Synchronization between Bidirectional Coupled Nonautonomous Delayed Cohen-Grossberg Neural Networks

Abstract

Based on using suitable Lyapunov function and the properties of M-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.

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 Section 2. 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: 𝑥̇_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)𝑏_𝑖𝑡,𝑥_𝑖(𝑡)−^𝑛_𝑗=1𝑢_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)−^𝑛_𝑗=1𝑣_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑤_𝑖𝑗(𝑡)₀^+∞𝑘_𝑖𝑗(𝑠)𝑓_𝑗𝑥_𝑗(𝑡−𝑠)𝑑𝑠−𝐼_𝑖(𝑡)+^𝑛_𝑗=1𝛼_𝑖𝑗(𝑡)𝑦_𝑗(𝑡)−𝑥_𝑗(𝑡,𝑦̇_𝑖)(𝑡)=−𝑎_𝑖𝑦_𝑖(𝑡)𝑏_𝑖𝑡,𝑦_𝑖(𝑡)−^𝑛_𝑗=1𝑢_𝑖𝑗(𝑡)𝑓_𝑗𝑦_𝑗(𝑡)−^𝑛_𝑗=1𝑣_𝑖𝑗(𝑡)𝑓_𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑤_𝑖𝑗(𝑡)₀^+∞𝑘_𝑖𝑗(𝑠)𝑓_𝑗𝑦_𝑗(𝑡−𝑠)𝑑𝑠−𝐼_𝑖(𝑡)+^𝑛_𝑗=1𝛽_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)−𝑦_𝑗(𝑡) (2.1)(2.2) for 1≤𝑖≤𝑛, 𝑡>0. 𝑥_𝑖(𝑡) and 𝑦_𝑖(𝑡) denote the state variable of the 𝑖th neuron in (2.1) and (2.2), 𝑓_𝑗(⋅) denote the signal functions of the 𝑗th neuron at time t; 𝐼_𝑖(𝑡) denote inputs of the 𝑖th neuron at time 𝑡; 𝑎_𝑖(⋅) represent amplification functions; 𝑏_𝑖(𝑡,⋅) are appropriately behaved functions; 𝑢_𝑖𝑗(𝑡), 𝑣_𝑖𝑗(𝑡), and 𝑤_𝑖𝑗(𝑡) are bounded connection weights of the neural networks, respectively; 𝜏_𝑖𝑗(𝑡) correspond to the finite speed of the axonal signal transmission, there exist positive 𝜏_𝑖𝑗 and 𝜏̇⁺_𝑖𝑗 such that 0≤𝜏_𝑖𝑗(𝑡)≤𝜏_𝑖𝑗 and 𝜏̇_𝑖𝑗(𝑡)≤𝜏̇⁺_𝑖𝑗<1; 𝑘_𝑖𝑗(⋅) correspond to the delay kernel functions, coupled matrix are (𝛼_𝑖𝑗(𝑡))_𝑛×𝑛 and (𝛽_𝑖𝑗(𝑡))_𝑛×𝑛 in which 𝛼_𝑖𝑖(𝑡)≥0 and 𝛽_𝑖𝑖(𝑡)≥0 and 𝛼_𝑖𝑗(𝑡),  𝛽_𝑖𝑗(𝑡) are bounded on [0,+∞).

Throughout this paper, we assume for system (2.1) and (2.2) that

(H₁) amplification functions 𝑎_𝑖(⋅) are continuous and there exist constants 𝑎_𝑖,𝑎_𝑖 such that 0<𝑎_𝑖≤𝑎_𝑖(⋅)≤𝑎_𝑖 for 1≤𝑖≤𝑛;

(H₂) there exist positive continuous and bounded functions 𝑏_𝑖(𝑡) such that 𝑏_𝑖(𝑡,𝑥)−𝑏_𝑖𝑡,𝑦𝑥−𝑦≥𝑏_𝑖(𝑡)>0 (2.3) for all 𝑥≠𝑦,1≤𝑖≤𝑛;

(H₃) for activation functions 𝑓_𝑗(⋅), there exist positive constants 𝐿_𝑗 such that 𝐿_𝑗=sup_𝑥≠𝑦𝑓_𝑗(𝑥)−𝑓_𝑗𝑦𝑥−𝑦 (2.4) for all 𝑥≠𝑦,1≤𝑗≤𝑛;

