混沌系统同步与复杂网络牵制控制研究
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摘要
复杂性科学以复杂系统为研究对象,非线性动力学和复杂网络的研究是目前复杂性科学研究领域比较瞩目的课题。混沌是非线性动力系统中特有的一种运动形式,是在确定性系统中出现的貌似无规则、类似随机的复杂现象。混沌在现代科学技术中,尤其是在复杂系统的研究中占有重要的地位。
网络无处不在,遍及自然界、生物系统和人类社会。随着网络规模的不断增大,如何对这些复杂的网络进行有效控制是一个重要课题,其中,牵制控制被证明是一种重要的控制策略。牵制控制最初被应用到由大量混沌节点以某种耦合方式连接在一起构成的时空混沌系统中,通过对网络中的一部分节点施加控制,使得整个网络的时空混沌行为得到有效的抑制,实现整个网络的同步。
本文研究非线性混沌系统同步和复杂网络的牵制控制问题。首先建立一类通用的混沌系统模型,分析不同结构混沌系统的同步方案,给出相应的稳定性证明。不同于目前已有文献的研究,本文考虑的通用模型中系统具有参数不确定、系统阶数不同、存在延时和随机噪声等不确定性,在此情况下,建立自适应反馈同步策略,实现混沌系统的广义投影同步和混合投影同步。其次,在研究混沌系统同步方法的同时,本文分析了在存在延时和噪声干扰情况下混沌同步方法在保密通信领域的应用。最后,本文以复杂网络的度相关性为例,分析了不同度相关网络对于牵制控制的影响。
本文的主要内容和创新之处可概括如下:
(1)一类参数不确定混沌系统的延时同步
由于实现混沌系统的电容、电阻、放大器等电子器件本身存在一定的参数不确定性,以及混沌实现电路运行过程中会受到外部的干扰,实际系统同步过程中不可避免地存在延时,因此需要建立含参数不确定项的混沌系统模型,进而实现混沌系统的延时同步。本文通过建立一类通用混沌系统模型,设计合适的自适应反馈控制器,在参数不确定情况下,考虑系统中的常数延时和时变延时,实现不同系统的同步。同时,通过Lyapunov稳定性理论对于同步系统的稳定性给出理论证明,数值仿真的结果证实了同步方法的有效性。
(2)不同阶数混沌系统的混合投影同步
在实现混沌系统同步的过程中,驱动系统和响应系统以一定的比例因子实现混合投影同步,即该比例因子可以通过对角阵形式描述。实际同步的系统经常具有不同的动力学行为,如呼吸系统和循环系统之间形成了某种同步,但它们的模型完全不同,甚至具有不同的阶数。本文建立了具有不同阶数的混沌系统模型,定义升阶混合投影同步和降阶混合投影同步,给出了实现升阶混合投影同步和降阶混合投影同步的策略。通过数值仿真实例验证了同步的有效性。
(3)随机干扰下混沌系统的广义投影同步
随机干扰对系统的同步行为有一定的影响,本文研究了参数不确定的混沌系统在随机干扰下的广义投影同步,通过设计自适应反馈控制器,利用伊藤微分理论实现随机干扰下混沌系统的广义投影同步。同时,分析了随机干扰下升阶和降阶混沌系统的广义投影同步,并给出了相应的同步策略。通过Lyapunov稳定性理论对系统的稳定性给出了证明,数值仿真实例证实了同步的有效性。
(4)噪声干扰下延时混沌系统同步在保密中的应用分析
本文研究了噪声干扰下延时混沌系统同步在保密通信领域的实现方案。在系统存在噪声干扰和延时情况下,对混沌遮掩和混沌参数调制保密通信方案进行了系统分析。在混沌遮掩保密通信中,设计了对有用信号进行函数变换的方法,加大了破译难度,增大了密钥空间,同时可以有效地解密。参数调制通信策略中考虑一类通用的混沌系统,通过调制系统中的多个参数传送信息,能够在系统中存在延时且信道中存在噪声的情况下很好地恢复传送信号。
(5)复杂网络度相关性对牵制控制的影响分析
研究表明现实的复杂网络存在多种度相关特征:许多社会关系网络具有正相关特征,然而,大多数技术网络和生物网络表现出负相关特征。牵制控制通过牵制网络中的部分少量节点实现整个网络的同步,本文研究了度相关性对于牵制控制的影响,实现了不同度相关特征的网络模型。在分析随机牵制和最大度牵制的同时,提出了混合牵制策略。发现在不同的度相关网络中,合适的控制增益增强网络的可控性;对于给定网络,负相关特征有助于控制作用的实现;网络的可控性对正相关特征的变化比较敏感。
Nonlinear dynamics system and complex networks are two brilliant aspects of complexity science.Chaos is a special moving form of nonlinear dynamics system, whose trajectory of the orbits in the phase plane is very complex,pseudorandom,and can be observed in fairly simple dynamical systems.In modem science and technology,chaos is very important especially in studying complex systems.
