复杂动力网络系统的同步控制研究
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摘要
复杂动力网络系统的同步控制是当今研究复杂网络动力学的重大课题之一,近年来受到了国内外许多学者的广泛关注.本文主要研究几类复杂动力网络在不同控制策略下的同步,包括神经网络的周期间歇控制,无向网络的同步控制,有向网络的自适应间歇控制和社团网络的聚类同步与完全同步控制.
第一部分讨论了两类神经网络模型在周期间歇控制下的同步.首先研究了一类具有混合时滞的神经网络的全局指数滞后同步性.通过引入周期间歇控制策略,运用一些典型的分析技巧如反证法、数学归纳法等,建立了系统在无穷范数意义下的全局指数滞后同步准则.其次,考虑了一类具有混合时滞和Dirichlet边界条件的反应扩散神经网络的全局指数同步性.通过对响应网络施加周期间歇控制,利用引进多参数法,Lyapunov泛函技巧,在p范数的基础上得到了系统全局指数同步的充分条件.特别地,在分散型间歇控制下,推出了一个关于控制增益和控制时间率的同步可行域.本节得到的同步判据考虑了扩散强度和扩散空间对网络同步的影响,并明确指出增强神经元的扩散强度或减小扩散空间有利于网络完成同步,反之则不利于网络同步的实现.本章处理的间歇控制策略去除了已有工作中对时滞、控制时间的苛刻限制,推广和改进了前人的工作.
第二部分讨论了无向网络的同步控制.首先研究了具有自适应耦合权重的无向网络.结合牵制控制和自适应反馈控制,利用不等式分析方法和Barbalat引理建立了保证网络实现完全同步的判定条件.然后讨论了一类具有分布耦合单时滞的等度网络的间歇控制同步.运用分析方法和不等式技巧,建立了网络全局指数同步的判据.并在此基础上,得到了关于控制增益和控制时间率的同步可行域.与传统的复杂网络同步研究不同,本节处理的同步态不再是网络孤立节点的状态,而是非解耦态,考虑了网络的内联矩阵和节点度对同步状态的影响.
第三部分解决了一个在已有工作中提出的开问题,即复杂网络的自适应间歇控制.首先建立了一个有向动力网络模型,指出了它与无向网络在模型表现上的差异.通过对网络部分节点施加分散型自适应间歇控制,运用不等式分析技巧,建立了有向网络实现全局指数同步的判别准则.并根据所得同步判据,推出了一个关于控制时间率的同步可行域.最后,通过两个实例验证了理论结果的正确性和有效性.
第四部分研究了两类社团网络模型.首先,考虑了有向社团网络的聚类同步.通过对部分社团分别施加反馈控制和自适应控制,利用Lyapunov稳定性理论,建立了网络实现聚类同步的充分条件.所得结论回答了如下几个具有挑战性的难题：(a)什么样的社团应该优先被控制?(b)至少控制多少个社团才能实现聚类同步?(c)对于受控制的社团,应该选取多大的控制增益才能完成聚类同步?本节研究工作与已有结果的最大不同之处在于在聚类同步处理过程中,每一个社团被视为一个整体来对待,所得聚类同步准则包含了网络社团结构的信息.其次,处理了一类具有传输时滞的加权社团网络的完全同步问题.结合开环控制和反馈控制,通过对耦合强度设计自适应更新律,利用不等式技巧和Barbalat引理,建立了社团网络完全同步到一个预先给定的光滑动力学状态的判定条件.最后,利用数值实例和仿真验证了理论结果的正确性和有效性.
Synchronization control of complex dynamical networks is a central topic in theinvestigation of the dynamics of complex networks and has been received much at-tention by a lot of scholars. The aim of this work is to investigate the synchronizationcontrol of several complex dynamic network models, which include the periodicallyintermittent control of neural networks, the synchronization control of undirectednetworks, the adaptive intermittent control for directed networks as well as thecluster synchronization and complete synchronization of community networks.
In the first part, the synchronization of two kinds of neural networks is discussedunder periodically intermittent control. First of all, the lag synchronization of neuralnetworks with mixed delays is proposed. By introducing periodically intermitten-t control and applying analysis techniques such as mathematical induction methodand the reduction to absurdity, the criteria for exponentially lag synchronization arederived in terms of the infinite norm. Secondly, the exponential synchronization fora class of reaction-difusion neural networks with mixed delays is considered. Basedon p-norm, the conditions of exponentially complete synchronization are obtainedvia introducing multi-parameters and Lyapunov theory. Especially, a feasible syn-chronization region concerning control gain and the rate of control time is derivedunder a decentralized intermittent control. Besides, the efects of reaction difusionson synchronization are considered and we pointed out that it is beneficial to realizethe synchronization of neural networks when the difusion strengths are strengthenedor the difusion spaces are reduced. It is noted that some traditional restrictions ondelays and work time are removed in our results.
The synchronization control of undirected networks is analyzed in the secondpart. First, the models of undirected network with adaptive coupling weights arestudied and the criteria of complete synchronization are given based on pinning con-trol, adaptive feedback laws and Barbalat lemma. In addition, the synchronizationof complex networks with node balance and single coupling distributed delays isdiscussed by using intermittent control. Some conditions are derived to ensure therealization of exponential synchronization by using analysis technique and inequalitymethods. Moreover, a feasible synchronization region for control gains and the rate of control time are also obtained. Diferent from traditional results, the proposedsynchronization states are un-decoupled states and the influence of inner couplingmatrix and the degree of nodes on the synchronized states is included.
An open problem, that is, the problem of adaptive intermittent control forcomplex network is solved in third part. First, a model of directed network is es-tablished and the diference on model representations between directed network andun-directed network is pointed out. Some criteria are established to ensure theglobally exponential synchronization by imposing decentralized adaptive intermit-tent control on partial nodes and using inequality techniques. Besides, a feasiblesynchronization region concerning the rate of control time is given. Finally, twonumerical examples are provided to show the validity and efectiveness of the theo-retical results.
In the forth part, two type of community networks are investigated. Firstly,the cluster synchronization is proposed via imposing feedback control and adaptivecontrol on partial communities and some sufcient conditions are obtained based onLyapunov theory. In all, this work answers several challenging problems in pinningcontrol of directed community networks:(a) What communities should be chosenas controlled candidates?(b) How many communities are needed to be controlled?(c) How large should the control gains be used in a given community network toachieve cluster synchronization? Unlike the previous results, each community isregarded as a whole and the informations of communities are included in the derivedcriteria. Additionally, the complete synchronization of a delayed community networkis considered in this part. By combining open loop control with feedback control anddesigning adaptive update law for coupling strength, some criteria are established toensure the community networks synchronize onto an any given smoothly dynamicalstate based on inequality techniques and Barbalat lemma. Finally, the validity andefectiveness of the theoretical results are approved by two numerical examples.
第一部分讨论了两类神经网络模型在周期间歇控制下的同步.首先研究了一类具有混合时滞的神经网络的全局指数滞后同步性.通过引入周期间歇控制策略,运用一些典型的分析技巧如反证法、数学归纳法等,建立了系统在无穷范数意义下的全局指数滞后同步准则.其次,考虑了一类具有混合时滞和Dirichlet边界条件的反应扩散神经网络的全局指数同步性.通过对响应网络施加周期间歇控制,利用引进多参数法,Lyapunov泛函技巧,在p范数的基础上得到了系统全局指数同步的充分条件.特别地,在分散型间歇控制下,推出了一个关于控制增益和控制时间率的同步可行域.本节得到的同步判据考虑了扩散强度和扩散空间对网络同步的影响,并明确指出增强神经元的扩散强度或减小扩散空间有利于网络完成同步,反之则不利于网络同步的实现.本章处理的间歇控制策略去除了已有工作中对时滞、控制时间的苛刻限制,推广和改进了前人的工作.