(H₄) the kernel functions 𝑘_𝑖𝑗(𝑠) are nonnegative continuous function on [0,+∞) and satisfy ₀^+∞𝑠𝑒^𝜆𝑠𝑘_𝑖𝑗(𝑠)𝐾𝑑𝑠<+∞,_𝑖𝑗(𝜆)=₀^+∞𝑒^𝜆𝑠𝑘_𝑖𝑗(𝑠)𝑑𝑠 (2.5) are differentiable functions for 𝜆∈[0,𝑟_𝑖𝑗),  0<𝑟_𝑖𝑗<+∞,  𝐾_𝑖𝑗(0)=1 and lim_𝜆→𝑟⁻_𝑖𝑗𝐾_𝑖𝑗(𝜆)=+∞.

Remark 2.1. A typical example of kernel function is given by 𝑘_𝑖𝑗(𝑠)=(𝑠^𝑟/𝑟!)𝑟^𝑟+1_𝑖𝑗𝑒^{−𝑟𝑖𝑗𝑠} for 𝑠∈[0,+∞), where 𝑟_𝑖𝑗∈(0,+∞),𝑟∈{0,1,…,𝑛}. These kernel functions are called as the gamma memory filter [24] and satisfy condition (H₄).
For any bounded function 𝑠(𝑡) on [0,+∞), 𝑠 and 𝑠 denote inf_{𝑡∈[0,+∞)}{|𝑠(𝑡)|} and sup_{𝑡∈[0,+∞)}{|𝑠(𝑡)|}, respectively.
For any 𝑥(𝑡)=(𝑥_𝑖(𝑡),𝑥₂(𝑡),…,𝑥_𝑘(𝑡))^𝑇∈𝑅^𝑘,𝑡>0, define ‖𝑥(𝑡)‖₁=∑^𝑘_𝑖=1|𝑥_𝑖(𝑡)|, and for any 𝜑(𝑠)=(𝜑₁(𝑠),𝜑₂(𝑠),…,𝜑_𝑘(𝑠))^𝑇∈𝑅^𝑘,𝑠∈(−∞,0], define ‖𝜑‖=sup_{𝑠∈(−∞,0]}∑^𝑘_𝑖=1|𝜑_𝑖(𝑠))|.
Denote 𝐶(−∞,0],𝑅^𝑘=𝜓∶(−∞,0]→𝑅^𝑘∣𝜓(𝑠)isboundedandcontinuousfor𝑠∈(−∞,0], (2.6) Then 𝐶((−∞,0],𝑅^𝑘) is a Banach space with respect to ‖⋅‖.
The initial conditions of system (2.1) and (2.2) are given by 𝑥_𝑖(𝑠)=𝜑_𝑖(𝑠)𝑦,−∞<𝑠≤0,1≤𝑖≤𝑛,_𝑖(𝑠)=𝜍_𝑖(𝑠),−∞<𝑠≤0,1≤𝑖≤𝑛. (2.7)

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 𝑦𝑡,𝜓₂−𝑥𝑡,𝜓₁₁≤𝑀𝜓₂−𝜓₁𝑒^−𝜆𝑡,𝑡≥0, (2.8) where 𝜓₂(𝑠)=(𝜍₁(𝑠),𝜍₂(𝑠),…,𝜍_𝑛(𝑠))^𝑇,𝜓₁(𝑠)=(𝜑₁(𝑠),𝜑₂(𝑠),…,𝜑_𝑛(𝑠))^𝑇∈𝐶((−∞,0],𝑅^𝑛).

Definition 2.3. A real matrix 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛 is said to be a nonsingular 𝑀-matrix if 𝑎_𝑖𝑗≤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 elements 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛 is a nonsingular 𝑀-matrix if and only if there exists a vector 𝑝=(𝑝_𝑖)_1×𝑛>0 such that 𝑝𝐴>0 or 𝐴𝑝^𝑇>0 holds.