Complex networks are everywhere,ranging from nature and biological system to society.With the increasing of the networks complexity,how to control the whole network is an interesting topic.Pinning control is one of effective method in controlling networks with large inter-connection chaos nodes.By applying an action on some nodes,we can get the synchronization of the whole networks.
This paper studies synchronization of nonlinear chaotic systems and pinning control of complex networks.The thesis contains the following(1)Develop a general chaotic system model and investigate the corresponding synchronization method. Considering the unknown system parameters,time-delay,different order and stochastic perturbation,we design an appropriative control method and realize general projective synchronization and hybrid projective synchronization.(2)Analyze the application of lag synchronization of chaotic systems with noise perturbation in secure communication.(3)Analyze the effects of degree correlation on the controllability of complex networks.
The main contents can be summarized as follows:
(1)Lag synchronization of a class of chaotic systems with unknown parameters
In real systems,the connection weights of the neurons depend on certain resistance and capacitance values which include uncertainties.On the other hand,the information storage and neurotransmission frequently suffer from the fluctuations. Therefore,when designing a chaos model,the parameter uncertainties and time-delay should be involved.By constructing a general chaos model and designing a proper nonlinear controller,lag synchronization is achieved.Also an appropriative Lyapunov function is established to verify the systems stability.Moreover,we illustrate the application of the proposed scheme by numerical simulation,which demonstrates the effectiveness and feasibility of the proposed synchronization method.
(2)Hybrid projective synchronization of chaotic systems with different-order
There is increasing interest in the study of chaotic synchronization with different structure and different order due to its wide existence in biological science and social science.One instance is the synchronization that occurs between heart and lung, where one can observe that circulatory and respiratory systems synchronize with different models and different order.In this paper,hybrid projective synchronization of chaotic systems is introduced.Reduced order projective synchronization and increased order projective synchronization of different chaotic systems with fully unknown parameters are considered in detail.By combining the adaptive control method and feedback control technique,the suitable controllers and parameters update laws are derived to achieve synchronization of chaotic systems.
(3)Generalized projective stochastic perturbation synchronization of chaotic systems
It is found that noise plays a significant role in chaotic synchronization.We investigate the generalized projective stochastic perturbation synchronization,where the drive and response systems are synchronized up to a scaling factor,which is a constant.We present general methods for achieving the projective synchronization of two chaotic systems with uncertain parameters.The conclusions are proved by Lyapunov stability theory.Moreover,the unknown parameters can be efficiently estimated according to a rigorous and systematic scheme.Finally,the corresponding simulation results are given to verify the effectiveness of the proposed methods.
(4)Application of lag synchronization of chaotic systems with noise perturbation in secure communication
As a special moving form of nonlinear dynamics system,chaos is a behavior of uncorrelated,board-band,noise-like and sensitivity to chaotic system initial conditions.Chaotic signal has the properties of ergodicity,aperiodic,determinate and random-like.These properties make the signal be fit for applying in secure communication and information encrypting.This paper researches the application of chaotic synchronization in secure communication.Based on the lag synchronization methods,the simulations of chaos masking and chaos parameter modulation are present.In the chaos masking secure communication scheme,some function are used to transform the useful signal,it will increase the difficulty of decryption.
(5)Effects of degree correlation on the controllability of complex networks
In real world,complex networks have different degree-mixing patterns:many social networks are usually assortative mixing where nodes with similar degrees are tending to interconnect with each other.However,a lot of technological and biological networks are mixed disassortatively,nodes with large degrees are willing to select small-degrees ones as their neighbors.Pinning control is a feedback control action to reach the extended networks synchronization.Through the Master Stability Function (MSF) theory,the network controllability can be estimated in terms of the stability of this synchronous state of the extended network.Random pinning and max-degree pinning strategies are presented to control the networks.Mix-degree pinning scheme is firstly introduced and applied,in which some nodes are selected by sorted degrees, and others are selected randomly.It is found that disassortative mixing feature enhances the network controllability contrast to assortative mixing,to which the network controllability is sensible.
