混沌系统前向同步及滞同步的研究
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摘要
近年来,混沌同步引起了人们相当大的兴趣,因为混沌同步广泛的应用在各个领域,如物理,化学以及生物系统。在Voss发现混沌系统的前向同步后,越来越多的人们致力于对其进行理论分析。但是,带有参数失配的混沌系统的同步研究的比较少,甚至可以说没有。一般来说,由于噪声或人为因素,参数失配是不可避免的。因此,考虑有参数失配的混沌同步是很重要的。在本文中,我们将得到一些关于耦合混沌系统前向同步以及滞同步的准则。这篇论文的主要贡献如下:
①研究了耦合混沌系统的前向同步。利用Lyapunov-Krasovskii泛函和线性矩阵不等式,通过模型转换得到了误差系统指数稳定和渐近稳定的一些准则。基于这些准则,根据系统参数和前向时间,设计了耦合强度。
②研究了带有参数失配的混沌系统的前向同步。通过严格的理论分析,估计出前向同步误差边界。
③研究了带有时滞和参数失配的混沌系统的前向同步。利用Lyapunov泛函和线性矩阵不等式,通过严格的理论分析,得到了误差系统的收敛准则,并且估计出前向同步误差边界。
④研究了带有时滞和参数失配的混沌系统的滞同步。根据误差系统的指数稳定条件,带有小参数失配耦合系统可以达到滞准同步。通过严格的理论分析估计出滞同步的误差边界。
In the last decade, synchronization of chaos has triggered considerable interest mainly because of a wide range of applications in physical, chemical and biological systems. Recently, much effort has been devoted to the theoretical analysis of anticipating synchronization after the observation of this kind of synchronization by Voss. However, synchronization of chaotic systems with parameter mismatches is rarely studied. In general, parameter mismatch is inevitable in the synchronization of practical chaotic systems due to noise or other artificial factors. It is relevant issue to investigate the effects of parameter mismatch on synchronization although there are only a few reports on this so far. In this dissertation, several criteria for the anticipating synchronization and lag synchronization of coupled chaotic systems are established. The main contributions of this dissertation are as follows:
①The anticipating synchronization of a class of coupled chaotic systems without-delay is studied. The asymptotic stability and exponential stability criteria for the involved error dynamical system are established by means of model transformation incorporated with Lyapunov-Krasovskii functional and linear matrix inequality. Based on the proposed stability conditions the coupling strength is then explicitly designed in terms of system parameters and anticipation time.
②The anticipating synchronization of a class of coupled chaotic systems with parameter mismatches is studied. We estimate the error bound of anticipating synchronization by rigorously theoretical analysis.
③The anticipating synchronization of a class of coupled chaotic systems with time delay and parameter mismatches is studied. The convergence criteria for the error dynamical system under study are established. The error bound of anticipating synchronization is estimated by rigorous theoretical analysis.
④The lag synchronization of a class of coupled chaotic systems with time delay and parameter mismatches is studied. Lag synchronization of coupled systems may weakly achieve in the presence of small parameter mismatches in terms of exponential stability criteria. The error bound of lag-synchronization arising from the parameter mismatches is also estimated by rigorously theoretical analysis.
