi(t,yi(t))-∑j=1n‍uij(t)fj(yj(t))-∑j=1n‍vij(t)fj(yj(t-τij(t)))-ai(xi(t))-∑j=1n‍wij(t)∫0+∞‍kij(s)fj(yj(t-s))ds-Ii(t)]+∑j=1n‍βij(t)[xj(t)-yj(t)]
for 1≤i≤n, t>0. xi(t) and yi(t) denote the state variable of the ith neuron in (2.1) and (2.2), fj(·) denote the signal functions of the jth neuron at time t; Ii(t) denote inputs of the ith neuron at time t; ai(·) represent amplification functions; bi(t,·) are appropriately behaved functions; uij(t), vij(t), and wij(t) are bounded connection weights of the neural networks, respectively; τij(t) correspond to the finite speed of the axonal signal transmission, there exist positive τij and τ˙ij+ such that 0≤τij(t)≤τij and τ˙ij(t)≤τ˙ij+<1; kij(·) correspond to the delay kernel functions, coupled matrix are (αij(t))n×n and (βij(t))n×n in which αii(t)≥0 and βii(t)≥0 and αij(t),  βij(t) are bounded on [0,+∞).Throughout this paper, we assume for system (2.1) and (2.2) that(H1) amplification functions ai(·) are continuous and there exist constants a_i,a-i such that 0<a_i≤ai(·)≤a-i for 1≤i≤n;(H2) there exist positive continuous and bounded functions bi(t) such that
(2.3)bi(t,x)-bi(t,y)x-y≥bi(t)>0
for all x≠y,1≤i≤n;(H3) for activation functions fj(·), there exist positive constants Lj such that
(2.4)Lj=supx≠y|fj(x)-fj(y)x-y|
for all x≠y,1≤j≤n;(H4) the kernel functions kij(s) are nonnegative continuous function on [0,+∞) and satisfy
(2.5)∫0+∞‍seλskij(s)ds<+∞,Kij(λ)=∫0+∞‍eλskij(s)ds
are differentiable functions for λ∈[0,rij),  0<rij<+∞,  Kij(0)=1 and limλ→rij-Kij(λ)=+∞.Remark 2.1.
A typical example of kernel function is given bykij(s)=(sr/r!)rijr+1e-rijs for s∈[0,+∞), where rij∈(0,+∞),r∈{0,1,…,n}. These kernel functions are called as the gamma memory filter [24] and satisfy condition (H4).
For any bounded functions(t) on [0,+∞),  s_ and s- denote inft∈[0,+∞){|s(t)|} and supt∈[0,+∞){|s(t)|}, respectively.
For anyx(t)=(xi(t),x2(t),…,xk(t))T∈Rk,t>0, define ∥x(t)∥1=∑i=1k‍|xi(t)|, and for any φ(s)=(φ1(s),φ2(s),…,φk(s))T∈Rk,s∈(-∞,0], define ∥φ∥=sups∈(-∞,0]∑i=1k‍|φi(s))|.
Denote(2.6)C((-∞,0],Rk)={ψ:(-∞,0]→Rk∣ψ(s)isboundedandcontinuousfors∈(-∞,0]},
Then C((-∞,0],Rk) is a Banach space with respect to ∥·∥.
The initial conditions of system (2.1) and (2.2) are given by
(2.7)xi(s)=φi(s),-∞<s≤0,1≤i≤n,yi(s)=ςi(s),-∞<s≤0,1≤i≤n.Definition 2.2.
System (2.1) and system (2.2) are said to achieve global exponential complete synchronization, if for any solution x(t,ψ1) of system (2.1) and any solution y(t,ψ2) of system (2.2), there exist positive constants λ and M such that
(2.8)∥y(t,ψ2)-x(t,ψ1)∥1≤M∥ψ2-ψ1∥e-λt,t≥0,
where ψ2(s)=(ς1(s),ς2(s),…,ςn(s))T,ψ1(s)=(φ1(s),φ2(s),…,φn(s))T∈C((-∞,0],Rn).Definition 2.3.
A real matrixA=(aij)n×n is said to be a nonsingular M-matrix if aij≤0(i,j=1,2,…,n,i≠j), and all successive principle minors of A are positive.Lemma 2.4 (see [25]).