Lemma 2.5 (see [26]). Let 𝑎<𝑏≤+∞. Suppose that 𝑣(𝑡)=(𝑣₁(𝑡),𝑣₂(𝑡),…,𝑣_𝑛(𝑡))^𝑇 satisfies the following differential equality: 𝐷⁺𝑣(𝑡)≤𝑃𝑣(𝑡)+(𝑄⊗𝑣(𝑡))𝑒_𝑛+₀^+∞((𝑠𝑣()[),𝑣()]𝑅⊗𝐾))𝑡−𝑠𝑑𝑠,𝑡∈𝑎,𝑏𝑎+𝑠∈𝐶(−∞,0,𝑅^𝑛,𝑠∈(−∞,0], (2.9) where 𝑃=𝑝_{𝑖𝑗𝑛×𝑛},𝑝_𝑖𝑗𝑞≥0for𝑖≠𝑗,𝑄=_{𝑖𝑗𝑛×𝑛},𝑞_𝑖𝑗≥0,𝑅=𝑟_{𝑖𝑗𝑛×𝑛},𝑟_𝑖𝑗𝑣(𝑡)=𝑣≥0,_𝑗𝑡−𝜏_𝑖𝑗(𝑡)_𝑛×𝑛,𝑒_𝑛=(1,1,…,1)^𝑇∈𝑅^𝑛,𝐾(𝑠)=𝑘_𝑖𝑗(𝑠)_𝑛×𝑛, (2.10) and ⊗ means Hadamard product. If initial conditions satisfy 𝑣(𝑡)≤𝜅𝜉𝑒^{−𝜆(𝑡−𝑎)},𝜅≥0,𝑡∈(−∞,𝑎], (2.11) where 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑛)^𝑇>0 and the positive number 𝜆 is determined by the following inequality: [𝜆𝐸+𝑃+𝑄⊗𝜀(𝜆)+(𝑅⊗𝒦(𝜆))]𝜉<0 (2.12) in which 𝜀(𝜆)=𝑒^{𝜆𝜏𝑖𝑗}_𝑛×𝑛,𝒦(𝜆)=𝒦_𝑖𝑗(𝜆)_𝑛×𝑛,𝒦_𝑖𝑗(𝜆)=₀^+∞𝑒^𝜆𝑠𝑘_𝑖𝑗(𝑠)𝑑𝑠, (2.13) 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 𝐴₁=diag𝑏₁+𝛾₁₁𝑎₁,𝑏₂+𝛾₂₂𝑎₂,…,𝑏_𝑛+𝛾_𝑛𝑛𝑎_𝑛,𝐶₁=𝑐_{𝑖𝑗𝑛×𝑛},𝑐_𝑖𝑗=𝑢_𝑖𝑗+𝑣_𝑖𝑗11−𝜏̇⁺_𝑖𝑗+𝑤_𝑖𝑗𝐿_𝑗+𝑞_𝑖𝑗, (3.1) where 𝑞_𝑖𝑗=𝛾_𝑖𝑗/𝑎_𝑖,𝑗≠𝑖 and 𝑞_𝑖𝑖=0,𝛾_𝑖𝑗(𝑡)=𝛼_𝑖𝑗(𝑡)+𝛽_𝑖𝑗(𝑡),𝑖,𝑗=1,2,…,𝑛.