网络无处不在,遍及自然界、生物系统和人类社会。随着网络规模的不断增大,如何对这些复杂的网络进行有效控制是一个重要课题,其中,牵制控制被证明是一种重要的控制策略。牵制控制最初被应用到由大量混沌节点以某种耦合方式连接在一起构成的时空混沌系统中,通过对网络中的一部分节点施加控制,使得整个网络的时空混沌行为得到有效的抑制,实现整个网络的同步。
本文研究非线性混沌系统同步和复杂网络的牵制控制问题。首先建立一类通用的混沌系统模型,分析不同结构混沌系统的同步方案,给出相应的稳定性证明。不同于目前已有文献的研究,本文考虑的通用模型中系统具有参数不确定、系统阶数不同、存在延时和随机噪声等不确定性,在此情况下,建立自适应反馈同步策略,实现混沌系统的广义投影同步和混合投影同步。其次,在研究混沌系统同步方法的同时,本文分析了在存在延时和噪声干扰情况下混沌同步方法在保密通信领域的应用。最后,本文以复杂网络的度相关性为例,分析了不同度相关网络对于牵制控制的影响。
本文的主要内容和创新之处可概括如下:
(1)一类参数不确定混沌系统的延时同步
由于实现混沌系统的电容、电阻、放大器等电子器件本身存在一定的参数不确定性,以及混沌实现电路运行过程中会受到外部的干扰,实际系统同步过程中不可避免地存在延时,因此需要建立含参数不确定项的混沌系统模型,进而实现混沌系统的延时同步。本文通过建立一类通用混沌系统模型,设计合适的自适应反馈控制器,在参数不确定情况下,考虑系统中的常数延时和时变延时,实现不同系统的同步。同时,通过Lyapunov稳定性理论对于同步系统的稳定性给出理论证明,数值仿真的结果证实了同步方法的有效性。
(2)不同阶数混沌系统的混合投影同步
在实现混沌系统同步的过程中,驱动系统和响应系统以一定的比例因子实现混合投影同步,即该比例因子可以通过对角阵形式描述。实际同步的系统经常具有不同的动力学行为,如呼吸系统和循环系统之间形成了某种同步,但它们的模型完全不同,甚至具有不同的阶数。本文建立了具有不同阶数的混沌系统模型,定义升阶混合投影同步和降阶混合投影同步,给出了实现升阶混合投影同步和降阶混合投影同步的策略。通过数值仿真实例验证了同步的有效性。
(3)随机干扰下混沌系统的广义投影同步
随机干扰对系统的同步行为有一定的影响,本文研究了参数不确定的混沌系统在随机干扰下的广义投影同步,通过设计自适应反馈控制器,利用伊藤微分理论实现随机干扰下混沌系统的广义投影同步。同时,分析了随机干扰下升阶和降阶混沌系统的广义投影同步,并给出了相应的同步策略。通过Lyapunov稳定性理论对系统的稳定性给出了证明,数值仿真实例证实了同步的有效性。
(4)噪声干扰下延时混沌系统同步在保密中的应用分析
本文研究了噪声干扰下延时混沌系统同步在保密通信领域的实现方案。在系统存在噪声干扰和延时情况下,对混沌遮掩和混沌参数调制保密通信方案进行了系统分析。在混沌遮掩保密通信中,设计了对有用信号进行函数变换的方法,加大了破译难度,增大了密钥空间,同时可以有效地解密。参数调制通信策略中考虑一类通用的混沌系统,通过调制系统中的多个参数传送信息,能够在系统中存在延时且信道中存在噪声的情况下很好地恢复传送信号。
(5)复杂网络度相关性对牵制控制的影响分析
研究表明现实的复杂网络存在多种度相关特征:许多社会关系网络具有正相关特征,然而,大多数技术网络和生物网络表现出负相关特征。牵制控制通过牵制网络中的部分少量节点实现整个网络的同步,本文研究了度相关性对于牵制控制的影响,实现了不同度相关特征的网络模型。在分析随机牵制和最大度牵制的同时,提出了混合牵制策略。发现在不同的度相关网络中,合适的控制增益增强网络的可控性;对于给定网络,负相关特征有助于控制作用的实现;网络的可控性对正相关特征的变化比较敏感。
Nonlinear dynamics system and complex networks are two brilliant aspects of complexity science.Chaos is a special moving form of nonlinear dynamics system, whose trajectory of the orbits in the phase plane is very complex,pseudorandom,and can be observed in fairly simple dynamical systems.In modem science and technology,chaos is very important especially in studying complex systems.