①研究了耦合混沌系统的前向同步。利用Lyapunov-Krasovskii泛函和线性矩阵不等式,通过模型转换得到了误差系统指数稳定和渐近稳定的一些准则。基于这些准则,根据系统参数和前向时间,设计了耦合强度。
②研究了带有参数失配的混沌系统的前向同步。通过严格的理论分析,估计出前向同步误差边界。
③研究了带有时滞和参数失配的混沌系统的前向同步。利用Lyapunov泛函和线性矩阵不等式,通过严格的理论分析,得到了误差系统的收敛准则,并且估计出前向同步误差边界。
④研究了带有时滞和参数失配的混沌系统的滞同步。根据误差系统的指数稳定条件,带有小参数失配耦合系统可以达到滞准同步。通过严格的理论分析估计出滞同步的误差边界。
In the last decade, synchronization of chaos has triggered considerable interest mainly because of a wide range of applications in physical, chemical and biological systems. Recently, much effort has been devoted to the theoretical analysis of anticipating synchronization after the observation of this kind of synchronization by Voss. However, synchronization of chaotic systems with parameter mismatches is rarely studied. In general, parameter mismatch is inevitable in the synchronization of practical chaotic systems due to noise or other artificial factors. It is relevant issue to investigate the effects of parameter mismatch on synchronization although there are only a few reports on this so far. In this dissertation, several criteria for the anticipating synchronization and lag synchronization of coupled chaotic systems are established. The main contributions of this dissertation are as follows:
①The anticipating synchronization of a class of coupled chaotic systems without-delay is studied. The asymptotic stability and exponential stability criteria for the involved error dynamical system are established by means of model transformation incorporated with Lyapunov-Krasovskii functional and linear matrix inequality. Based on the proposed stability conditions the coupling strength is then explicitly designed in terms of system parameters and anticipation time.
②The anticipating synchronization of a class of coupled chaotic systems with parameter mismatches is studied. We estimate the error bound of anticipating synchronization by rigorously theoretical analysis.
③The anticipating synchronization of a class of coupled chaotic systems with time delay and parameter mismatches is studied. The convergence criteria for the error dynamical system under study are established. The error bound of anticipating synchronization is estimated by rigorous theoretical analysis.
④The lag synchronization of a class of coupled chaotic systems with time delay and parameter mismatches is studied. Lag synchronization of coupled systems may weakly achieve in the presence of small parameter mismatches in terms of exponential stability criteria. The error bound of lag-synchronization arising from the parameter mismatches is also estimated by rigorously theoretical analysis.
引文
[1] Akihiro Yamaguchi, Hirotaka Watanabe, Sadayoshi Mikami, Mitsuo Wada. Characterization of biological internal dynamics by the synchronization of coupled chaotic system[J]. Robotics and Autonomous Systems, 1999, 28:195-206.
[2] William G. Baxt. Complexity, chaos and human physiology: the justification for non-linear neural computational analysis[J]. Cancer Letters, 1994, 77:85-93.
[3] Scott Barton. Chaos, self-organization, and psychology[J]. America Psychologist, 1994, 49:5-14.
[4] L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems[J]. Physical Review Letters, 1990, 64:821-824.
[5] Alexandra S., Landsman and Ira B. Schwartz. Complete chaotic synchronization in mutually coupled time-delay systems[J]. Physical review E, 2007, 75:026201.
[6] Meng Zhan, Xingang Wang, Xiaofeng Gong, G. W. Wei,and C.-H. Lai. Complete synchronization and generalized synchronization of one-way coupled time-delay systems[J]. Physical review E, 2003, 68:036208.
[7] I. P. Marino, E. A.llaria, M. A. F. Sanjuan, R. Meucci and F. T. Arecchi. Coupling scheme for complete synchronization of periodically forced chaotic CO2 lasers[J]. Physical review E, 2004, 70:036208.
[8] Shuguang Guan, Ying-Cheng Lai and Choy-Heng Lai. Effect of noise on generalized chaotic synchronization. Physical Review E, 2006, 73:046201.
[9] Jianfeng Lu. Generalized (complete, lag, anticpated) synchronization of discrete-time chaotic systems[J]. Communications in Nonlinear Science and Numerical Simulation, 2008, 13:1851-1859.
[10] Guo Hui Li, ShiPing Zhou and Kui Yang. Generalized projective synchronization between different chaotic systems using active backstepping control[J]. Physics Letters A, 2006, 355:326-330.
[11] Michael G. Rosenblum, Arkady S. Pikovsky and Jurgen Kurths. Phase sychronization in driven and coupled chaotic oscillators[J]. IEEE Transactions on Circutts and Systems-Ⅰ:Fundaental Theory and Applications, 1997, 44:874-881.
[12] Arkady S. Pikovsky, Michael G. Rsenblum, Grigory V. Osipov, Jurgen Kurths. Phase synchronization of chaotic oscillators by external driving[J]. Physica D, 1997, 104:219-238.