第二部分讨论了无向网络的同步控制.首先研究了具有自适应耦合权重的无向网络.结合牵制控制和自适应反馈控制,利用不等式分析方法和Barbalat引理建立了保证网络实现完全同步的判定条件.然后讨论了一类具有分布耦合单时滞的等度网络的间歇控制同步.运用分析方法和不等式技巧,建立了网络全局指数同步的判据.并在此基础上,得到了关于控制增益和控制时间率的同步可行域.与传统的复杂网络同步研究不同,本节处理的同步态不再是网络孤立节点的状态,而是非解耦态,考虑了网络的内联矩阵和节点度对同步状态的影响.
第三部分解决了一个在已有工作中提出的开问题,即复杂网络的自适应间歇控制.首先建立了一个有向动力网络模型,指出了它与无向网络在模型表现上的差异.通过对网络部分节点施加分散型自适应间歇控制,运用不等式分析技巧,建立了有向网络实现全局指数同步的判别准则.并根据所得同步判据,推出了一个关于控制时间率的同步可行域.最后,通过两个实例验证了理论结果的正确性和有效性.
第四部分研究了两类社团网络模型.首先,考虑了有向社团网络的聚类同步.通过对部分社团分别施加反馈控制和自适应控制,利用Lyapunov稳定性理论,建立了网络实现聚类同步的充分条件.所得结论回答了如下几个具有挑战性的难题：(a)什么样的社团应该优先被控制?(b)至少控制多少个社团才能实现聚类同步?(c)对于受控制的社团,应该选取多大的控制增益才能完成聚类同步?本节研究工作与已有结果的最大不同之处在于在聚类同步处理过程中,每一个社团被视为一个整体来对待,所得聚类同步准则包含了网络社团结构的信息.其次,处理了一类具有传输时滞的加权社团网络的完全同步问题.结合开环控制和反馈控制,通过对耦合强度设计自适应更新律,利用不等式技巧和Barbalat引理,建立了社团网络完全同步到一个预先给定的光滑动力学状态的判定条件.最后,利用数值实例和仿真验证了理论结果的正确性和有效性.
Synchronization control of complex dynamical networks is a central topic in theinvestigation of the dynamics of complex networks and has been received much at-tention by a lot of scholars. The aim of this work is to investigate the synchronizationcontrol of several complex dynamic network models, which include the periodicallyintermittent control of neural networks, the synchronization control of undirectednetworks, the adaptive intermittent control for directed networks as well as thecluster synchronization and complete synchronization of community networks.
In the first part, the synchronization of two kinds of neural networks is discussedunder periodically intermittent control. First of all, the lag synchronization of neuralnetworks with mixed delays is proposed. By introducing periodically intermitten-t control and applying analysis techniques such as mathematical induction methodand the reduction to absurdity, the criteria for exponentially lag synchronization arederived in terms of the infinite norm. Secondly, the exponential synchronization fora class of reaction-difusion neural networks with mixed delays is considered. Basedon p-norm, the conditions of exponentially complete synchronization are obtainedvia introducing multi-parameters and Lyapunov theory. Especially, a feasible syn-chronization region concerning control gain and the rate of control time is derivedunder a decentralized intermittent control. Besides, the efects of reaction difusionson synchronization are considered and we pointed out that it is beneficial to realizethe synchronization of neural networks when the difusion strengths are strengthenedor the difusion spaces are reduced. It is noted that some traditional restrictions ondelays and work time are removed in our results.
The synchronization control of undirected networks is analyzed in the secondpart. First, the models of undirected network with adaptive coupling weights arestudied and the criteria of complete synchronization are given based on pinning con-trol, adaptive feedback laws and Barbalat lemma. In addition, the synchronizationof complex networks with node balance and single coupling distributed delays isdiscussed by using intermittent control. Some conditions are derived to ensure therealization of exponential synchronization by using analysis technique and inequalitymethods. Moreover, a feasible synchronization region for control gains and the rate of control time are also obtained. Diferent from traditional results, the proposedsynchronization states are un-decoupled states and the influence of inner couplingmatrix and the degree of nodes on the synchronized states is included.
An open problem, that is, the problem of adaptive intermittent control forcomplex network is solved in third part. First, a model of directed network is es-tablished and the diference on model representations between directed network andun-directed network is pointed out. Some criteria are established to ensure theglobally exponential synchronization by imposing decentralized adaptive intermit-tent control on partial nodes and using inequality techniques. Besides, a feasiblesynchronization region concerning the rate of control time is given. Finally, twonumerical examples are provided to show the validity and efectiveness of the theo-retical results.
In the forth part, two type of community networks are investigated. Firstly,the cluster synchronization is proposed via imposing feedback control and adaptivecontrol on partial communities and some sufcient conditions are obtained based onLyapunov theory. In all, this work answers several challenging problems in pinningcontrol of directed community networks:(a) What communities should be chosenas controlled candidates?(b) How many communities are needed to be controlled?(c) How large should the control gains be used in a given community network toachieve cluster synchronization? Unlike the previous results, each community isregarded as a whole and the informations of communities are included in the derivedcriteria. Additionally, the complete synchronization of a delayed community networkis considered in this part. By combining open loop control with feedback control anddesigning adaptive update law for coupling strength, some criteria are established toensure the community networks synchronize onto an any given smoothly dynamicalstate based on inequality techniques and Barbalat lemma. Finally, the validity andefectiveness of the theoretical results are approved by two numerical examples.
引文
[1]胡守仁.神经网络应用技术[M].长沙：国防科技大学出版社,1998.
[2]W.S. McCulloch, W. Pitts. A logical calculus of the ideas immanent in neuron activity [J]. Bull Math Biophys.1943,5:115-133.
[3]J. Hopfield. Meuron networks and physical systems with emergent collective comu-tational abilities [J]. Proc Natl Acad Sci.1982,79:2554-2558.
[4]L.O. Chua, L. Yang. Cellular neural networks, Theory [J]. IEEE Trans Circuits Syst.1998,35:1257-1272.
[5]M. Gohen, S. Grossberg. Absolute stability of global parttern formation and parallel memory storage by compititive neural networks [J]. IEEE Trans Syst Man Cyber.1983,13:815-826.
[6]B. Kosko. Bidirectional associative memories [J]. IEEE Trans Syst Man Cyber.1988,18:49-60.
[7]H. Yang, T.S. Dillon. Exponential stability and oscillation of Hopfield graded re-sponse neural network [J]. IEEE Trans Neural Networks.1994,5:719-729.
[8]Y. Fang, T.G. Kincaid. Stability analysis of dynamical neural networks [J]. IEEE Trans Neural Networks.1996,7:996-1006.
[9]J.D. Cao. Global stability analysis in delayed cellular neural networks [J]. Phys Rev E.1999,59:5940-5944.
[10]J.J. Wei, S. Ruan. Stability and bifurcation in a neural network model with two delays [J]. Physica D.1999,130:255-272.
[11]J.D. Cao. On stability of delayed cellular neural networks [J]. Phys Lett A.1999,261:303-308.
[12]T.L. Liao, F.C. Wang. Global stability for cellular neural networks with time delay [J]. IEEE Trans Neural Networks.2000,11:1481-1484.
[13]Z.H. Guan, G.R. Chen, Y. Qin. On equilibria, stability, and instability of hopfield neural networks [J]. IEEE Trans Neural Networks.2000,11:534-540.
[14] S. Arik. Global asymptotic stability of a class of dynamical neural networks [J].IEEE Trans Circuits Syst I.2000,47:568-571.