Complex networks are everywhere,ranging from nature and biological system to society.With the increasing of the networks complexity,how to control the whole network is an interesting topic.Pinning control is one of effective method in controlling networks with large inter-connection chaos nodes.By applying an action on some nodes,we can get the synchronization of the whole networks.
This paper studies synchronization of nonlinear chaotic systems and pinning control of complex networks.The thesis contains the following(1)Develop a general chaotic system model and investigate the corresponding synchronization method. Considering the unknown system parameters,time-delay,different order and stochastic perturbation,we design an appropriative control method and realize general projective synchronization and hybrid projective synchronization.(2)Analyze the application of lag synchronization of chaotic systems with noise perturbation in secure communication.(3)Analyze the effects of degree correlation on the controllability of complex networks.
The main contents can be summarized as follows:
(1)Lag synchronization of a class of chaotic systems with unknown parameters
In real systems,the connection weights of the neurons depend on certain resistance and capacitance values which include uncertainties.On the other hand,the information storage and neurotransmission frequently suffer from the fluctuations. Therefore,when designing a chaos model,the parameter uncertainties and time-delay should be involved.By constructing a general chaos model and designing a proper nonlinear controller,lag synchronization is achieved.Also an appropriative Lyapunov function is established to verify the systems stability.Moreover,we illustrate the application of the proposed scheme by numerical simulation,which demonstrates the effectiveness and feasibility of the proposed synchronization method.
(2)Hybrid projective synchronization of chaotic systems with different-order
There is increasing interest in the study of chaotic synchronization with different structure and different order due to its wide existence in biological science and social science.One instance is the synchronization that occurs between heart and lung, where one can observe that circulatory and respiratory systems synchronize with different models and different order.In this paper,hybrid projective synchronization of chaotic systems is introduced.Reduced order projective synchronization and increased order projective synchronization of different chaotic systems with fully unknown parameters are considered in detail.By combining the adaptive control method and feedback control technique,the suitable controllers and parameters update laws are derived to achieve synchronization of chaotic systems.
(3)Generalized projective stochastic perturbation synchronization of chaotic systems
It is found that noise plays a significant role in chaotic synchronization.We investigate the generalized projective stochastic perturbation synchronization,where the drive and response systems are synchronized up to a scaling factor,which is a constant.We present general methods for achieving the projective synchronization of two chaotic systems with uncertain parameters.The conclusions are proved by Lyapunov stability theory.Moreover,the unknown parameters can be efficiently estimated according to a rigorous and systematic scheme.Finally,the corresponding simulation results are given to verify the effectiveness of the proposed methods.
(4)Application of lag synchronization of chaotic systems with noise perturbation in secure communication
As a special moving form of nonlinear dynamics system,chaos is a behavior of uncorrelated,board-band,noise-like and sensitivity to chaotic system initial conditions.Chaotic signal has the properties of ergodicity,aperiodic,determinate and random-like.These properties make the signal be fit for applying in secure communication and information encrypting.This paper researches the application of chaotic synchronization in secure communication.Based on the lag synchronization methods,the simulations of chaos masking and chaos parameter modulation are present.In the chaos masking secure communication scheme,some function are used to transform the useful signal,it will increase the difficulty of decryption.
(5)Effects of degree correlation on the controllability of complex networks
In real world,complex networks have different degree-mixing patterns:many social networks are usually assortative mixing where nodes with similar degrees are tending to interconnect with each other.However,a lot of technological and biological networks are mixed disassortatively,nodes with large degrees are willing to select small-degrees ones as their neighbors.Pinning control is a feedback control action to reach the extended networks synchronization.Through the Master Stability Function (MSF) theory,the network controllability can be estimated in terms of the stability of this synchronous state of the extended network.Random pinning and max-degree pinning strategies are presented to control the networks.Mix-degree pinning scheme is firstly introduced and applied,in which some nodes are selected by sorted degrees, and others are selected randomly.It is found that disassortative mixing feature enhances the network controllability contrast to assortative mixing,to which the network controllability is sensible.