[13] E. M. Shahverdiev, S. Sivaprakasam and K. A. Shore. Lag synchronization in time-delayedsystems[J]. Physics Letters A, 2002, 292:320-324.
[14] Alexander N. Pisarchik and Rider Jaimes-Reategui. Intermittent lag synchronization in a nonautonomous system of coupled oscillators[J]. Physics Letters A, 2005, 338:141-149.
[15] Yun Chen, Xiaoxiang Chen, Sengshi Gu. Lag synchronization of struturally nonequivalent chaotic systems with time delays[J]. Nonlinear Analysis, 2007, 66:1929-1937.
[16] Saeed Taherion and Ying-Cheng Lai. Observability of lag synchronization of coupled chaotic oscillators[J]. Physical Review E, 59:6247-6250.
[17] Zhongkui Sun, Wei Xu and Xiaoli Yang. Adaptive scheme for time-varying anticipating synchronization of certain or uncertain chaotic dynamical systems[J]. Mathematical and Computer Modelling, 2008, 48:1018-1032.
[18] Marcin Kostur, Peter Hanggi, Peter Talkner and Jose L. Mateos. Anticipated synchronization in coupled inertial ratchets with time-delayed feedback: A numerical study[J]. Physical Review E, 2005, 72:036201.
[19] Thang Manh Hoang, and Masahiro Nakagawa. Anticipating and projective-anticipating synchronization of coupled multidelay feedback systems[J]. Physics Letters A, 2007, 365:407-411.
[20] Kenji Kusumoto and Junji Ohtsubo. Anticipating synchronization based on optical Injection-locking in chaotic semiconductor lasers[J]. IEEE Journal of Quantum Electronics, 2003, 39:1531-1536.
[21] Qi Han, Chuandong Li, Junjian Huang, Xiaofeng Liao and Tingwen Huang. Anticipating synchronization of chaotic systems with parameter mismatch[J]. The 9th International Conference for Young Computer Scientists. 2008, 2931-2936.
[22] F. Rogister, D. Pieroux, M. Sciamanna, P. Megret and M. Blondel. Anticipating synchronization of two chaotic laser diodes by incoherent optical coupling and its application to secure communications[J]. Optics Communications, 2002, 207:295-306.
[23] C. Masoller. Anticipating in the synchronization of chaotic semiconductor lasers with optical feedback[J]. Physical Review Letters, 2001, 86:2782-2785.
[24] M. Ciszak, J. M. Gutierrez, A.S. Cofino, and C.Mirasso. Approch to predictability via anticipted synchronization[J]. Physical Review E, 2005, 72:046218.
[25] Liang Wu and Shiqun Zhu. Coexistence and switching of anticipating synchronization and lag synchronization in an optical system[J]. Physics Letters A, 2003, 315:101-108.
[26] H.J. Wang, H.B. Huang and G.X. Qi. Coexistence of anticipated and layered chaotic synchronization in time-delay systems[J]. Physical Review E, 2005, 72:037203.
[27] Kestutis Pyragas and Tatjana Pyragiene. Coupling design for a long-term anticipatingsynchronization of chaos[J]. Physical Review E, 2008, 78:046217.
[28] S. Sivaprakasam, E.M. Shahverdiev, P.S. Spencer and K.A. Shore. Experimental demonstration of anticipating synchronization in chaotic semiconductor lasers with optical feedback[J]. Physical Review Letters, 2001, 87:104101.
[29] S. Tang and J.M. Liu. Experimental verification of anticipated and retarded synchronization in chaotic semiconductor lasers. Physical Review Letters, 2003, 90:194101.
[30] Shawn K. Pethel, Ned J. Corron, Quitisha R. Underwood and Krishna Myneni. Information flow in chaos synchronization: fundamental tradeoffs in precision, delay, and anticipation. Physical Review Letters, 2003, 90:254101.