Proof. Let 𝑥(𝑡),𝑦(𝑡) be two solutions of system (2.1) with initial value 𝜓₁=(𝜑₁,𝜑₂,…,𝜑_𝑛) and system (2.2) with 𝜓₂=(𝜍₁,𝜍₂,…,𝜍_𝑛)∈𝐶((−∞,0],𝑅^𝑛), respectively.
Denote 𝑒_𝑖(𝑡)=𝑦_𝑖(𝑡)−𝑥_𝑖(𝑡).
Note that conditions (H₅), ℳ₁ is a nonsingular 𝑀-matrix implies that ℳ^𝑇₁ is a nonsingular 𝑀-matrix. From Lemma 2.4, we know that there exists a vector 𝑝=(𝑝₁,𝑝₂,…,𝑝_𝑛)^𝑇 such that ℳ^𝑇₁𝑝>0, that is 𝑝_𝑖𝑏_𝑖+𝛾_𝑖𝑖𝑎_𝑖−^𝑛_𝑗=1𝑝_𝑗𝑢_𝑗𝑖+𝑣_𝑗𝑖1−𝜏̇⁺_𝑗𝑖+𝑤_𝑗𝑖𝐿_𝑖−^𝑛_{𝑗=1,𝑗≠𝑖}𝑝_𝑗𝛾_𝑗𝑖𝑎_𝑗>0 (3.2) for 1≤𝑖≤𝑛.
Denote 𝐹_𝑖(𝜃)=𝑝_𝑖−𝜃𝑎_𝑖+𝑏_𝑖+𝛾_𝑖𝑖𝑎_𝑖−^𝑛_𝑗=1𝑝_𝑗𝑢_𝑗𝑖𝐿_𝑖+𝑣_𝑗𝑖𝑒^{𝜃𝜏𝑗𝑖}1−𝜏̇⁺_𝑗𝑖𝐿_𝑖+𝑤_𝑗𝑖𝐿_𝑖0^+∞𝑘_𝑗𝑖(𝑠)𝑒^𝜃𝑠−𝑑𝑠_{𝑛𝑗=1,𝑗≠𝑖}𝑝_𝑗𝛾_𝑗𝑖𝑎_𝑗, (3.3) for 1≤𝑖≤𝑛, which indicates 𝐹_𝑖(0)>0. Since 𝐹_𝑖(𝜃) are continuous and differential on [0,𝑟_𝑗𝑖) on [0,𝑟_𝑗𝑖) in which 𝑟_𝑗𝑖 are some positive constants, and lim_𝜃→𝑟⁻_𝑗𝑖𝐹_𝑖(𝜃)=−∞ according to condition (H₄), furthermore, 𝐹^′_𝑖(𝜃)<0 for 𝜃∈[0,𝑟_𝑗𝑖). There exist constants 𝜃_𝑖 such that 𝐹_𝑖(𝜃_𝑖)=0 for 𝑖=1,2,…,𝑛. So we can choose 0<𝜆≤min_{1≤𝑖≤𝑛}𝜃_𝑖 (3.4) such that 𝐹_𝑖(𝜆)≥0. (3.5) Let 𝑉_𝑖(𝑡)=𝑒^𝜆𝑡sign𝑒_𝑖(𝑡)^𝑦𝑖_𝑥^(𝑡)_𝑖(𝑡)1𝑎_𝑖(𝑠)𝑑𝑠for𝑖=1,2,…,𝑛, (3.6) where 𝑒_𝑖(𝑡)=𝑦_𝑖(𝑡)−𝑥_𝑖(𝑡).
Calculating the upper right derivative of 𝑉_𝑖(𝑡) along solutions of (2.1) and (2.2), we get 𝐷⁺𝑉_𝑖(𝑡)≤𝑒^𝜆𝑡𝜆𝑎_𝑖𝑒_𝑖(𝑡)−sign𝑒_𝑖(𝑡)𝑏_𝑖𝑡,𝑦_𝑖(𝑡)−𝑏_𝑖𝑡,𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑢_𝑖𝑗(𝑡)𝑓_𝑗𝑦_𝑗(𝑡)−𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑣_𝑖𝑗(𝑡)𝑓_𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝑓_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝑤_𝑖𝑗(𝑡)₀^+∞𝑘_𝑖𝑗(𝑠)𝑓_𝑗𝑦_𝑗(𝑡−𝑠)𝑑𝑠−₀^+∞𝑘_𝑖𝑗(𝑠)𝑓_𝑗𝑥_𝑗()−𝛾𝑡−𝑠𝑑𝑠_𝑖𝑖𝑎_𝑖𝑒_𝑖(𝑡)+^𝑛_{𝑗=1,𝑗≠𝑖}𝛾_𝑖𝑗𝑎_𝑖𝑒_𝑗(𝑡)≤𝑒^𝜆𝑡𝜆𝑎_𝑖−𝑏_𝑖−𝛾_𝑖𝑖𝑎_𝑖𝑒_𝑖(𝑡)+^𝑛_𝑗=1𝑢_𝑖𝑗𝐿_𝑗𝑒_𝑗(𝑡)+^𝑛_𝑗=1𝑣_𝑖𝑗𝐿_𝑗𝑒_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝑤_𝑖𝑗𝐿_𝑗0^+∞𝑘_𝑖𝑗(𝑠)𝑒_𝑗(𝑡−𝑠)𝑑𝑠+^𝑛_{𝑗=1,𝑗≠𝑖}𝛾_𝑖𝑗𝑎_𝑖𝑒_𝑗(𝑡), (3.7) where 𝛾_𝑖𝑗(𝑡)=𝛼_𝑖𝑗(𝑡)+𝛽_𝑖𝑗(𝑡).
Now we define a Lyapunov function 𝑉(𝑡) by 𝑉(𝑡)=^𝑛_𝑖=1𝑝_𝑖𝑉_𝑖(𝑡)+^𝑛_𝑗=1𝑣_𝑖𝑗𝐿_𝑗𝑒^{𝜆𝜏𝑖𝑗}1−𝜏̇⁺_𝑖𝑗^𝑡_{𝑡−𝜏𝑖𝑗(𝑡)}𝑒_𝑗(𝑠)𝑒^𝜆𝑠+𝑑𝑠_𝑛𝑗=1𝑤_𝑖𝑗𝐿_𝑗0^+∞𝑘_𝑖𝑗(𝑠)^𝑡_𝑡−𝑠𝑒_𝑗𝜇𝑒^{𝜆(𝑠+𝜇)}𝑑𝜇𝑑𝑠. (3.8) We can obtain that 𝐷⁺𝑉(𝑡)≤𝑒^𝑛𝜆𝑡_𝑖=1𝑝_𝑖𝜆𝑎_𝑖−𝑏_𝑖−𝛾_𝑖𝑖𝑎_𝑖𝑒_𝑖(𝑡+_𝑛𝑗=1)𝑢_𝑖𝑗+𝑣_𝑖𝑗𝑒^{𝜆𝜏𝑖𝑗}1−𝜏̇⁺_𝑖𝑗+𝑤_𝑖𝑗0^+∞𝑘_𝑖𝑗(𝑠)𝑒^𝜆𝑠𝑑𝑠×𝐿_𝑗𝑒_𝑗(𝑡)+^𝑛_{𝑗=1,𝑗≠𝑖}𝛾_𝑖𝑗𝑎_𝑖𝑒_𝑗(𝑡)=−𝑒^𝑛𝜆𝑡_𝑖=1𝐹_𝑖(𝜆)𝑒_𝑖(𝑡)≤0, (3.9) which together with (3.6) and (3.8) leads to 𝑚₀𝑒^𝑛𝜆𝑡_𝑖=1𝑒_𝑖(𝑡)≤𝑉(𝑡)≤𝑉(0)≤𝑀₀𝜓₂(𝑙)−𝜓₁(𝑙), (3.10) where 𝑚₀=min_{1≤𝑖≤𝑛}𝑝_𝑖𝑎_𝑖,𝑀₀=max𝑀₁,𝑀₂,𝑀₃,𝑀₁=max_{1≤𝑖≤𝑛}𝑝_𝑖𝑎_𝑖,𝑀₂=max_{1≤𝑖≤𝑛}𝑒^𝜆𝜏−1𝜆^𝑛_𝑗=1𝑝_𝑗𝑣_𝑗𝑖𝐿_𝑖1−𝜏̇⁺_𝑗𝑖,𝑀₃=^𝑛_𝑗=1𝑝_𝑗max_{1≤𝑖≤𝑛}𝑤_𝑗𝑖𝐿_𝑖0^+∞𝑠𝑒^𝜆𝑠max_{1≤𝑖≤𝑛}𝑘_𝑗𝑖(𝑠)𝑑𝑠. (3.11) Hence, we obtain that the following inequality holds: ^𝑛_𝑖=1𝑒_𝑖(𝑡≤𝑀)⁰𝑚₀𝜓₂(𝑙)−𝜓₁(𝑙)𝑒^−𝜆𝑡,𝑡≥0, (3.12) that is, 𝑦(𝑡)−𝑥(𝑡)₁≤𝑀₀𝑚₀𝜓₂(𝑙)−𝜓₁(𝑙)𝑒^−𝜆𝑡,𝑡≥0. (3.13) 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 𝐴₂=diag𝑎₁𝑏₁+𝛾₁₁𝑎₁,𝑎₂𝑏₂+𝛾₂₂𝑎₂,…,𝑎_𝑛𝑏_𝑛+𝛾_𝑛𝑛𝑎_𝑛,𝐶₂=𝑐_{𝑖𝑗𝑛×𝑛},𝑐_𝑖𝑗=𝑎_𝑗𝑢_𝑖𝑗+𝑣_𝑖𝑗+𝑤_𝑖𝑗𝐿_𝑗+𝑞_𝑖𝑗, (3.14) in which 𝑞_𝑖𝑗=𝛾_𝑖𝑗/𝑎_𝑖,𝑗≠𝑖,𝑞_𝑖𝑖=0 and 𝛾_𝑖𝑗(𝑡)=𝛼_𝑖𝑗(𝑡)+𝛽_𝑖𝑗(𝑡),𝑖,𝑗=1,2,…,𝑛.