A matrix with nonpositive offdiagonal elementsA=(aij)n×n is a nonsingular M-matrix if and only if there exists a vector p=(pi)1×n>0 such that pA>0 or ApT>0 holds.Lemma 2.5 (see [26]).
Leta<b≤+∞. Suppose that v(t)=(v1(t),v2(t),…,vn(t))T satisfies the following differential equality:
(2.9)D+v(t)≤Pv(t)+(Q⊗v~(t))en+∫0+∞‍(R⊗K(s))v(t-s)ds,t∈[a,b),v(a+s)∈C((-∞,0],Rn),s∈(-∞,0],
where
(2.10)P=(pij)n×n,pij≥0fori≠j,Q=(qij)n×n,qij≥0,R=(rij)n×n,rij≥0,v~(t)=(vj(t-τij(t)))n×n,en=(1,1,…,1)T∈Rn,K(s)=(|kij(s)|)n×n,
and ⊗ means Hadamard product. If initial conditions satisfy
(2.11)v(t)≤κξe-λ(t-a),κ≥0,t∈(-∞,a],
where ξ=(ξ1,ξ2,…,ξn)T>0 and the positive number λ is determined by the following inequality:
(2.12)[λE+P+Q⊗ε(λ)+(R⊗𝒦(λ))]ξ<0
in which
(2.13)ε(λ)=(eλτij)n×n,𝒦(λ)=(𝒦ij(λ))n×n,𝒦ij(λ)=∫0+∞‍eλs|kij(s)|ds,
and E is an identity matrix. Then v(t)≤κξe-λ(t-a) for t∈[a,b).


## 3. Main Results

Theorem 3.1.
Under assumptions (H1)–(H4), system (2.1) and system (2.2) will achieve global exponential synchronization, if the following conditions hold:
(H5)   ℳ1=A1-C1 is a nonsingular M-matrix, where
(3.1)A1=diag(b_1+γ_11a-1,b_2+γ_22a-2,…,b_n+γ_nna-n),C1=(cij)n×n,cij=(u-ij+v-ij11-τ˙ij++w-ij)Lj+qij,
where qij=γ-ij/a_i,j≠i and qii=0,γij(t)=αij(t)+βij(t),i,j=1,2,…,n.Proof.
Letx(t),y(t) be two solutions of system (2.1) with initial value ψ1=(φ1,φ2,…,φn) and system (2.2) with ψ2=(ς1,ς2,…,ςn)∈C((-∞,0],Rn), respectively.
Denoteei(t)=yi(t)-xi(t).
Note that conditions (H5), ℳ1 is a nonsingular M-matrix implies that ℳ1T is a nonsingular M-matrix. From Lemma 2.4, we know that there exists a vector p=(p1,p2,…,pn)T such that ℳ1Tp>0, that is
(3.2)pi(b_i+γ_iia-i)-∑j=1n‍pj(u-ji+v-ji1-τ˙ji++w-ji)Li-∑j=1,j≠in‍pjγ-jia_j>0
for 1≤i≤n.
Denote(3.3)Fi(θ)=pi(-θa_i+b_i+γ_iia-i)-∑j=1n‍pj(u-jiLi+v-jieθτji1-τ˙ji+Li+w-jiLi∫0+∞‍kji(s)eθsds)-∑j=1,j≠in‍pjγ-jia_j,
for 1≤i≤n, which indicates Fi(0)>0. Since Fi(θ) are continuous and differential on [0,rji) on [0,rji) in which rji are some positive constants, and limθ→rji-Fi(θ)=-∞ according to condition (H4), furthermore, Fi′(θ)<0 for θ∈[0,rji). There exist constants θi such that Fi(θi)=0 for i=1,2,…,n. So we can choose
(3.4)0<λ≤min1≤i≤n{θi}
such that
(3.5)Fi(λ)≥0.
Let
(3.6)Vi(t)=eλtsign(ei(t))∫xi(t)yi(t)‍1ai(s)dsfori=1,2,…,n,
where ei(t)=yi(t)-xi(t).