[15] S. Arik, V. Tavsanoglu. On the global asymptotic stability of delayed cellular neuralnetworks [J]. IEEE Trans Circuits Syst I.2000,47:571-574.
[16] M. Joy. Results concerning the absolute stability of delayed neural networks [J].Neural Networks.2000,13:613-616.
[17] T.P. Chen. Global exponential stability of delayed Hopfield neural networks [J].Neural Networks.2001,14:977-980.
[18] S. Mohamad. Global exponential stability in continuous-time and discrete-time de-layed bidirectional neural networks [J]. Physica D.2001,159:233-251.
[19] L. Wang, X.F. Zou. Exponential stability of Cohen-Grossberg neural networks [J].Neural Networks.2002,15:415-422.
[20] S. Mohamad, K. Gopalsamy. Exponential stability of continuous-time and discrete-time cellular neural networks with delays [J]. Appl Math Comput.2002,135:17-38.
[21] H.J. Jiang, Z.D. Teng. Global exponential stability of cellular neural networks withtime-varying coefcients and delay [J]. Neural Networks.2004,17:1415-1425.
[22] W.L. Lu, T.P. Chen. New conditions on global stability of Cohen-Grossberg neuralnetworks [J]. Neural Computation.2003,15:1173-1189.
[23] H.J. Jiang, Z.D. Teng. A new criterion on the global exponential stability for cellularnetworks with multiple time-varying delays [J]. Phys Lett A.2005,338:461-471.
[24] H.J. Jiang, Z.D. Teng. Some new results for recurrent neural networks with varying-time coefcients and delays [J]. Phys Lett A.2005,338:446-460.
[25] H.J. Jiang, Z.D. Teng. Boundedness and stability for nonautonomous cellular neuralnetworks with delay [J]. Phys Lett A.2003,306:313-325.
[26] J.L. Liang, J.D. Cao. Boundedness and stability for recurrent neural networks withvariable coefcients and time-varying delays [J]. Phys Lett A.2003,318:53-64.
[27]H. Jiang, Z. Teng. Boundedness and stability for nonautonomous bidirectional as-sociative neural networks with delay [J]. IEEE Trans Circuits Syst Ⅱ.2004,51:174-180.
[28]M. Rehim, H. Jiang, Z. Teng. Boundedness and stability for nonautonomous cellular neural networks with delay [J]. Neural Networks.2004,17:1017-1025.
[29]蒋海军,滕志东.非自治神经网络的有界性和稳定性[J].黑龙江大学学报.2006,23：437-440.
[30]H. Jiang, Z. Teng. Boundedness and global stability for nonautonomous recurrent neural networks with distributed delays [J]. Chaos Solitons Fract.2006,30:83-93.
[31]J.D. Cao. Periodic solutions and exponential stability in delayed cellular neural networks [J]. Phys Rev E.1999,60:3244-3248.
[32]J. Cao. Global exponential stability and periodic solutions of delayed cellular neural networks [J]. J Coumput Syst Sci.2000,60:38-46.
[33]S. Guo, L. Huang. Exponential stability and periodic solutions of neural networks with continuously distributed delays [J]. Phys Rev E.2003,67:011902.
[34]D. Papini, V. Taddei. Global exponential stability of the periodic solution of a delayed neural network with discontinuous activations [J]. Phys Lett A.2005,343:117-128.
[35]H. Jiang, J. Cao. Global exponential stability of periodic neural networks with time-varying delays [J]. Neurocomputing.2006,70:343-350.
[36]Z. Gui, X. Yang, W. Ge. Periodic solution for nonautonomous bidirectional as-sociative memory neural networks with impulses [J]. Neurocomputing.2007,70:2517-2527.
[37]H. Gu, H. Jiang, Z. Teng. Stability and periodic in High-order neural networks with impulsive effects [J]. Nonlinear Anal TMA.2008,68:3186-3200.
[38]F. Zou, J.A. Nossek. Bifurcation and chaos in cellular neural networks [J]. IEEE Trans Circuits Syst Ⅰ.1993,40:166-173.
[39]H.T. Lu. Chaotic attractors in delayed neural networks [J]. Phys Lett A.2002,298:109-119.
[40] G.R. Chen, J. Zhou, Z.R. Liu. Global synchronization of coupled delayed neuralnetworks and applications to chaotic CNN models [J]. Inter J Bifur Chaos.2004,14:2229-2240.
[41] W.L. Lu, T.P. Chen. Synchronization of coupled connected neural networks withdelays [J]. IEEE Trans Circuits Syst I.2004,51:2491-2503.
[42] C.J. Cheng, T.L. Liao, C.C. Hwang. Exponential synchronization of a class of chaoticneural networks [J]. Chaos Solitons Fract.2005,24:197-206.
[43] J.D. Cao, J.Q. Lu. Adaptive synchronization of neural networks with or withouttime-varying delay [J]. Chaos.2006,16:013133.
[44] J.D. Cao, P. Li, W.W. Wang. Global synchronization in arrays of delayed neuralnetworks with constant and delayed coupling [J]. Phys Lett A.2006,353:318-325.
[45] J. Meng, X.Y. Wang. Robust anti-synchronization of a class of delayed chaotic neuralnetworks [J]. Chaos.2007.17:023113.
[46] X.Y. Lou, B.T. Cui. Anti-synchronization of chaotic delayed neural networks [J].Acta Physica Sinica.2008,57:2060-2067.
[47] J. Zhou, T.P. Chen, L. Xiang. Chaotic Lag synchronization of coupled delayedneural networks and its applications in secure communication [J]. Circuits Systemsand Signal Processing.2005,24:599-613.
[48] Y. Sun, J. Cao. Adaptive lag synchronization of unknown chaotic delayed neuralnetworks with noise perturbation [J]. Phys Lett A.2007,364:277-285.
[49] Y. Yang, J. Cao. Exponential lag synchronization of a class of chaotic delayed neuralnetworks with impulsive efects [J]. Physica A.2007,386:492-502.
[50] Y. Tang, R. Qiu, J. Fang, Q. Miao, M. Xia. Adaptive lag synchronization in unknownstochastic chaotic neural networks with discrete and distributed time-varying delays[J]. Phys Lett A.2008,372:4425-4433.
[51] W. Ding, M. Han, M. Li. Exponential lag synchronization of delayed fuzzy cellularneural networks with impulses [J]. Phys Lett A.2009,373:832-837.
[52]W. Yu, J. Cao. Adaptive synchronization and lag synchronization of uncertain dy-namical system with time delay based on parameter identification [J]. Physica A.2007,375:467-482.
[53]M. Zochowski. Intermittent dynamical control [J]. Physica D.2000,145:181-190.
[54]C. Li, G. Feng, X. Liao. Stabilization of nonlinear systems via periodically intermit-tent control [J]. IEEE Trans Circuits Syst II.2007,54:1019-1023.
[55]C.D. Li, X.F. Liao, T.W. Huang. Exponential stabilization of chaotic systems with delay by periodically intermittent control [J]. Chaos.2007,17:013103.
[56]T. Huang, C. Li, X. Liu. Sychronization of chaotic systems with delay using inter-mittent linear state feedback [J]. Chaos.2008,18:033122.
[57]T. Huang, C. Li, W. Yu, G. Chen. Synchronization of delayed chaotic systems with parameter mismatches by using intermittent linear state feedback [J]. Nonlinearity.2009,22:569-584.
[58]J.J. Huang, C.D. Li, Q. Han. Stabilization of delayed chaotic neural networks by periodically intermittent control [J]. Circuits Syst Signal Process.2009,28:567-579.