Proof. Let 𝑥(𝑡),𝑦(𝑡) be two solutions of system (2.1) with initial value 𝜓₁=(𝜑₁,𝜑₂,…,𝜑_𝑛) and system (2.2) with 𝜓₂=(𝜍₁,𝜍₂,…,𝜍_𝑛)∈𝐶((−∞,0],𝑅^𝑛), respectively.
Denote 𝑒_𝑖(𝑡)=𝑦_𝑖(𝑡)−𝑥_𝑖(𝑡).
Let 𝑉_𝑖(𝑡)=sign𝑒_𝑖(𝑡)^𝑦𝑖_𝑥^(𝑡)_𝑖(𝑡)1𝑎_𝑖(𝑠)𝑑𝑠for𝑖=1,2,…,𝑛 (3.15) and note that 1𝑎_𝑖𝑒_𝑖(𝑡)≤𝑉_𝑖(𝑡)≤1𝑎_𝑖𝑒_𝑖(𝑡). (3.16) Calculating the upper right derivative of 𝑉_𝑖(𝑡) along solutions of (2.1) and (2.2), we can get 𝐷⁺𝑉_𝑖(𝑡)≤−𝑏_𝑖𝑒_𝑖(𝑡)+^𝑛_𝑗=1𝑢_𝑖𝑗𝐿_𝑗𝑒_𝑗(𝑡)+^𝑛_𝑗=1𝑣_𝑖𝑗𝐿_𝑗𝑒_𝑗𝑡−𝜏_𝑖𝑗(𝑡+_𝑛𝑗=1)𝑤_𝑖𝑗𝐿_𝑗0^+∞𝑘_𝑖𝑗(𝑠)𝑒_𝑗(𝛾𝑡−𝑠)𝑑𝑠−^𝑖𝑖(𝑡)𝑎_𝑖𝑒_𝑖(𝑡)+^𝑛_{𝑗=1,𝑗≠𝑖}𝛾_𝑖𝑗𝑎_𝑖𝑒_𝑗(𝑡)≤−𝑏_𝑖𝑎_𝑖𝑉_𝑖(𝑡)+^𝑛_𝑗=1𝑢_𝑖𝑗𝐿_𝑗𝑎_𝑗𝑉_𝑗(𝑡)+^𝑛_𝑗=1𝑣_𝑖𝑗𝐿_𝑗𝑎_𝑗𝑉_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝑤_𝑖𝑗𝐿_𝑗𝑎_𝑗0^+∞𝑘_𝑖𝑗(𝑠)𝑉_𝑗()𝛾𝑡−𝑠𝑑𝑠−^𝑖𝑖𝑎_𝑖𝑎_𝑖𝑉_𝑖(𝑡)+^𝑛_{𝑗=1,𝑗≠𝑖}𝛾_𝑖𝑗𝑎_𝑗𝑎_𝑖𝑉_𝑗(𝑡), (3.17) that is, 𝐷⁺𝑉(𝑡)≤𝐴𝑉(𝑡)+𝐶⊗𝑉(𝑡)𝐸_𝑛+₀^+∞(𝐷⊗𝐾(𝑠))𝑉(𝑡−𝑠)𝑑𝑠, (3.18) where 𝑉(𝑡)=𝑉₁(𝑡),𝑉₂(𝑡),…,𝑉_𝑛(𝑡)^𝑇,𝑉(𝑡)=(𝑉_𝑗𝑡−𝜏_𝑖𝑗(𝑡)_𝑛×𝑛,𝐴=diag−𝑎₁𝑏₁+𝛾₁₁𝑎₁,−𝑎₂𝑏₂+𝛾₂₂𝑎₂,…,−𝑎_𝑛𝑏_𝑛+𝛾_𝑛𝑛𝑎_𝑛ℎ+𝐻,𝐻=_{𝑖𝑗𝑛×𝑛},ℎ_𝑖𝑗=𝑎_𝑗𝑢_𝑖𝑗𝐿_𝑗+𝑞_𝑖𝑗,𝑞_𝑖𝑗=𝑎_𝑗𝛾_𝑖𝑗𝑎_𝑖,𝑗≠𝑖,𝑞_𝑖𝑖𝑐̃=0,𝑖,𝑗=1,2,…,𝑛,𝐶=_{𝑖𝑗𝑛×𝑛},𝑐̃_𝑖𝑗=𝑎_𝑗𝑣_𝑖𝑗𝐿_𝑗,𝐷=𝑑_{𝑖𝑗𝑛×𝑛},𝑑_𝑖𝑗=𝑎_𝑗𝑤_𝑖𝑗𝐿_𝑗,𝐸_𝑛=(1,1,…,1)^𝑇∈𝑅^𝑛,𝐾(𝑠)=𝑘_𝑖𝑗(𝑠)_𝑛×𝑛. (3.19) 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,…,𝜉_𝑛)^𝑇>0 such that −(𝐴+𝐷+𝐶)𝜉>0, consequently, −𝜉_𝑖𝑎_𝑖𝑏_𝑖+𝛾_𝑖𝑖𝑎_𝑖𝑎_𝑖+^𝑛_𝑗=1𝜉_𝑗𝑎_𝑗𝑢_𝑖𝑗+𝑣_𝑖𝑗+𝑤_𝑖𝑗𝐿_𝑗+^𝑛_{𝑗=1,𝑗≠𝑖}𝜉_𝑗𝑎_𝑗𝛾_𝑖𝑗𝑎_𝑖>0. (3.20) Consider the function 𝐹_𝑖(𝜃)=𝜉_𝑖𝜃−𝑎_𝑖𝑏_𝑖+𝛾_𝑖𝑖𝑎_𝑖𝑎_𝑖+^𝑛_𝑗=1𝜉_𝑗𝑎_𝑗𝑢_𝑖𝑗+𝑣_𝑖𝑗𝑒^{𝜃𝜏𝑖𝑗}+𝑤_𝑖𝑗𝐾_𝑖𝑗(𝜃)𝐿_𝑗+^𝑛_{𝑗=1,𝑗≠𝑖}𝜉_𝑗𝑎_𝑗𝛾_𝑖𝑗𝑎_𝑖, (3.21) in which 𝐾_𝑖𝑗(𝜃)=∫₀^+∞𝑒^{𝜃𝜏𝑖𝑗}𝑘_𝑖𝑗(𝑠)𝑑𝑠.
We obtain from condition (H₄) that 𝐾_𝑖𝑗(0)=1, which, together with (3.20), leads to 𝐹_𝑖(0)<0, and similar to proof of Theorem 3.1 above, there exist 0<𝜆<min_{1≤𝑖≤𝑛}𝜃_𝑖 (3.22) such that 𝐹_𝑖(𝜆)<0, (3.23) that is, 𝜆𝐸+𝐴+𝐶⊗𝑒^{𝜆𝜏𝑖𝑗}_𝑛×𝑛+𝐷⊗𝒦(𝜆)𝜉<0, (3.24) where 𝐾(𝜆)=(𝐾_𝑖𝑗(𝜆))_𝑛×𝑛 in which 𝐾_𝑖𝑗(𝜆) shown in (3.21).
Note that 𝑉_𝑖(𝑠)≤1/𝑎_𝑖‖𝜓₂−𝜓₁‖ and denote 𝜅=‖𝜓₂−𝜓₁‖/min_{1≤𝑖≤𝑛}{𝑎_𝑖𝜉_𝑖}, we have 𝑉(𝑡)≤𝜅𝜉𝑒^−𝜆𝑡,𝑡∈(−∞,0]. (3.25) We have from Lemma 2.5, (3.18), (3.24), and (3.25) that 𝑉(𝑡)≤𝜅𝜉𝑒^−𝜆𝑡,𝑡≥0. (3.26) It follows from (3.16) that 𝑒_𝑖(𝑡)≤𝑎_𝑖𝑉_𝑖(𝑡)≤𝑎_𝑖𝜅𝜉_𝑖𝜓₂−𝜓₁𝑒^−𝜆𝑡 (3.27) in which 𝜅=1/min_{1≤𝑖≤𝑛}{𝑎_𝑖𝜉_𝑖}.
Hence 𝑦(𝑡)−𝑥(𝑡)₁≤𝑀𝜓₂−𝜓₁𝑒^−𝜆𝑡,𝑡≥0, (3.28) where 𝑀=(∑^𝑛_𝑖=1𝑎_𝑖𝜉_𝑖)/min_{1≤𝑖≤𝑛}{𝑎_𝑖𝜉_𝑖}. 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 𝛼_𝑖𝑗=0,𝑖,𝑗=1,2,…,𝑛, if 𝑀₁ in Theorem 3.1 and 𝑀₂ in Theorem 3.2 are nonsingular 𝑀-matrices with 𝛾_𝑖𝑗(𝑡)=𝛽_𝑖𝑗(𝑡).

