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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 481582, 16 pages
Synchronization between Bidirectional Coupled Nonautonomous Delayed Cohen-Grossberg Neural Networks
Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China
Received 15 May 2012; Accepted 3 September 2012
Academic Editor: Wing-Sum Cheung
Copyright © 2012 Qiming Liu and Rui Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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 , 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  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  and Chen discussed the synchronization of impulsive delayed CGNNs under noise perturbation , in 2010, Zhu and Cao discussed adaptive synchronization of delayed CGNNs  and in 2012, Gan also discussed adaptive synchronization of delayed CGNNs , and in 2011, Yu et al. discuss synchronization of delayed CGNNs via periodically intermittent control .
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 -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: for , . and denote the state variable of the th neuron in (2.1) and (2.2), denote the signal functions of the th neuron at time t; denote inputs of the th neuron at time ; represent amplification functions; are appropriately behaved functions; , , and are bounded connection weights of the neural networks, respectively; correspond to the finite speed of the axonal signal transmission, there exist positive and such that and ; correspond to the delay kernel functions, coupled matrix are and in which and and , are bounded on .
() amplification functions are continuous and there exist constants , such that for ;
() there exist positive continuous and bounded functions such that for all ;
() for activation functions , there exist positive constants such that for all ;
() the kernel functions are nonnegative continuous function on and satisfy are differentiable functions for , , and .
Remark 2.1. A typical example of kernel function is given by for , where . These kernel functions are called as the gamma memory filter  and satisfy condition (H4).
For any bounded function on , and denote and , respectively.
For any , define , and for any , define .
Denote Then is a Banach space with respect to .
The initial conditions of system (2.1) and (2.2) are given by
Definition 2.2. System (2.1) and system (2.2) are said to achieve global exponential complete synchronization, if for any solution of system (2.1) and any solution of system (2.2), there exist positive constants and such that where .
Definition 2.3. A real matrix is said to be a nonsingular -matrix if , and all successive principle minors of are positive.
Lemma 2.4 (see ). A matrix with nonpositive offdiagonal elements is a nonsingular -matrix if and only if there exists a vector such that or holds.
Lemma 2.5 (see ). Let . Suppose that satisfies the following differential equality: where and means Hadamard product. If initial conditions satisfy where and the positive number is determined by the following inequality: in which and is an identity matrix. Then for .
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) is a nonsingular -matrix, where where and .
Proof. Let be two solutions of system (2.1) with initial value and system (2.2) with , respectively.
Note that conditions (H5), is a nonsingular -matrix implies that is a nonsingular -matrix. From Lemma 2.4, we know that there exists a vector such that , that is for .
Denote for , which indicates . Since are continuous and differential on on in which are some positive constants, and according to condition (H4), furthermore, for . There exist constants such that for . So we can choose such that Let where .
Calculating the upper right derivative of along solutions of (2.1) and (2.2), we get where .
Now we define a Lyapunov function by We can obtain that which together with (3.6) and (3.8) leads to where Hence, we obtain that the following inequality holds: that is, 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) is a nonsingular -matrix, where in which , and .
Proof. Let be two solutions of system (2.1) with initial value and system (2.2) with , respectively.
Let and note that Calculating the upper right derivative of along solutions of (2.1) and (2.2), we can get that is, where We know from (H6) that is nonsingular -matrix, which implies is also a nonsingular -matrix. From Lemma 2.4, there exists a vector such that , consequently, Consider the function in which .
We obtain from condition (H4) that , which, together with (3.20), leads to , and similar to proof of Theorem 3.1 above, there exist such that that is, where in which shown in (3.21).
Note that and denote , we have We have from Lemma 2.5, (3.18), (3.24), and (3.25) that It follows from (3.16) that in which .
Hence where . 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 , if in Theorem 3.1 and in Theorem 3.2 are nonsingular -matrices with .
Remark 3.5. If , in Theorem 3.1 and in Theorem 3.2 still be nonsingular -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:
where we write system (4.1) and (4.2) in the vector-matrix form and
Let . Figure 1 shows the dynamical behaviors of (4.1) with .
Let and , it is easy to know is still a -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 and .
For driving system (4.4), we construct the response system as follows: where .
Let . It is easy to know is a -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 and .
Based on using suitable Lyapunov function and the properties of -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 and must be -matrix if we let , that is, be big enough to a certain extent such that and are strongly diagonally dominant matrix, this shows that our criteria are easy to verify and are useful for synchronization of CGNNs.
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.
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