[59]T.W. Huang, C.D. Li. Chaotic Synchronization by intermittent feedback method [J]. Comput Appl Math.2010,234:1097-1104.
[60]廖晓昕,傅予力,高健,赵新泉.具有反应扩散的Hopfield神经网络的稳定性_J].电子学报.2000,28：78-82.
[61]J.L. Liang, J.D. Cao. Global exponential stability of reaction-diffusion recurrent neural networks with time-varying delays [J]. Phys Lett A.2003,314:434-442.
[62]王林山,徐道义.变时滞反应扩散Hopfield神经网络的全局指数稳定性[J].中国科学(E).2003,33：488-495.
[63]W. Allegretto, D. Papini. Stability for delayed reaction-diffusion neural networks [J]. Phys Lett A.2007,360:669-680.
[64]J. Lu. Robust Global Exponential Stability for Interval Reaction-Diffusion Hopfield Neural Networks With Distributed Delays [J]. IEEE Trans Circuits Syst II.2007,54:1115-1119.
[65] J.G. Lu. Global exponential stability and periodicity of reaction-difusion delayedrecurrent neural networks with Dirichlet boundary conditions [J]. Chaos Sol Frac.2008,35:116-125.
[66] J. Wang, J. Lu. Global exponential stability of fuzzy cellular neural networks withdelays and reaction-difusion terms [J]. Chaos Sol Frac.2008,38:878-885.
[67] J. Lu, L. Lu. Global exponential stability and periodicity of reaction-difusion re-current neural networks with distributed delays and Dirichlet boundary conditions[J]. Chaos Sol Frac.2009,39:1538-1549.
[68] X. Lou, B. Cui. Asymptotic synchronization of a class of neural networks withreaction-difusion terms and time-varying delays [J]. Comput Math Appl.2006,52:897-904.
[69] Y. Wang, J. Cao. Synchronization of a class of delayed neural networks with reaction-difusion terms [J]. Phys Lett A.2007,369:201-211.
[70] L. Sheng, H. Yang, X. Lou. Adaptive exponential synchronization of delayed neuralnetworks with reaction-difusion terms [J]. Chaos Sol Frac.2009,40:930-939.
[71] K. Wang, Z. Teng, H. Jiang. Global exponential synchronization in delayed reaction-difusion cellular neural networks with the Dirichlet boundary conditions [J]. MathComput Model.2010,52:12-24.
[72] C. Hu§H. Jiang, Z. Teng. Impulsive control and synchronization for delayed neuralnetworks with reaction-difusion terms [J]. IEEE Trans Neural Networks.2010,21:67-81.
[73] L.O. Chua, T. Roska. Cellular neural networks and visual computing: foundationand applications [M]. New York: Cambridge University Press,2002.
[74] M. Faloutsos, P. Faloutsos, C. Faloutsos. On power-law relationships of the Interacttopology [J]. Computer Communication Revivew.1999,29:29-51
[75] N. Guelzim, S. Bottani, P. Bourgine, et a1. Topological and causal structure of theyeast transcriptional regulatory network [J]. Nature Genetics.2002,31:60-63
[76] M.E.J. Newman. Scientific collaboration networks I: Network construction and fun-damental results [J]. Phys Rev E.2001,64:016131
[77]A. Cardillo, S. Scellato, V. Latora, S. Porta. Structural properties of planar graph of urban street patterns [J]. Phys Rev E.2006,73:066107.
[78]D.J. Watts, S.H. Strogatz. Collective dynamics of'small-world'networks [J]. Nature.1998,393:440-442.
[79]A.L. Barabasi, R. Albert. Emergence of scaling in random networks [J]. Science.1999,286:509-512.
[80]方锦清,汪小帆,刘曾荣.略论复杂性问题和非线性复杂网络系统的研究[J].基础科学,科技导报.2004,2：9-12.
[81]M.E.J. Newman, D. J. Watts. Scaling and percolation in the small-world network model [J]. Phys Rev E.1999,60:7332-7342.
[82]X. Li, G. Chen. Alocal-world evolving network model [J]. Physica A.2003,328:274-286.
[83]K. Klemm, V.M. Eguiluz. Highly clustered scale-free networks [J]. Phys Rev E.2002,65:6123-6127.
[84]D.J. Watts. The'new'science of networks [J]. Annual Review of Sociology.2004,1:243-270.
[85]S.H. Strogatz, I. Stewart. Coupled oscillators and biological synchronization [J]. Scientific American.1993,269:102-109.
[86]C。M. Gray. Synchronous oscillations in neuronal systems:mechanisms and functions [J]. Journal of Computer Neurosci.1994,1:11-38.
[87]R.Q. Wang, L.N. Chen. Synchronizing genetic oscillators by signaling molecules [J]. Journal of Biological Rhythms.2005,20:257-269.
[88]Z. Nfda, E. Ravasz, T. Vicsek, et al. Physics of the rhythmic applause [J]. Phys Rev E.2000,6:6987-6992.
[89]X. Wang, G. Chen. Synchronization in scale-free dynamical networks:robustness and fragility [J]. IEEE Trans Circuits Syst I.2002,1:54-62.
[90]X. Wang, G. Chen. Synchronization in small-world dynamical networks [J]. Inter J Bifur Chaos.2002,1:187-192.
[91] X. Wang, G. Chen. Pinning control of scale-free dynamical networks [J]. Physica A.2002,310:521-531.
[92] T. Chen, X. Liu, W. Lu. Pinning Complex Networks by a Single Controller [J].IEEE Trans Circuits Syst I.2007,54:1317-1326.
[93] W. Lu, X. Li, Z. Rong. Global stabilization of complex networks with digraphtopologies via a local pinning algorithm [J]. Automatica.2010,46:116-121.
[94] W. Yu, G. Chen, J. Lu¨. On pinning synchronization of complex dynamical networks[J]. Automatica.2009,45:429-435.
[95] W. Wu, T. Chen. Global Synchronization Criteria of Linearly Coupled Neural Net-work Systems With Time-Varying Coupling [J]. IEEE Trans Neural Networks.2008,19:319-332.
[96] J. Zhao, J. Lu, Q. Zhang. Pinning a Complex Delayed Dynamical Network to aHomogenous Trajectory [J]. IEEE Trans Circuits Syst II.2009,56:514-518.
[97] X. Li, X. Wang, G. Chen. Pinning a Complex Dynamical Network to Its Equilibrium[J]. IEEE Trans Circuits Syst I.2004,51:2074-2087.
[98] J. Zhou, J. Lu, J. Lu¨. Pinning adaptive synchronization of a general complex dy-namical networ [J]. Automatica.2008,44:996-1003.
[99] Z.X. Liu, Z.Q. Chen, Z.Z. Yuan. Pinning control of weighted general complex dy-namical networks with time delay [J]. Physica A.2007,375:345-354.
[100] B. Wang, Z. Guan. Chaos synchronization in general complex dynamical networkswith coupling delays [J]. Nonlinear Anal RWA.2010,11:1925-1932.
[101] Z. Li, G. Chen. Global Synchronization and Asymptotic Stability of Complex Dy-namical Networks [J]. IEEE Trans Circuits Syst II.2006,53:28-33.
[102] Y. Xu, W. Zhou, J. Fang, W. Sun. Adaptive synchronization of the complex dy-namical network with non-derivative and derivative coupling [J]. Phys Lett A.2010,374:1673-1677.
[103] Z. Li, L. Jiao, J. Lee. Robust adaptive global synchronization of complex dynamicalnetworks by adjusting time-varying coupling strength [J]. Physica A.2008,387:1369-1380.