Remark 3.4. If 𝑎_𝑖(⋅)=1,𝑏_𝑖(𝑡,𝑥_𝑖(𝑡))=𝑏_𝑖(𝑡)𝑥_𝑖(𝑡) and 𝑏_𝑖(𝑡,𝑦_𝑖(𝑡))=𝑏_𝑖(𝑡)𝑦_𝑖(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. If 𝛼_𝑖𝑗(𝑡)=𝛽_𝑖𝑗(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: 𝑥̇(𝑡)=−𝑎(𝑥(𝑡))𝐴𝑥(𝑡)−𝐵₀^+∞𝐾(𝑠)tanh(𝑥(𝑡−𝑠))𝑑𝑠−𝐼(𝑡)+𝐾₁(𝑡)𝑦(𝑡)(𝑡),𝑦̇(𝑡)𝑦(𝑡)(𝑡)−𝑥=−𝑎𝐴𝑦−𝐵₀^+∞𝐾(𝑠)tanh𝑦(𝑡−𝑠)𝑑𝑠−𝐼(𝑡)+𝐾₂(𝑡)𝑥(𝑡)−𝑦(𝑡), (4.1)(4.2) where we write system (4.1) and (4.2) in the vector-matrix form and 𝑥(𝑡)=𝑥₁(𝑡),𝑥₂(𝑡)^𝑇,𝑦(𝑡)=𝑦₁(𝑡),𝑦₂(𝑡)^𝑇,𝐼(𝑡)=(2,2)^𝑇,tanh(𝑥(𝑡))=tanh𝑥₁(𝑡),tanh𝑥₂(𝑡)^𝑇,tanh𝑦(𝑡)=tanh𝑦₁(𝑡),tanh𝑦₂(𝑡)^𝑇,𝑎(𝑥(𝑡))=diag2+sin𝑥₁(𝑡),2+sin𝑥₂(𝑡),𝑎𝑦(𝑡)=diag2+sin𝑦₁(𝑡),2+sin𝑦₂(𝑡),(𝑡)𝐴=(𝑠)=1002,𝐵=0−2cos2−0,𝐾0e^−𝑡e^−𝑡,𝐾,0₁(𝑡)=𝛼₁(𝑡)00𝛼₂(𝑡),𝐾₂(𝑡)=𝛽₁(𝑡)00𝛽₂(𝑡). (4.3)
Let 𝐾₁(𝑡)=0. Figure 1 shows the dynamical behaviors of (4.1) with 𝐾₁(𝑡)=0.
Let 𝐾₁(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 state 𝑥1,𝑥2 of system (4.1) with initial value (0.1,0.1) and 𝐾1=0.