Calculating the upper right derivative ofVi(t) along solutions of (2.1) and (2.2), we get
(3.7)D+Vi(t)≤eλt{{∑j=1,j≠in‍γ-ija_i}λa_i|ei(t)|-sign(ei(t))(bi(t,yi(t))-bi(t,xi(t)))eλt+|∑j=1n‍uij(t)(fj(yj(t))-fj(xj(t)))|eλt+|∑j=1n‍vij(t)(fj(yj(t-τij(t)))-fj(xj(t-τij(t))))|eλt+|∑j=1n‍wij(t)(∫0+∞‍kij(s)fj(yj(t-s))ds-∫0+∞‍kij(s)fj(xj(t-s))ds)|eλt-γ_iia-i|ei(t)|+∑j=1,j≠in‍γ-ija_i|ej(t)|}≤eλt{(λa_i-b_i-γ_iia-i)|ei(t)|+∑j=1n‍u-ijLj|ej(t)|+∑j=1n‍v-ijLj|ej(t-τij(t))|eλt+∑j=1n‍w-ijLj∫0+∞‍kij(s)|ej(t-s)|ds+∑j=1,j≠in‍γ-ija_i|ej(t)|},
where γij(t)=αij(t)+βij(t).
Now we define a Lyapunov functionV(t) by
(3.8)V(t)=∑i=1n‍pi{Vi(t)+∑j=1n‍v-ijLjeλτij1-τ˙ij+∫t-τij(t)t‍|ej(s)|eλsds∑i=1n‍pi+∑j=1n‍w-ijLj∫0+∞‍kij(s)∫t-st‍|ej(μ)|eλ(s+μ)dμds}.
We can obtain that
(3.9)D+V(t)≤eλt∑i=1n‍pi{{∑j=1,j≠in‍γ-ija_i}(λa_i-b_i-γ_iia-i)|ei(t)|eλt∑i=1n‍pi+∑j=1n‍(u-ij+v-ijeλτij1-τ˙ij++w-ij∫0+∞‍kij(s)eλsds)eλt∑i=1n‍pi×Lj|ej(t)|+∑j=1,j≠in‍γ-ija_i|ej(t)|}=-eλt∑i=1n‍Fi(λ)|ei(t)|≤0,
which together with (3.6) and (3.8) leads to
(3.10)m0eλt∑i=1n‍|ei(t)|≤V(t)≤V(0)≤M0∥ψ2(l)-ψ1(l)∥,
where
(3.11)m0=min1≤i≤n{pia-i},M0=max{M1,M2,M3},M1=max1≤i≤n(pia_i),M2=max1≤i≤n((eλτ-1)λ∑j=1n‍pjv-jiLi1-τ˙ji+),M3=∑j=1n‍pjmax1≤i≤n(w-jiLi)∫0+∞‍seλsmax1≤i≤nkji(s)ds.
Hence, we obtain that the following inequality holds:
(3.12)∑i=1n‍|ei(t)|≤M0m0∥ψ2(l)-ψ1(l)∥e-λt,t≥0,
that is,
(3.13)∥y(t)-x(t)∥1≤M0m0∥ψ2(l)-ψ1(l)∥e-λt,t≥0.
This completes the proof.Theorem 3.2.
Under assumptions (H1)–(H4), system (2.2) and system (2.1) will achieve global exponential synchronization, if the following conditions hold:
(H6)   ℳ2=A2-C2 is a nonsingular M-matrix, where
(3.14)A2=diag(a_1(b_1+γ_11a-1),a_2(b_2+γ_22a-2),…,a_n(b_n+γ_nna-n)),C2=(cij)n×n,cij=a-j[(u-ij+v-ij+w-ij)Lj+qij],
in which qij=γ-ij/a_i,j≠i,qii=0 and γij(t)=αij(t)+βij(t),i,j=1,2,…,n.Proof.
Letx(t),y(t) be two solutions of system (2.1) with initial value ψ1=(φ1,φ2,…,φn) and system (2.2) with ψ2=(ς1,ς2,…,ςn)∈C((-∞,0],Rn), respectively.
Denoteei(t)=yi(t)-xi(t).
Let(3.15)Vi(t)=sign(ei(t))∫xi(t)yi(t)‍1ai(s)dsfori=1,2,…,n
and note that
(3.16)1a-i|ei(t)|≤Vi(t)≤1a_i|ei(t)|.