[104] L. Huang, Z. Wang, Y. Wang, Y. Zuo. Synchronization analysis of delayed complexnetworks via adaptive time-varying coupling strengths [J]. Phys Lett A.2009,373:3952-3958.
[105] H. Liu, J. Chen, J. Lu, M. Cao. Generalized synchronization in complex dynamicalnetworks via adaptive couplings [J]. Physica A.2010,389:1759-1770.
[106] H. Tang, L. Chen, J. Lu, C. Tse. Adaptive synchronization between two complexnetworks with nonidentical topological structures [J]. Physica A.2008,387:5623-5630.
[107] X. Wu. Synchronization-based topology identification of weighted general complexdynamical networks with time-varying coupling delay [J]. Physica A.2008,387:997-1008.
[108] G. Zhang, Z. Liu, Z. Ma. Synchronization of complex dynamical networks via im-pulsive control [J]. Chaos.2007,17:043126.
[109] Q. Zhang, J. Lu, J. Zhao. Impulsive synchronization of general continuous anddiscrete-time complex dynamical networks [J]. Commun Nonlinear Sci Numer Sim-ulat.2010,15:1063-1070.
[110] G. Wang, J. Cao, J. Lu. Outer synchronization between two nonidentical networkswith circumstance noise [J]. Physica A.2010,389:1480-1488.
[111] W. Lu, T. Chen. New approach to synchronization analysis of linearly coupledordinary diferential systems [J]. Physica D.2006,213:214-230.
[112] W. Guo, F. Austin, S. Chen. Global synchronization of nonlinearly coupled complexnetworks with non-delayed and delayed coupling [J]. Commun Nonlinear Sci NumerSimulat.2010,15:1631-1639.
[113] P.D. Lellis, M. Bernardo, F. Garofalo. Synchronization of complex networks throughlocal adaptive coupling [J]. Chaos.2008,18:037110.
[114] H. Su, Z. Rong, X. Wang, G. Chen. On Decentralized Adaptive Pinning Synchroniza-tion of Complex Dynamical Networks [R]. Proceedings of2010IEEE InternationalSymposium on Circuits and Systems. IEEE Press: New Jersey,2010,417-420.
[115] Z. Yuan, J. Cai, M. Lin. Global Synchronization in Complex Networks with AdaptiveCoupling [J]. Mathematical Problems in Engineering.2010,826721.
[116] W.G. Xia, J.D. Cao. Pinning synchronization of delayed dynamical networks viaperiodically intermittent control [J]. Chaos.2009,19:013120.
[117] X. Yang, J. Cao. Stochastic synchronization of coupled neural networks with inter-mittent control [J]. Phys Lett A.2009,373:3259-3272.
[118] S. Cai, Z. Liu, F. Xua, J. Shen. Periodically intermittent controlling complex dy-namical networks with time-varying delays to a desired orbit [J]. Phys Lett A.2009,373:3846-3854.
[119] X. Liu, T. Chen. Cluster Synchronization in Directed Networks Via IntermittentPinning Control [J]. IEEE Trans Neural Networks.2011,22:1009-1020.
[120] C.W. Wu. Localization of efective pinning control in complex networks of dynamicalsystems [R]. Proc IEEE Int Symp Circuits Syst. IEEE Press: New Jersey,2008,2530-2533.
[121] I. Belykh, V. Belykh, M. Hasler. Synchronization in asymmetrically coupled net-works with node balance [J]. Chaos.2006,16:015102.
[122] Y.Y. Lu, X.F. Wang. Pinning control of directed dynamical networks based onControlRank [J]. Int J Comput Math.2008,85:1279-1286.
[123] Q. Song, J. Cao. On Pinning Synchronization of Directed and Undirected ComplexDynamical Networks [J]. IEEE Trans Circuits Syst I.2010,57:672-680.
[124] M. Yang, Y. Liu, Z. You, P. Sheng. Global synchronization for directed complexnetworks [J]. Nonlinear Anal RWA.2010,11:2127-2135.
[125] F. Nian, X. Wang. Optimal pinning synchronization on directed complex network[J]. Chaos.2011,21:043131.
[126] W. Lu, T. Chen. Synchronisation in complex networks of coupled systems withdirected topologies [J]. Int J Syst Sci.2009,40:909-921.
[127] K. Kaneko. Relevance of dynamic clustering to biological networks [J]. Physica D.1994,75:55-73.
[128]V.N. Belykh, I.V. Belykh, E. Mosekilde. Cluster synchronization modes in an en-semble of coupled chaotic oscillators [J]. Phys Rev E,2001,63:036216.
[129]W. Qin, G. Chen. Coupling schemes for cluster synchronization in coupled Josephson equations [J]. Physica D.2004,197:375-391.
[130]X.B. Lu, X.F. Wang, J.Q. Fang. Controlling a complex dynamical network to attain an inhomogeneous equilibrium [J]. Physica D.2010,239:341-347.
[131]X. Lu, B. Qin. Adaptive cluster synchronization in complex dynamical networks [J]. Phys Lett A.2009,373:3650-3658.
[132]W. Wu, W. Zhou, T. Chen. Cluster Synchronization of Linearly Coupled Complex Networks Under Pinning Control [J]. IEEE Trans Circuits Syst I.2009,56:829-839.
[133]L. Chen, J. Lu. Cluster synchronization in a complex network with two nonidentical clusters [J]. Jrl Syst Sci&Complexity,2008,21:20-33.
[134]K. Wang, X. Fu, K. Li. Cluster synchronization in community networks with non-identical nodes [J]. Chaos.2009,19:023106.
[135]X. Wu, H. Lu. Cluster synchronization in the adaptive complex dynamical networks via a novel approach [J]. Phys Lett A.2011,375:1559-1565.
[136]W. Lu, B. Liu, T. Chen. Cluster synchronization in networks of coupled nonidentical dynamical systems [J]. Chaos.2010,20:013120.
[137]Q. Song, J. Cao, F. Liu. Synchronization of complex dynamical networks with non-identical nodes [J]. Phys Lett A.2010,374:544-551.
[138]S. Cai, Q. He, J. Hao, Z. Liu. Exponential synchronization of complex networks with nonidentical time-delayed dynamical nodes [J]. Phys Lett A.2010,374:2539-2550.
[139]R.A. Horn, C.R. Johnson. Matrix Analysis [M]. New York: Cambridge University Press,2005.
[140]W. Lu, T. Chen, G. Chen. Synchronization analysis of linearly coupled systems described by differential equations with a coupling delay [J]. Physica D.2006,221:118-134.
[141]程云鹏.矩阵论[M].第2版.西安：西北工业大学出版社,2005.
[142] C. Wu. Synchronization in Complex Networks of Nonlinear Dynamical Systems [M].Singapore: World Scientific Publishing,2007.
[2]W.S. McCulloch, W. Pitts. A logical calculus of the ideas immanent in neuron activity [J]. Bull Math Biophys.1943,5:115-133.
[3]J. Hopfield. Meuron networks and physical systems with emergent collective comu-tational abilities [J]. Proc Natl Acad Sci.1982,79:2554-2558.
[4]L.O. Chua, L. Yang. Cellular neural networks, Theory [J]. IEEE Trans Circuits Syst.1998,35:1257-1272.
[5]M. Gohen, S. Grossberg. Absolute stability of global parttern formation and parallel memory storage by compititive neural networks [J]. IEEE Trans Syst Man Cyber.1983,13:815-826.
[6]B. Kosko. Bidirectional associative memories [J]. IEEE Trans Syst Man Cyber.1988,18:49-60.
[7]H. Yang, T.S. Dillon. Exponential stability and oscillation of Hopfield graded re-sponse neural network [J]. IEEE Trans Neural Networks.1994,5:719-729.