(a)

(b)

Figure 2 

Complete synchronization of coupled systems (4.1) and (4.2).

Example 4.2. Consider the following Cohen-Grossberg neural network with discrete delay: 𝑥̇₁(𝑡)=−3𝑥₁(𝑡)−3.5sin𝑥₂(),𝑥̇𝑡−0.4₂(𝑡)=−22𝑥₂(𝑡)−3sin𝑥₁(𝑡−0.4). (4.4) System (4.4) is chaotic system. Figure 3 shows the chaotic behaviors of system (4.4) with initial condition (0.2,0.2).

(a)

(b)

Figure 3 

(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).

For driving system (4.4), we construct the response system as follows: 𝑦̇₁(𝑡)=−3𝑦₁(𝑡)−3.5sin𝑦₂(𝑡−0.4)+𝛼₁₁𝑥₁(𝑡)−𝑦₁(𝑡),𝑦̇₂(𝑡)=−22𝑦₂(𝑡)−3sin𝑦₁(𝑡−0.4)+𝛼₂₂𝑥₂(𝑡)−𝑦₂(𝑡), (4.5) where 𝛼₁₁>0,𝛼₂₂>0.

Let 𝛼₁₁=9,𝛼₂₂=0. It is easy to know 𝑀₁=3−3.5−34 (4.6) 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).

(a)

(b)

(c)

Figure 4 

Complete synchronization of the response system (4.5) and the driving system (4.4).

5. Conclusions

Based on using suitable Lyapunov function and the properties of 𝑀-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 𝛼_𝑖𝑖(𝑡)+𝛽_𝑖𝑖(𝑡), that is, 𝛾_𝑖𝑖(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.

Acknowledgments

The authors are grateful to the editor and two reviewers for their careful reading of this paper, helpful comments, and valuable suggestions. This research was supported by the National Natural Science Foundation of China under Grant no. 11071254 and the Hebei Provincial Natural Science Foundation of China under Grant no. A2012205028.