[31] E.M.Shahverdiev, S. Sivaprakasam and K.A. Shore. Inverse anticipting chaos synchronization. Physical Review E, 2002, 66:017204.
[32] E.M.Shahverdiev, S. Sivaprakasam and K.A. Shore. Parameter mismatchs and perfect anticipating synchronization in bidirectionally coupled external cavity laser diodes. Physical Review E, 2002, 66:017206.
[33] Qi Han, Chuandong Li and Junjian Huang. Anticipating synchronization of chaotic sysems with time delay and parameter mismatch[J]. Chaos, 2009, 19:013104.
[34] Henning U. Voss. Anticipating chaotic synchronization[J]. Physical Review E, 2000, 61:5115-5119.
[35] Chuandong Li, Guanrong Chen, Xiaofeng Liao and Zhengping Fan. Chaos quasisyn chronization induced by impulses with parameter mismatches[J]. Chaos, 2006, 16:023102.
[36] Chuandong Li and Xiaofeng Liao. Complete and lag synchronization of hyperchaotic systems using small impulses[J]. Chaos, Solitons and Fractals, 2004, 22:857-867.
[37] Chuandong Li, Xiaofeng Liao and Rong Zhang. Delay-dependent exponential stability analysis of bi-directional associative memory neural networks with time delay: an LMI approach[J]. Chaos, Solitons and Fractals, 2005, 24:1119-1134.
[38] Chuandong Li, Xiaofeng Liao, Rong Zhang and Ashutosh Prasad. Global robust exponential stability analysis for interval neural networks with time-varying delays[J]. Chaos, Solitons and Fractals, 2005, 25:751-757.
[39]黄军建,间歇控制在神经网络中的应用研究[D]。重庆:重庆大学,2008.
[40] Tingwen Huang, Chuandong Li and Xiaofeng Liao. Synchronization of a class of coupled chaotic delayed systems with parameter mismatch[J]. Chaos, 2007, 17:033121.
[41] E.N.Sanchez and J.P. Perez, Input-to-state stability (ISS) analysis for dynamic neural networks[J]. IEEE Transactions on Circuits Systems-Ⅰ:Fundaental Theory and Applications, 1999, 46:1395-1398.
[42] S.Boyd, L.E.Ghaoui, E. Feron and V. Balakrishnan, Linear matrix inequalities in systems and control theory(M). SIAM Studies in Applied Mathematics, Philadephia, 1994.
[43] K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics(M). Kluwer Academic, Dordrecht, 1992.
[44] M.S. Fofana, Asymptotic stability of a stochatic delay equation(J). Probabilistic Engineering Mechanics, 2002, 17:385-392.
[45] Xiaofeng Liao, Yanbing Liu, Songtao and Huanhuan Mai, Asymptotic stability of delayed neural networks: A descriptor system approach(J). Commun Nonlinear Sci Numer Simulat, 2009, 14:3120-3133.
[46] Zhi Li and Guanrong Chen, Global synchronization and asymptotic stability of complex dynamical networks(J). IEEE Transactions on Circuits and Systems-Ⅱ: Express Briefs, 2006, 53:28-33.
[47] Sabri Arik, Global asymptotic stability of a larger class of neural networks with constant time delay(J). Physics Letters A, 2003, 311:504-511.
[48] R.Samidurai, R.Sakthivel and S.M.Anthoni, Global asymptotic stability of BAM neural networks with mixed delays and impulses(J). Applied Mathematics and Computation, 2009, 212:113-119.
[49] Hongyong Zhao, Global asymptotic stability of Hopfield neural network involving distributed delays(J). Neural Networks, 2004, 17:47-53.
[50] Xuyang Lou and Baotong Cui, New LMI conditions for delay-dependent asymptotic stability of delayed Hopfield neural networks(J). Neurocomputing, 2006, 69:2374-2378.
[51] Junkang Tian, Dongsheng Xu and Jian Zu, Novel delay-dependent asyptotic stability criteria for neural networks with time-varying delays(J). Journal of Computational and Applied Mathematics, 2009, 228:133-138.