[8]Y. Fang, T.G. Kincaid. Stability analysis of dynamical neural networks [J]. IEEE Trans Neural Networks.1996,7:996-1006.
[9]J.D. Cao. Global stability analysis in delayed cellular neural networks [J]. Phys Rev E.1999,59:5940-5944.
[10]J.J. Wei, S. Ruan. Stability and bifurcation in a neural network model with two delays [J]. Physica D.1999,130:255-272.
[11]J.D. Cao. On stability of delayed cellular neural networks [J]. Phys Lett A.1999,261:303-308.
[12]T.L. Liao, F.C. Wang. Global stability for cellular neural networks with time delay [J]. IEEE Trans Neural Networks.2000,11:1481-1484.
[13]Z.H. Guan, G.R. Chen, Y. Qin. On equilibria, stability, and instability of hopfield neural networks [J]. IEEE Trans Neural Networks.2000,11:534-540.
[14] S. Arik. Global asymptotic stability of a class of dynamical neural networks [J].IEEE Trans Circuits Syst I.2000,47:568-571.
[15] S. Arik, V. Tavsanoglu. On the global asymptotic stability of delayed cellular neuralnetworks [J]. IEEE Trans Circuits Syst I.2000,47:571-574.
[16] M. Joy. Results concerning the absolute stability of delayed neural networks [J].Neural Networks.2000,13:613-616.
[17] T.P. Chen. Global exponential stability of delayed Hopfield neural networks [J].Neural Networks.2001,14:977-980.
[18] S. Mohamad. Global exponential stability in continuous-time and discrete-time de-layed bidirectional neural networks [J]. Physica D.2001,159:233-251.
[19] L. Wang, X.F. Zou. Exponential stability of Cohen-Grossberg neural networks [J].Neural Networks.2002,15:415-422.
[20] S. Mohamad, K. Gopalsamy. Exponential stability of continuous-time and discrete-time cellular neural networks with delays [J]. Appl Math Comput.2002,135:17-38.
[21] H.J. Jiang, Z.D. Teng. Global exponential stability of cellular neural networks withtime-varying coefcients and delay [J]. Neural Networks.2004,17:1415-1425.
[22] W.L. Lu, T.P. Chen. New conditions on global stability of Cohen-Grossberg neuralnetworks [J]. Neural Computation.2003,15:1173-1189.
[23] H.J. Jiang, Z.D. Teng. A new criterion on the global exponential stability for cellularnetworks with multiple time-varying delays [J]. Phys Lett A.2005,338:461-471.
[24] H.J. Jiang, Z.D. Teng. Some new results for recurrent neural networks with varying-time coefcients and delays [J]. Phys Lett A.2005,338:446-460.
[25] H.J. Jiang, Z.D. Teng. Boundedness and stability for nonautonomous cellular neuralnetworks with delay [J]. Phys Lett A.2003,306:313-325.
[26] J.L. Liang, J.D. Cao. Boundedness and stability for recurrent neural networks withvariable coefcients and time-varying delays [J]. Phys Lett A.2003,318:53-64.
[27]H. Jiang, Z. Teng. Boundedness and stability for nonautonomous bidirectional as-sociative neural networks with delay [J]. IEEE Trans Circuits Syst Ⅱ.2004,51:174-180.
[28]M. Rehim, H. Jiang, Z. Teng. Boundedness and stability for nonautonomous cellular neural networks with delay [J]. Neural Networks.2004,17:1017-1025.
[29]蒋海军,滕志东.非自治神经网络的有界性和稳定性[J].黑龙江大学学报.2006,23：437-440.
[30]H. Jiang, Z. Teng. Boundedness and global stability for nonautonomous recurrent neural networks with distributed delays [J]. Chaos Solitons Fract.2006,30:83-93.
[31]J.D. Cao. Periodic solutions and exponential stability in delayed cellular neural networks [J]. Phys Rev E.1999,60:3244-3248.
[32]J. Cao. Global exponential stability and periodic solutions of delayed cellular neural networks [J]. J Coumput Syst Sci.2000,60:38-46.
[33]S. Guo, L. Huang. Exponential stability and periodic solutions of neural networks with continuously distributed delays [J]. Phys Rev E.2003,67:011902.
[34]D. Papini, V. Taddei. Global exponential stability of the periodic solution of a delayed neural network with discontinuous activations [J]. Phys Lett A.2005,343:117-128.
[35]H. Jiang, J. Cao. Global exponential stability of periodic neural networks with time-varying delays [J]. Neurocomputing.2006,70:343-350.
[36]Z. Gui, X. Yang, W. Ge. Periodic solution for nonautonomous bidirectional as-sociative memory neural networks with impulses [J]. Neurocomputing.2007,70:2517-2527.
[37]H. Gu, H. Jiang, Z. Teng. Stability and periodic in High-order neural networks with impulsive effects [J]. Nonlinear Anal TMA.2008,68:3186-3200.
[38]F. Zou, J.A. Nossek. Bifurcation and chaos in cellular neural networks [J]. IEEE Trans Circuits Syst Ⅰ.1993,40:166-173.
[39]H.T. Lu. Chaotic attractors in delayed neural networks [J]. Phys Lett A.2002,298:109-119.
[40] G.R. Chen, J. Zhou, Z.R. Liu. Global synchronization of coupled delayed neuralnetworks and applications to chaotic CNN models [J]. Inter J Bifur Chaos.2004,14:2229-2240.
[41] W.L. Lu, T.P. Chen. Synchronization of coupled connected neural networks withdelays [J]. IEEE Trans Circuits Syst I.2004,51:2491-2503.
[42] C.J. Cheng, T.L. Liao, C.C. Hwang. Exponential synchronization of a class of chaoticneural networks [J]. Chaos Solitons Fract.2005,24:197-206.
[43] J.D. Cao, J.Q. Lu. Adaptive synchronization of neural networks with or withouttime-varying delay [J]. Chaos.2006,16:013133.
[44] J.D. Cao, P. Li, W.W. Wang. Global synchronization in arrays of delayed neuralnetworks with constant and delayed coupling [J]. Phys Lett A.2006,353:318-325.
[45] J. Meng, X.Y. Wang. Robust anti-synchronization of a class of delayed chaotic neuralnetworks [J]. Chaos.2007.17:023113.
[46] X.Y. Lou, B.T. Cui. Anti-synchronization of chaotic delayed neural networks [J].Acta Physica Sinica.2008,57:2060-2067.
[47] J. Zhou, T.P. Chen, L. Xiang. Chaotic Lag synchronization of coupled delayedneural networks and its applications in secure communication [J]. Circuits Systemsand Signal Processing.2005,24:599-613.
[48] Y. Sun, J. Cao. Adaptive lag synchronization of unknown chaotic delayed neuralnetworks with noise perturbation [J]. Phys Lett A.2007,364:277-285.
[49] Y. Yang, J. Cao. Exponential lag synchronization of a class of chaotic delayed neuralnetworks with impulsive efects [J]. Physica A.2007,386:492-502.
[50] Y. Tang, R. Qiu, J. Fang, Q. Miao, M. Xia. Adaptive lag synchronization in unknownstochastic chaotic neural networks with discrete and distributed time-varying delays[J]. Phys Lett A.2008,372:4425-4433.
[51] W. Ding, M. Han, M. Li. Exponential lag synchronization of delayed fuzzy cellularneural networks with impulses [J]. Phys Lett A.2009,373:832-837.
[52]W. Yu, J. Cao. Adaptive synchronization and lag synchronization of uncertain dy-namical system with time delay based on parameter identification [J]. Physica A.2007,375:467-482.
[53]M. Zochowski. Intermittent dynamical control [J]. Physica D.2000,145:181-190.