Calculating the upper right derivative of Vi(t) along solutions of (2.1) and (2.2), we can get
(3.17)D+Vi(t)≤-b_i|ei(t)|+∑j=1n‍u-ijLj|ej(t)|+∑j=1n‍v-ijLj|ej(t-τij(t))|+∑j=1n‍w-ijLj∫0+∞‍kij(s)|ej(t-s)|ds-γii(t)a-i|ei(t)|+∑j=1,j≠in‍γ-ija_i|ej(t)|≤-b_ia_iVi(t)+∑j=1n‍u-ijLja-jVj(t)+∑j=1n‍v-ijLja-jVj(t-τij(t))+∑j=1n‍w-ijLja-j∫0+∞‍kij(s)Vj(t-s)ds-γ_iia_ia-iVi(t)+∑j=1,j≠in‍γ-ija-ja_iVj(t),
that is,
(3.18)D+V(t)≤AV(t)+(C⊗V~(t))En+∫0+∞‍(D⊗K(s))V(t-s)ds,
where
(3.19)V(t)=(V1(t),V2(t),…,Vn(t))T,V~(t)=(Vj(t-τij(t))n×n,A=diag(-a_1(b_1+γ_11a-1),-a_2(b_2+γ_22a-2),…,-a_n(b_n+γ_nna-n))+H,H=(hij)n×n,hij=a-ju-ijLj+q~ij,q~ij=a-jγ-ija_i,j≠i,q~ii=0,i,j=1,2,…,n,C=(c~ij)n×n,c~ij=a-jv-ijLj,D=(dij)n×n,dij=a-jw-ijLj,En=(1,1,…,1)T∈Rn,K(s)=(kij(s))n×n.
We know from (H6) that M2 is nonsingular M-matrix, which implies -(A+D+C) is also a nonsingular M-matrix. From Lemma 2.4, there exists a vector ξ=(ξ1,ξ2,…,ξn)T>0 such that -(A+D+C)ξ>0, consequently,
(3.20)-ξia_i(b_i+γ_iia_ia-i)+∑j=1n‍ξja-j(u-ij+v-ij+w-ij)Lj+∑j=1,j≠in‍ξja-jγ-ija_i>0.
Consider the function
(3.21)Fi(θ)=ξi(θ-a_i(b_i+γ_iia_ia-i))+∑j=1n‍ξja-j(u-ij+v-ijeθτij+w-ijKij(θ))Lj+∑j=1,j≠in‍ξja-jγ-ija_i,
in which Kij(θ)=∫0+∞‍eθτijkij(s)ds.
We obtain from condition (H4) that Kij(0)=1, which, together with (3.20), leads to Fi(0)<0, and similar to proof of Theorem 3.1 above, there exist
(3.22)0<λ<min1≤i≤n{θi}
such that
(3.23)Fi(λ)<0,
that is,
(3.24)[λE+A+C⊗(eλτij)n×n+D⊗𝒦(λ)]ξ<0,
where K(λ)=(Kij(λ))n×n in which Kij(λ) shown in (3.21).
Note thatVi(s)≤1/a_i∥ψ2-ψ1∥ and denote κ=∥ψ2-ψ1∥/min1≤i≤n{a_iξi}, we have
(3.25)V(t)≤κξe-λt,t∈(-∞,0].
We have from Lemma 2.5, (3.18), (3.24), and (3.25) that
(3.26)V(t)≤κξe-λt,t≥0.
It follows from (3.16) that
(3.27)|ei(t)|≤a-iVi(t)≤a-iκ~ξi∥ψ2-ψ1∥e-λt
in which κ~=1/min1≤i≤n{a_iξi}.
Hence(3.28)∥y(t)-x(t)∥1≤M∥ψ2-ψ1∥e-λt,t≥0,
where M=(∑i=1n‍a-iξi)/min1≤i≤n{a_iξi}. This completes the proof.Corollary 3.3.
Under assumptions (H1)–(H4), response system (2.2) will be globally exponentially synchronized with master system (2.1) with αij=0,i,j=1,2,…,n, if M1 in Theorem 3.1 and M2 in Theorem 3.2 are nonsingular M-matrices with γij(t)=βij(t).Remark 3.4.
Ifai(·)=1,bi(t,xi(t))=bi(t)xi(t) and bi(t,yi(t))=bi(t)yi(t), system (2.1) and (2.2) reduce to coupled Hopfield neural networks. Both Theorems 3.1 and 3.2 reduce to the simpler cases in which a-i=a_i=1.Remark 3.5.