[52] Xiaopeng Wei, Dongsheng Zhou and Qiang Zhang, On asymptotic stability of discrete-time non-autonomous delayed Hopfield neural networks(J). Computers and Mathematics with Applications, 2009, 57:1938-1942.
[53] O.M.Kwon and Ju H.Park, Exponential stability for uncertain cellualr neural networks with discrete and distributed time-varying delays(J). Applied Mathematics and Computation, 2008, 203:813-823.
[54] S. Mohamad and K. Gopalsamy, Exponential stability of continuous-time and discrete-time cellular neural networks with delays(J). Applied Mathematics and Computation, 2003, 135:17-38.
[55] Sanny Mohamad, Global exponential stability in continuous-time and discrete-time delayedbidirectional neural networks(J). Physica D, 2001, 159:233-151.
[56] Baoxian Wang, Jigui Jian and Chuande Guo, Global exponential stability of a class of BAM networks with time-varying delays and continuously distributed delays(J). Neyrocomputing, 2008, 71:495-501.
[2] William G. Baxt. Complexity, chaos and human physiology: the justification for non-linear neural computational analysis[J]. Cancer Letters, 1994, 77:85-93.
[3] Scott Barton. Chaos, self-organization, and psychology[J]. America Psychologist, 1994, 49:5-14.
[4] L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems[J]. Physical Review Letters, 1990, 64:821-824.
[5] Alexandra S., Landsman and Ira B. Schwartz. Complete chaotic synchronization in mutually coupled time-delay systems[J]. Physical review E, 2007, 75:026201.
[6] Meng Zhan, Xingang Wang, Xiaofeng Gong, G. W. Wei,and C.-H. Lai. Complete synchronization and generalized synchronization of one-way coupled time-delay systems[J]. Physical review E, 2003, 68:036208.
[7] I. P. Marino, E. A.llaria, M. A. F. Sanjuan, R. Meucci and F. T. Arecchi. Coupling scheme for complete synchronization of periodically forced chaotic CO2 lasers[J]. Physical review E, 2004, 70:036208.
[8] Shuguang Guan, Ying-Cheng Lai and Choy-Heng Lai. Effect of noise on generalized chaotic synchronization. Physical Review E, 2006, 73:046201.
[9] Jianfeng Lu. Generalized (complete, lag, anticpated) synchronization of discrete-time chaotic systems[J]. Communications in Nonlinear Science and Numerical Simulation, 2008, 13:1851-1859.
[10] Guo Hui Li, ShiPing Zhou and Kui Yang. Generalized projective synchronization between different chaotic systems using active backstepping control[J]. Physics Letters A, 2006, 355:326-330.
[11] Michael G. Rosenblum, Arkady S. Pikovsky and Jurgen Kurths. Phase sychronization in driven and coupled chaotic oscillators[J]. IEEE Transactions on Circutts and Systems-Ⅰ:Fundaental Theory and Applications, 1997, 44:874-881.
[12] Arkady S. Pikovsky, Michael G. Rsenblum, Grigory V. Osipov, Jurgen Kurths. Phase synchronization of chaotic oscillators by external driving[J]. Physica D, 1997, 104:219-238.
[13] E. M. Shahverdiev, S. Sivaprakasam and K. A. Shore. Lag synchronization in time-delayedsystems[J]. Physics Letters A, 2002, 292:320-324.
[14] Alexander N. Pisarchik and Rider Jaimes-Reategui. Intermittent lag synchronization in a nonautonomous system of coupled oscillators[J]. Physics Letters A, 2005, 338:141-149.
[15] Yun Chen, Xiaoxiang Chen, Sengshi Gu. Lag synchronization of struturally nonequivalent chaotic systems with time delays[J]. Nonlinear Analysis, 2007, 66:1929-1937.
[16] Saeed Taherion and Ying-Cheng Lai. Observability of lag synchronization of coupled chaotic oscillators[J]. Physical Review E, 59:6247-6250.