[54]C. Li, G. Feng, X. Liao. Stabilization of nonlinear systems via periodically intermit-tent control [J]. IEEE Trans Circuits Syst II.2007,54:1019-1023.
[55]C.D. Li, X.F. Liao, T.W. Huang. Exponential stabilization of chaotic systems with delay by periodically intermittent control [J]. Chaos.2007,17:013103.
[56]T. Huang, C. Li, X. Liu. Sychronization of chaotic systems with delay using inter-mittent linear state feedback [J]. Chaos.2008,18:033122.
[57]T. Huang, C. Li, W. Yu, G. Chen. Synchronization of delayed chaotic systems with parameter mismatches by using intermittent linear state feedback [J]. Nonlinearity.2009,22:569-584.
[58]J.J. Huang, C.D. Li, Q. Han. Stabilization of delayed chaotic neural networks by periodically intermittent control [J]. Circuits Syst Signal Process.2009,28:567-579.
[59]T.W. Huang, C.D. Li. Chaotic Synchronization by intermittent feedback method [J]. Comput Appl Math.2010,234:1097-1104.
[60]廖晓昕,傅予力,高健,赵新泉.具有反应扩散的Hopfield神经网络的稳定性_J].电子学报.2000,28：78-82.
[61]J.L. Liang, J.D. Cao. Global exponential stability of reaction-diffusion recurrent neural networks with time-varying delays [J]. Phys Lett A.2003,314:434-442.
[62]王林山,徐道义.变时滞反应扩散Hopfield神经网络的全局指数稳定性[J].中国科学(E).2003,33：488-495.
[63]W. Allegretto, D. Papini. Stability for delayed reaction-diffusion neural networks [J]. Phys Lett A.2007,360:669-680.
[64]J. Lu. Robust Global Exponential Stability for Interval Reaction-Diffusion Hopfield Neural Networks With Distributed Delays [J]. IEEE Trans Circuits Syst II.2007,54:1115-1119.
[65] J.G. Lu. Global exponential stability and periodicity of reaction-difusion delayedrecurrent neural networks with Dirichlet boundary conditions [J]. Chaos Sol Frac.2008,35:116-125.
[66] J. Wang, J. Lu. Global exponential stability of fuzzy cellular neural networks withdelays and reaction-difusion terms [J]. Chaos Sol Frac.2008,38:878-885.
[67] J. Lu, L. Lu. Global exponential stability and periodicity of reaction-difusion re-current neural networks with distributed delays and Dirichlet boundary conditions[J]. Chaos Sol Frac.2009,39:1538-1549.
[68] X. Lou, B. Cui. Asymptotic synchronization of a class of neural networks withreaction-difusion terms and time-varying delays [J]. Comput Math Appl.2006,52:897-904.
[69] Y. Wang, J. Cao. Synchronization of a class of delayed neural networks with reaction-difusion terms [J]. Phys Lett A.2007,369:201-211.
[70] L. Sheng, H. Yang, X. Lou. Adaptive exponential synchronization of delayed neuralnetworks with reaction-difusion terms [J]. Chaos Sol Frac.2009,40:930-939.
[71] K. Wang, Z. Teng, H. Jiang. Global exponential synchronization in delayed reaction-difusion cellular neural networks with the Dirichlet boundary conditions [J]. MathComput Model.2010,52:12-24.
[72] C. Hu§H. Jiang, Z. Teng. Impulsive control and synchronization for delayed neuralnetworks with reaction-difusion terms [J]. IEEE Trans Neural Networks.2010,21:67-81.
[73] L.O. Chua, T. Roska. Cellular neural networks and visual computing: foundationand applications [M]. New York: Cambridge University Press,2002.
[74] M. Faloutsos, P. Faloutsos, C. Faloutsos. On power-law relationships of the Interacttopology [J]. Computer Communication Revivew.1999,29:29-51
[75] N. Guelzim, S. Bottani, P. Bourgine, et a1. Topological and causal structure of theyeast transcriptional regulatory network [J]. Nature Genetics.2002,31:60-63
[76] M.E.J. Newman. Scientific collaboration networks I: Network construction and fun-damental results [J]. Phys Rev E.2001,64:016131
[77]A. Cardillo, S. Scellato, V. Latora, S. Porta. Structural properties of planar graph of urban street patterns [J]. Phys Rev E.2006,73:066107.
[78]D.J. Watts, S.H. Strogatz. Collective dynamics of'small-world'networks [J]. Nature.1998,393:440-442.
[79]A.L. Barabasi, R. Albert. Emergence of scaling in random networks [J]. Science.1999,286:509-512.
[80]方锦清,汪小帆,刘曾荣.略论复杂性问题和非线性复杂网络系统的研究[J].基础科学,科技导报.2004,2：9-12.
[81]M.E.J. Newman, D. J. Watts. Scaling and percolation in the small-world network model [J]. Phys Rev E.1999,60:7332-7342.
[82]X. Li, G. Chen. Alocal-world evolving network model [J]. Physica A.2003,328:274-286.
[83]K. Klemm, V.M. Eguiluz. Highly clustered scale-free networks [J]. Phys Rev E.2002,65:6123-6127.
[84]D.J. Watts. The'new'science of networks [J]. Annual Review of Sociology.2004,1:243-270.
[85]S.H. Strogatz, I. Stewart. Coupled oscillators and biological synchronization [J]. Scientific American.1993,269:102-109.
[86]C。M. Gray. Synchronous oscillations in neuronal systems:mechanisms and functions [J]. Journal of Computer Neurosci.1994,1:11-38.
[87]R.Q. Wang, L.N. Chen. Synchronizing genetic oscillators by signaling molecules [J]. Journal of Biological Rhythms.2005,20:257-269.
[88]Z. Nfda, E. Ravasz, T. Vicsek, et al. Physics of the rhythmic applause [J]. Phys Rev E.2000,6:6987-6992.
[89]X. Wang, G. Chen. Synchronization in scale-free dynamical networks:robustness and fragility [J]. IEEE Trans Circuits Syst I.2002,1:54-62.
[90]X. Wang, G. Chen. Synchronization in small-world dynamical networks [J]. Inter J Bifur Chaos.2002,1:187-192.
[91] X. Wang, G. Chen. Pinning control of scale-free dynamical networks [J]. Physica A.2002,310:521-531.
[92] T. Chen, X. Liu, W. Lu. Pinning Complex Networks by a Single Controller [J].IEEE Trans Circuits Syst I.2007,54:1317-1326.
[93] W. Lu, X. Li, Z. Rong. Global stabilization of complex networks with digraphtopologies via a local pinning algorithm [J]. Automatica.2010,46:116-121.
[94] W. Yu, G. Chen, J. Lu¨. On pinning synchronization of complex dynamical networks[J]. Automatica.2009,45:429-435.
[95] W. Wu, T. Chen. Global Synchronization Criteria of Linearly Coupled Neural Net-work Systems With Time-Varying Coupling [J]. IEEE Trans Neural Networks.2008,19:319-332.
[96] J. Zhao, J. Lu, Q. Zhang. Pinning a Complex Delayed Dynamical Network to aHomogenous Trajectory [J]. IEEE Trans Circuits Syst II.2009,56:514-518.
[97] X. Li, X. Wang, G. Chen. Pinning a Complex Dynamical Network to Its Equilibrium[J]. IEEE Trans Circuits Syst I.2004,51:2074-2087.
[98] J. Zhou, J. Lu, J. Lu¨. Pinning adaptive synchronization of a general complex dy-namical networ [J]. Automatica.2008,44:996-1003.