引文
1.Lorenz E.N.“Deterministic nonperodic flow,”J Atmos Sci.1963,20:130-141.
2.Li T.Y.and Yorke J.A.“Period three implies chaos,”Amer.Math.Mont.1975,82:985-992.
3.Ott E.Grebogi C.and Yorke J.A.“Controlling Chaos,”Phys.Rev.Lett.1990,64:1196-1199.
4.Qammar H.K.and Mossagchi F.“Adaptive Control Chaos,”Phys.Rev.Lett.1993,178:279.
5.Sinha S.and Biswas D.“Adaptive Dynamics on a Chaotic Lattice,”Phys.Rev.Lett.1993,71:2010.
6.Vassiliadis D.and Parametric,“Adaptive Control and Parameter Identification of Lows Dimensional Chaotic System,”Phys.D,1994,71:319.
7.Ni W.S.and Qin T.F.“Controlling Chaos by Method of Adaptive,”Chin.Phys.Lett.1994,11:325.
8.Tian Y.C.and Gao F.R.“Adaptive Control of Chaotic Continuous-Time Systems with Delay,”Phys.D,1998,117:1.
9.Wu C.W.Yang T.and Chua L.O.“On adaptive synchronization and control of nonlinear dynamical systems,”International Journal of Bifurcation and Chaos 1996,6:455-471.
10.Chen M.and Han Z.“Controlling and synchronizing chaotic Genesio system via nonlinear feedback control,”Chaos,Solitons and Fractals,2003,17:709-716.
11.Perora L.M.and Carroll T.L.“Synchronization in chaotic systems,”Phys.Rev.Lett.1990,64:821-825.
12.Boccaletti S.Bragard J.Arecchi F.T.and Mancini H.“Synchronization in nonidentical extended systems,”Phys.Rev.Lett.1999,83:536-539.
13.Boccaletti S.Valladares D.L.Kurths J.Maza D.and Mancini H.“Synchronization of chaotic structurally nonequivalent systems,”Phys.Rev.E,2000,61:3712-3715
14.Boccalettia S.Kurths J.Osipov G.Valladaresb D.L.and Zhou C.S.“The synchronization of chaotic systems,”Phys.Rep.2002,366:1-101.
15.Nijmeijer H.“A dynamical control view on synchronization,”Phys.D,2001,154:219-228.
16.Gonzalez M.Iranda J.M.Bistable.“Generalized synchronization of chaotic systems,” Computer Physics Communications,1999,121:429-431.
17.Maritan A.and Banavar J.“Chaos,noise and synchronization,” Phys.Rev.Lett.1994,72:1451.
18.Zhang H.G.Huang W.Wang Z.L.and Chai T.Y.“Adaptive synchronization between two different chaotic systems with unknown parameters,” Phys.Lett.A,2006,350:363-366.
19.Wu X.Y.Guan Z.H.Wu Z.P.et al,“Chaos synchronization between Chen system and Genesio system,” Phys.Lett.A,2007,364:484-487.
20.Rosenblum M.G.Pikovsky A.S.and Kurths J.“Phase synchronization of chaotic oscillators,” Phys.Rev.Lett.1996,76:1804
21.Rosenblum M.G.Pikovsky A.S.and Kurths J.“From phase to lag synchronization in coupled chaotic oscillators,” Phys.Rev.Lett.1997,78:4193.
22.Kocarev L.and Parlitz U.“From phase to lag synchronization in coupled chaotic oscillators,” Phys.Rev.Lett.1996,76:1816.
23.Mainieri R.and Rehacek J.“Projective Snchronization in Three-demensional Chaotic System,” Phys.Rev.Lett.1999,82:3042.
24.Kolcarev L.and Parlitz U.,“General Approach for Chaotic Synchronization with Applications to Communication,” Phys.Rev.Lett.1995,74(25):5028.
25.Park Ju H.“Chaos synchronization between two different chaotic dynamical systems,” Chaos,Solitons and Fractals,2006,27:549-554.
26.Agiza H.N.and Yassen M.T.“Synchronization of Rossler and Chen chaotic dynamical systems using active control,” Phys.Lett.A,2001,278:191-197.
27.Bai E.W.and Lonngren K.E.“Sequential synchronization of two Lorenz systems using active control,” Chaos,Solitons and Fractals,2000,11:1041-1044.