[17] Zhongkui Sun, Wei Xu and Xiaoli Yang. Adaptive scheme for time-varying anticipating synchronization of certain or uncertain chaotic dynamical systems[J]. Mathematical and Computer Modelling, 2008, 48:1018-1032.
[18] Marcin Kostur, Peter Hanggi, Peter Talkner and Jose L. Mateos. Anticipated synchronization in coupled inertial ratchets with time-delayed feedback: A numerical study[J]. Physical Review E, 2005, 72:036201.
[19] Thang Manh Hoang, and Masahiro Nakagawa. Anticipating and projective-anticipating synchronization of coupled multidelay feedback systems[J]. Physics Letters A, 2007, 365:407-411.
[20] Kenji Kusumoto and Junji Ohtsubo. Anticipating synchronization based on optical Injection-locking in chaotic semiconductor lasers[J]. IEEE Journal of Quantum Electronics, 2003, 39:1531-1536.
[21] Qi Han, Chuandong Li, Junjian Huang, Xiaofeng Liao and Tingwen Huang. Anticipating synchronization of chaotic systems with parameter mismatch[J]. The 9th International Conference for Young Computer Scientists. 2008, 2931-2936.
[22] F. Rogister, D. Pieroux, M. Sciamanna, P. Megret and M. Blondel. Anticipating synchronization of two chaotic laser diodes by incoherent optical coupling and its application to secure communications[J]. Optics Communications, 2002, 207:295-306.
[23] C. Masoller. Anticipating in the synchronization of chaotic semiconductor lasers with optical feedback[J]. Physical Review Letters, 2001, 86:2782-2785.
[24] M. Ciszak, J. M. Gutierrez, A.S. Cofino, and C.Mirasso. Approch to predictability via anticipted synchronization[J]. Physical Review E, 2005, 72:046218.
[25] Liang Wu and Shiqun Zhu. Coexistence and switching of anticipating synchronization and lag synchronization in an optical system[J]. Physics Letters A, 2003, 315:101-108.
[26] H.J. Wang, H.B. Huang and G.X. Qi. Coexistence of anticipated and layered chaotic synchronization in time-delay systems[J]. Physical Review E, 2005, 72:037203.
[27] Kestutis Pyragas and Tatjana Pyragiene. Coupling design for a long-term anticipatingsynchronization of chaos[J]. Physical Review E, 2008, 78:046217.
[28] S. Sivaprakasam, E.M. Shahverdiev, P.S. Spencer and K.A. Shore. Experimental demonstration of anticipating synchronization in chaotic semiconductor lasers with optical feedback[J]. Physical Review Letters, 2001, 87:104101.
[29] S. Tang and J.M. Liu. Experimental verification of anticipated and retarded synchronization in chaotic semiconductor lasers. Physical Review Letters, 2003, 90:194101.
[30] Shawn K. Pethel, Ned J. Corron, Quitisha R. Underwood and Krishna Myneni. Information flow in chaos synchronization: fundamental tradeoffs in precision, delay, and anticipation. Physical Review Letters, 2003, 90:254101.
[31] E.M.Shahverdiev, S. Sivaprakasam and K.A. Shore. Inverse anticipting chaos synchronization. Physical Review E, 2002, 66:017204.
[32] E.M.Shahverdiev, S. Sivaprakasam and K.A. Shore. Parameter mismatchs and perfect anticipating synchronization in bidirectionally coupled external cavity laser diodes. Physical Review E, 2002, 66:017206.
[33] Qi Han, Chuandong Li and Junjian Huang. Anticipating synchronization of chaotic sysems with time delay and parameter mismatch[J]. Chaos, 2009, 19:013104.
[34] Henning U. Voss. Anticipating chaotic synchronization[J]. Physical Review E, 2000, 61:5115-5119.
[35] Chuandong Li, Guanrong Chen, Xiaofeng Liao and Zhengping Fan. Chaos quasisyn chronization induced by impulses with parameter mismatches[J]. Chaos, 2006, 16:023102.
[36] Chuandong Li and Xiaofeng Liao. Complete and lag synchronization of hyperchaotic systems using small impulses[J]. Chaos, Solitons and Fractals, 2004, 22:857-867.