[99] Z.X. Liu, Z.Q. Chen, Z.Z. Yuan. Pinning control of weighted general complex dy-namical networks with time delay [J]. Physica A.2007,375:345-354.
[100] B. Wang, Z. Guan. Chaos synchronization in general complex dynamical networkswith coupling delays [J]. Nonlinear Anal RWA.2010,11:1925-1932.
[101] Z. Li, G. Chen. Global Synchronization and Asymptotic Stability of Complex Dy-namical Networks [J]. IEEE Trans Circuits Syst II.2006,53:28-33.
[102] Y. Xu, W. Zhou, J. Fang, W. Sun. Adaptive synchronization of the complex dy-namical network with non-derivative and derivative coupling [J]. Phys Lett A.2010,374:1673-1677.
[103] Z. Li, L. Jiao, J. Lee. Robust adaptive global synchronization of complex dynamicalnetworks by adjusting time-varying coupling strength [J]. Physica A.2008,387:1369-1380.
[104] L. Huang, Z. Wang, Y. Wang, Y. Zuo. Synchronization analysis of delayed complexnetworks via adaptive time-varying coupling strengths [J]. Phys Lett A.2009,373:3952-3958.
[105] H. Liu, J. Chen, J. Lu, M. Cao. Generalized synchronization in complex dynamicalnetworks via adaptive couplings [J]. Physica A.2010,389:1759-1770.
[106] H. Tang, L. Chen, J. Lu, C. Tse. Adaptive synchronization between two complexnetworks with nonidentical topological structures [J]. Physica A.2008,387:5623-5630.
[107] X. Wu. Synchronization-based topology identification of weighted general complexdynamical networks with time-varying coupling delay [J]. Physica A.2008,387:997-1008.
[108] G. Zhang, Z. Liu, Z. Ma. Synchronization of complex dynamical networks via im-pulsive control [J]. Chaos.2007,17:043126.
[109] Q. Zhang, J. Lu, J. Zhao. Impulsive synchronization of general continuous anddiscrete-time complex dynamical networks [J]. Commun Nonlinear Sci Numer Sim-ulat.2010,15:1063-1070.
[110] G. Wang, J. Cao, J. Lu. Outer synchronization between two nonidentical networkswith circumstance noise [J]. Physica A.2010,389:1480-1488.
[111] W. Lu, T. Chen. New approach to synchronization analysis of linearly coupledordinary diferential systems [J]. Physica D.2006,213:214-230.
[112] W. Guo, F. Austin, S. Chen. Global synchronization of nonlinearly coupled complexnetworks with non-delayed and delayed coupling [J]. Commun Nonlinear Sci NumerSimulat.2010,15:1631-1639.
[113] P.D. Lellis, M. Bernardo, F. Garofalo. Synchronization of complex networks throughlocal adaptive coupling [J]. Chaos.2008,18:037110.
[114] H. Su, Z. Rong, X. Wang, G. Chen. On Decentralized Adaptive Pinning Synchroniza-tion of Complex Dynamical Networks [R]. Proceedings of2010IEEE InternationalSymposium on Circuits and Systems. IEEE Press: New Jersey,2010,417-420.
[115] Z. Yuan, J. Cai, M. Lin. Global Synchronization in Complex Networks with AdaptiveCoupling [J]. Mathematical Problems in Engineering.2010,826721.
[116] W.G. Xia, J.D. Cao. Pinning synchronization of delayed dynamical networks viaperiodically intermittent control [J]. Chaos.2009,19:013120.
[117] X. Yang, J. Cao. Stochastic synchronization of coupled neural networks with inter-mittent control [J]. Phys Lett A.2009,373:3259-3272.
[118] S. Cai, Z. Liu, F. Xua, J. Shen. Periodically intermittent controlling complex dy-namical networks with time-varying delays to a desired orbit [J]. Phys Lett A.2009,373:3846-3854.
[119] X. Liu, T. Chen. Cluster Synchronization in Directed Networks Via IntermittentPinning Control [J]. IEEE Trans Neural Networks.2011,22:1009-1020.
[120] C.W. Wu. Localization of efective pinning control in complex networks of dynamicalsystems [R]. Proc IEEE Int Symp Circuits Syst. IEEE Press: New Jersey,2008,2530-2533.
[121] I. Belykh, V. Belykh, M. Hasler. Synchronization in asymmetrically coupled net-works with node balance [J]. Chaos.2006,16:015102.
[122] Y.Y. Lu, X.F. Wang. Pinning control of directed dynamical networks based onControlRank [J]. Int J Comput Math.2008,85:1279-1286.
[123] Q. Song, J. Cao. On Pinning Synchronization of Directed and Undirected ComplexDynamical Networks [J]. IEEE Trans Circuits Syst I.2010,57:672-680.
[124] M. Yang, Y. Liu, Z. You, P. Sheng. Global synchronization for directed complexnetworks [J]. Nonlinear Anal RWA.2010,11:2127-2135.
[125] F. Nian, X. Wang. Optimal pinning synchronization on directed complex network[J]. Chaos.2011,21:043131.
[126] W. Lu, T. Chen. Synchronisation in complex networks of coupled systems withdirected topologies [J]. Int J Syst Sci.2009,40:909-921.
[127] K. Kaneko. Relevance of dynamic clustering to biological networks [J]. Physica D.1994,75:55-73.
[128]V.N. Belykh, I.V. Belykh, E. Mosekilde. Cluster synchronization modes in an en-semble of coupled chaotic oscillators [J]. Phys Rev E,2001,63:036216.
[129]W. Qin, G. Chen. Coupling schemes for cluster synchronization in coupled Josephson equations [J]. Physica D.2004,197:375-391.
[130]X.B. Lu, X.F. Wang, J.Q. Fang. Controlling a complex dynamical network to attain an inhomogeneous equilibrium [J]. Physica D.2010,239:341-347.
[131]X. Lu, B. Qin. Adaptive cluster synchronization in complex dynamical networks [J]. Phys Lett A.2009,373:3650-3658.
[132]W. Wu, W. Zhou, T. Chen. Cluster Synchronization of Linearly Coupled Complex Networks Under Pinning Control [J]. IEEE Trans Circuits Syst I.2009,56:829-839.
[133]L. Chen, J. Lu. Cluster synchronization in a complex network with two nonidentical clusters [J]. Jrl Syst Sci&Complexity,2008,21:20-33.
[134]K. Wang, X. Fu, K. Li. Cluster synchronization in community networks with non-identical nodes [J]. Chaos.2009,19:023106.
[135]X. Wu, H. Lu. Cluster synchronization in the adaptive complex dynamical networks via a novel approach [J]. Phys Lett A.2011,375:1559-1565.
[136]W. Lu, B. Liu, T. Chen. Cluster synchronization in networks of coupled nonidentical dynamical systems [J]. Chaos.2010,20:013120.
[137]Q. Song, J. Cao, F. Liu. Synchronization of complex dynamical networks with non-identical nodes [J]. Phys Lett A.2010,374:544-551.
[138]S. Cai, Q. He, J. Hao, Z. Liu. Exponential synchronization of complex networks with nonidentical time-delayed dynamical nodes [J]. Phys Lett A.2010,374:2539-2550.
[139]R.A. Horn, C.R. Johnson. Matrix Analysis [M]. New York: Cambridge University Press,2005.
[140]W. Lu, T. Chen, G. Chen. Synchronization analysis of linearly coupled systems described by differential equations with a coupling delay [J]. Physica D.2006,221:118-134.
[141]程云鹏.矩阵论[M].第2版.西安：西北工业大学出版社,2005.
[142] C. Wu. Synchronization in Complex Networks of Nonlinear Dynamical Systems [M].Singapore: World Scientific Publishing,2007.