Adaptive Synchronization Between Two Different ... - Semantic Scholar
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 18, NO. 6, 2007
Adaptive Synchronization Between Two Different Chaotic Neural Networks With Time Delay Huaguang Zhang, Yinghui Xie, Zhiliang Wang, and Chengde Zheng
Abstract—This letter presents an adaptive synchronization scheme between two different kinds of delayed chaotic neural networks (NNs) with partly unknown parameters. An adaptive controller is designed to guarantee the global asymptotic synchronization of state trajectories for two different chaotic NNs with time delay. An illustrative example is given to demonstrate the effectiveness of the present method. Index Terms—Adaptive synchronization, chaotic neural networks (NNs), time delay.
I. INTRODUCTION Chaotic neural networks (NNs) which employ a chaotic-dynamic mechanism are well known for solving optimization problems [7], [16]. They have a rich range and flexible dynamics, which give them the ability to search for global optimal solutions. Milanovic´ et al. [12] propose a synchronization method of chaotic NNs and applications to communications. Tan et al. [15] propose an associative memory model using synchronization in a chaotic NN. Because of their potential applications in these fields, chaotic NNs have drawn more and more attention. Recently, there have been many efforts [10], [17] for the study of dynamical properties of delayed neural networks (DNNs). Most previous studies have predominantly concentrated on the stability analysis and periodic oscillations of this kind of DNNs [1], [5], [14]. It has been shown that such DNNs can exhibit complicated dynamics and even chaotic behavior if the parameters and time delays are appropriately chosen for the DNNs [2], [6], [11]. There are a few studies in the synchronization of this class of chaotic NNs with or without delays [3], [12], [15]. But the underlying assumption in those papers is that the drive and the response systems have identical dynamic structure and the same parameters. There is no study in the synchronization between two time-delayed chaotic NNs with different structure and partly unknown parameters to the best of our knowledge. On the other hand, the synchronization of chaotic systems has been extensively studied over the past two decades due to its potential applications in creating secure communication systems [4], [9]. Recently, there were some results on the synchronization of different chaotic
Manuscript received October 3, 2006; revised March 1, 2007 and April 16, 2007; accepted May 1, 2007. This work was supported by the National Natural Science Foundation of China under Grants 60325311, 60534010, and 60572070 and the Program for Changjiang Scholars and Innovative Research Group of China under Grant 60521003. H. Zhang and Z. Wang are with the College of Information Science and Engineering, Northeastern University, Shenyang, Liaonong 110004, P. R. China, and the Key Laboratory of Integrated Automation of Process Industry of Education Ministry, Shenyang, Liaonong 110004, P. R. China (e-mail: hgzhang@ieee. org). Y. Xie is with the College of Information Science and Engineering, Northeastern University, Shenyang, Liaonong 110004, P. R. China, and the Department of Foundation, ShenYang Artillery Institute, Shenyang, Liaoning 110163, P. R. China (e-mail: [email protected]). C. Zheng is with the Department of Mathematics, Dalian Jiaotong University, Dalian, Liaoning 116028, P. R. China (e-mail: [email protected]). Digital Object Identifier 10.1109/TNN.2007.902958
1841
systems with unknown parameters in the literature [8], [18]. But, the chaotic systems considered in these results all are not the time-delay chaotic systems. In real-world applications, however, it is hardly the case that the parameters of the drive and response chaotic NNs are identical. In engineering applications, time delays always exist, parameters of the system are inevitably perturbed by external factors, and the value of delays and parameters is often unknown in advance; so synchronizing two different time-delayed chaotic NNs with partly unknown parameters is more essential and useful in real-world applications. In this letter, we propose a method to adaptively synchronize two time-delayed chaotic NNs with different structures and partly unknown parameters. This class of delayed chaotic NNs can include several wellknown delayed chaotic NNs, such as delayed chaotic Hopfield NNs and delayed chaotic cellular NNs. By using the drive-response concept [13] and the Lyapunov stability theory, an adaptive synchronization controller is designed and a parameter update law is developed. This letter is organized as follows. In Section II, the problem studied in this letter is described. In Section III, an adaptive synchronization controller is designed and a parameter update law is developed. In Section IV, a simulation example is shown. In Section V, conclusions are given. The following notation will be used throughout this letter. For a 2 1=2 denote the Euclidean vector vector x 2
Adaptive Synchronization Between Two Different Chaotic Neural Networks With Time Delay Huaguang Zhang, Yinghui Xie, Zhiliang Wang, and Chengde Zheng
Abstract—This letter presents an adaptive synchronization scheme between two different kinds of delayed chaotic neural networks (NNs) with partly unknown parameters. An adaptive controller is designed to guarantee the global asymptotic synchronization of state trajectories for two different chaotic NNs with time delay. An illustrative example is given to demonstrate the effectiveness of the present method. Index Terms—Adaptive synchronization, chaotic neural networks (NNs), time delay.