Ifαij(t)=βij(t)=0,M1 in Theorem 3.1 and M2 in Theorem 3.2 still be nonsingular M-matrix, both theorems imply the global asymptotical stability of solutions of system (2.1), consequently, system (2.2) and (2.1) achieve complete synchronization for sure. In addition, if system (2.1) and (2.2) reduce to autonomous systems, the results in both theorems above still hold.


## 4. Two Simple Examples

Example 4.1.
Consider the following coupled CGNNs:(4.1)x˙(t)=-a(x(t))[Ax(t)-B∫0+∞‍K(s)tanh(x(t-s))ds-I(t)]+K1(t)(y(t)-x(t)),(4.2)y˙(t)=-a(y(t))[Ay(t)-B∫0+∞‍K(s)tanh(y(t-s))ds-I(t)]+K2(t)(x(t)-y(t)),
where we write system (4.1) and (4.2) in the vector-matrix form and
(4.3)x(t)=(x1(t),x2(t))T,y(t)=(y1(t),y2(t))T,I(t)=(2,2)T,tanh(x(t))=(tanh(x1(t)),tanh(x2(t)))T,tanh(y(t))=(tanh(y1(t)),tanh(y2(t)))T,a(x(t))=diag(2+sin(x1(t)),2+sin(x2(t))),a(y(t))=diag(2+sin(y1(t)),2+sin(y2(t))),A=(1002),B=(0-2cos(t)2-0),K(s)=(0e-te-t,0),K1(t)=(α1(t)00α2(t)),K2(t)=(β1(t)00β2(t)).
LetK1(t)=0. Figure 1 shows the dynamical behaviors of (4.1) with K1(t)=0.
LetK1(t)=diag(1,1) and K2(t)=diag(5,0), it is easy to know M1=(5-2-21) is still a M-matrix. From Theorem 3.1, we know coupled system (4.1) and system (4.2) achieve complete synchronization, and Figure 2 shows the dynamical behaviors of complete synchronization with initial conditions (y1(s),y2(s))=(2,2) and (x1(s),x2(s))=(0.1,0.1).Figure 1
Time response of statex1,x2 of system (4.1) with initial value (0.1,0.1) and K1=0.Complete synchronization of coupled systems (4.1) and (4.2).
(a)
(b)Example 4.2.
Consider the following Cohen-Grossberg neural network with discrete delay:(4.4)x˙1(t)=-3[x1(t)-3.5sin(x2(t-0.4))],x˙2(t)=-2[2x2(t)-3sin(x1(t-0.4))].
System (4.4) is chaotic system. Figure 3 shows the chaotic behaviors of system (4.4) with initial condition (0.2,0.2).(a) Phase plot of system (4.4) with initial condition (0.2,0.2). (b) Power spectrums of time series of x1 and x2 for system (4.4).
(a)
(b)For driving system (4.4), we construct the response system as follows:
(4.5)y˙1(t)=-3[y1(t)-3.5sin(y2(t-0.4))]+α11(x1(t)-y1(t)),y˙2(t)=-2[2y2(t)-3sin(y1(t-0.4))]+α22(x2(t)-y2(t)),
where α11>0,α22>0.Letα11=9,α22=0. It is easy to know
(4.6)M1=(3-3.5-34)
is a M-matrix. From Corollary 3.3, we know the response system (4.5) achieves complete synchronization with system (4.4), and Figure 4 shows the dynamical behaviors of complete synchronization with initial conditions (y1(s),y2(s))=(2,2) and (x1(s),x2(s))=(0.2,0.2).Complete synchronization of the response system (4.5) and the driving system (4.4).
(a)
(b)
(c)


## 5. Conclusions

Based on using suitable Lyapunov function and the properties ofM-matrix, sufficient conditions for complete synchronization of bidirectional coupled CGNNs are directly obtained without writing the explicit error system. Two examples show the effectiveness of the proposed method. Note that M1 and M2 must be M-matrix if we let αii(t)+βii(t), that is, γii(t) be big enough to a certain extent such that M1 and M2 are strongly diagonally dominant matrix, this shows that our criteria are easy to verify and are useful for synchronization of CGNNs.



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