[37] Chuandong Li, Xiaofeng Liao and Rong Zhang. Delay-dependent exponential stability analysis of bi-directional associative memory neural networks with time delay: an LMI approach[J]. Chaos, Solitons and Fractals, 2005, 24:1119-1134.
[38] Chuandong Li, Xiaofeng Liao, Rong Zhang and Ashutosh Prasad. Global robust exponential stability analysis for interval neural networks with time-varying delays[J]. Chaos, Solitons and Fractals, 2005, 25:751-757.
[39]黄军建,间歇控制在神经网络中的应用研究[D]。重庆:重庆大学,2008.
[40] Tingwen Huang, Chuandong Li and Xiaofeng Liao. Synchronization of a class of coupled chaotic delayed systems with parameter mismatch[J]. Chaos, 2007, 17:033121.
[41] E.N.Sanchez and J.P. Perez, Input-to-state stability (ISS) analysis for dynamic neural networks[J]. IEEE Transactions on Circuits Systems-Ⅰ:Fundaental Theory and Applications, 1999, 46:1395-1398.
[42] S.Boyd, L.E.Ghaoui, E. Feron and V. Balakrishnan, Linear matrix inequalities in systems and control theory(M). SIAM Studies in Applied Mathematics, Philadephia, 1994.
[43] K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics(M). Kluwer Academic, Dordrecht, 1992.
[44] M.S. Fofana, Asymptotic stability of a stochatic delay equation(J). Probabilistic Engineering Mechanics, 2002, 17:385-392.
[45] Xiaofeng Liao, Yanbing Liu, Songtao and Huanhuan Mai, Asymptotic stability of delayed neural networks: A descriptor system approach(J). Commun Nonlinear Sci Numer Simulat, 2009, 14:3120-3133.
[46] Zhi Li and Guanrong Chen, Global synchronization and asymptotic stability of complex dynamical networks(J). IEEE Transactions on Circuits and Systems-Ⅱ: Express Briefs, 2006, 53:28-33.
[47] Sabri Arik, Global asymptotic stability of a larger class of neural networks with constant time delay(J). Physics Letters A, 2003, 311:504-511.
[48] R.Samidurai, R.Sakthivel and S.M.Anthoni, Global asymptotic stability of BAM neural networks with mixed delays and impulses(J). Applied Mathematics and Computation, 2009, 212:113-119.
[49] Hongyong Zhao, Global asymptotic stability of Hopfield neural network involving distributed delays(J). Neural Networks, 2004, 17:47-53.
[50] Xuyang Lou and Baotong Cui, New LMI conditions for delay-dependent asymptotic stability of delayed Hopfield neural networks(J). Neurocomputing, 2006, 69:2374-2378.
[51] Junkang Tian, Dongsheng Xu and Jian Zu, Novel delay-dependent asyptotic stability criteria for neural networks with time-varying delays(J). Journal of Computational and Applied Mathematics, 2009, 228:133-138.
[52] Xiaopeng Wei, Dongsheng Zhou and Qiang Zhang, On asymptotic stability of discrete-time non-autonomous delayed Hopfield neural networks(J). Computers and Mathematics with Applications, 2009, 57:1938-1942.
[53] O.M.Kwon and Ju H.Park, Exponential stability for uncertain cellualr neural networks with discrete and distributed time-varying delays(J). Applied Mathematics and Computation, 2008, 203:813-823.
[54] S. Mohamad and K. Gopalsamy, Exponential stability of continuous-time and discrete-time cellular neural networks with delays(J). Applied Mathematics and Computation, 2003, 135:17-38.
[55] Sanny Mohamad, Global exponential stability in continuous-time and discrete-time delayedbidirectional neural networks(J). Physica D, 2001, 159:233-151.
[56] Baoxian Wang, Jigui Jian and Chuande Guo, Global exponential stability of a class of BAM networks with time-varying delays and continuously distributed delays(J). Neyrocomputing, 2008, 71:495-501.