I. INTRODUCTION Chaotic neural networks (NNs) which employ a chaotic-dynamic mechanism are well known for solving optimization problems [7], [16]. They have a rich range and flexible dynamics, which give them the ability to search for global optimal solutions. Milanovic´ et al. [12] propose a synchronization method of chaotic NNs and applications to communications. Tan et al. [15] propose an associative memory model using synchronization in a chaotic NN. Because of their potential applications in these fields, chaotic NNs have drawn more and more attention. Recently, there have been many efforts [10], [17] for the study of dynamical properties of delayed neural networks (DNNs). Most previous studies have predominantly concentrated on the stability analysis and periodic oscillations of this kind of DNNs [1], [5], [14]. It has been shown that such DNNs can exhibit complicated dynamics and even chaotic behavior if the parameters and time delays are appropriately chosen for the DNNs [2], [6], [11]. There are a few studies in the synchronization of this class of chaotic NNs with or without delays [3], [12], [15]. But the underlying assumption in those papers is that the drive and the response systems have identical dynamic structure and the same parameters. There is no study in the synchronization between two time-delayed chaotic NNs with different structure and partly unknown parameters to the best of our knowledge. On the other hand, the synchronization of chaotic systems has been extensively studied over the past two decades due to its potential applications in creating secure communication systems [4], [9]. Recently, there were some results on the synchronization of different chaotic
Manuscript received October 3, 2006; revised March 1, 2007 and April 16, 2007; accepted May 1, 2007. This work was supported by the National Natural Science Foundation of China under Grants 60325311, 60534010, and 60572070 and the Program for Changjiang Scholars and Innovative Research Group of China under Grant 60521003. H. Zhang and Z. Wang are with the College of Information Science and Engineering, Northeastern University, Shenyang, Liaonong 110004, P. R. China, and the Key Laboratory of Integrated Automation of Process Industry of Education Ministry, Shenyang, Liaonong 110004, P. R. China (e-mail: hgzhang@ieee. org). Y. Xie is with the College of Information Science and Engineering, Northeastern University, Shenyang, Liaonong 110004, P. R. China, and the Department of Foundation, ShenYang Artillery Institute, Shenyang, Liaoning 110163, P. R. China (e-mail: [email protected]). C. Zheng is with the Department of Mathematics, Dalian Jiaotong University, Dalian, Liaoning 116028, P. R. China (e-mail: [email protected]). Digital Object Identifier 10.1109/TNN.2007.902958
1841
systems with unknown parameters in the literature [8], [18]. But, the chaotic systems considered in these results all are not the time-delay chaotic systems. In real-world applications, however, it is hardly the case that the parameters of the drive and response chaotic NNs are identical. In engineering applications, time delays always exist, parameters of the system are inevitably perturbed by external factors, and the value of delays and parameters is often unknown in advance; so synchronizing two different time-delayed chaotic NNs with partly unknown parameters is more essential and useful in real-world applications. In this letter, we propose a method to adaptively synchronize two time-delayed chaotic NNs with different structures and partly unknown parameters. This class of delayed chaotic NNs can include several wellknown delayed chaotic NNs, such as delayed chaotic Hopfield NNs and delayed chaotic cellular NNs. By using the drive-response concept [13] and the Lyapunov stability theory, an adaptive synchronization controller is designed and a parameter update law is developed. This letter is organized as follows. In Section II, the problem studied in this letter is described. In Section III, an adaptive synchronization controller is designed and a parameter update law is developed. In Section IV, a simulation example is shown. In Section V, conclusions are given. The following notation will be used throughout this letter. For a 2 1=2 denote the Euclidean vector vector x 2