28.Li Z.Han C.Z.and Shi S.J.“Modification for synchronization of Rossler and Chen chaotic systems,” Phys.Lett.A,2002,301.
29.Li D.M.Wang Z.D.etc.“A note on chaotic synchronization of time-delay secure communication systems,”Chaos,Solitons and Fractals,2007,27:549-554.
30.于娜,丁群等,“异结构系统混沌同步及其在保密通信中的应用,”通信学报,2007,28(10):73-78.
31.Corron N.J.and Hahs D.W.“A new approach to communication using chaotic signals,” IEEE Trans.Circuits and Systems-I,1997,44:373-382.
32.Chua L.O.and Kocarev L.“Transmission of digital signals by chaotic synchronization,” International Journal of Bifurcation and Chaos,1992,2:973-977.
33.Wang X.F.Wang Z.Q.and Chen G.“A new criterion for synchronization of coupled chaotic oscillators with application to Chua's circuits,” International Journal of Bifurcation and Chaos,1999,1169-1174.
34.赵耿,方锦清.“混沌通信分类及其保密通信的研究,”自然杂志,2003,25:21-30.
35.关新平,范正平,陈彩莲,华长春,“混沌控制及其在保密通信中的应用,”国防工业出版社,2002.
36.Haier M.“Synchronization of Chaotic and Transmission of Information,”International Journa of Bifureation and Chaos,1998,8(4):647-659.
37.Milanovic V.and zagloul M.E.“Improved Masking Algorithm for Chaotic Communications systems,” Electronics Letters,1996,32(1):11-12.
38.Yan T.and Chua L.O.“Secure communication via chaotic parameter modulation,” IEEE Transactions on Circuits and Systems I,1996,43:817-819.
39.Fang J.A.and Miao Q.Y.“Chaotic parameter modulation and secure communication,” Proceedings of the 4th World Congress on Intelligent Control and Automation,2002.
40.关新平,范正平,彭海鹏,李丽香,“一种新的多级混沌系统在保密通信中的应用,”燕山大学学报,2001,25(4):287-289.
41.赵耿,方锦清,“现代信息安全与混沌保密通信应用研究与进展,”物理学进展,2003,23(2):212-255.
42.Milanovic V.and zagloul M.E.“Improved Masking Algorithm for Chaotic Communications systems,” Electronics Letters,1996,32(1):11-12.
43.Albert R.and Barab(?)si A.L.“Statistical mechanics of complex networks,” Rev.Mod.Phys.2002,74:47-97.
44.Newman M.“The structure and function of complex networks,” SIAM Rev.2003,45:167-256.
45.Wang X.F.“Complex networks:Topology,dynamics and synchronization,”International Journa of Bifureation and Chaos,2002,12:885-916.
46.Dorogovtsev S.N.and Mendes J.F.“Evolution of networks,” Adv.Phys.2002,51:1079-1187.
47.吴金闪,狄增如,“从统计物理学看复杂网络研究,”物理学进展,2004,24:18-46.
48.周涛,柏文洁,汪秉宏,刘之景,严钢,“复杂网络研究概述,”物理,2005,34:31-36.
49.Watts D.J.and Strogatz S.H.“Collective dynamics of small-world networks,”Nature,1998,393:440-442.
50.Faloutsos M.Faloutsos P.and Faloutsos C.“On power-law relationships of the internet topology,” Computer Communications Review,1999,29:251-262.
51.Liljeros F.Rdling C.R.Amaral L.A.N.Stanley H.E.and Aberg Y.“Random Walks and Scale-Free Networks,” Nature,2001,411:907-908.
52.Ebel H.Mielsch L.I.and Borbholdt S.“Scale-free topology of E-mail networks,”Phys.Rev.E,2002,66,035103.
53.Li W.and Cai X.“Statistical analysis of airport network of China,” Phys.Rev.E,2004,69:046106.
54.Watts D.J.“Networks,dynamics,and the small world phenomenon,” Am.J.Sociol.1999,105:493-592.
55.Watts D.J.“Small Worlds,” Princeton University Press,Princeton,1999.
56.Barab(?)si A.L.and Albert R.“Emergence of scaling in random networks,” Science,1999,286:509-512.
57.Barab(?)si A.L.Albert R.and Jeong H.“Mean-field theory for scale-free random networks,” Physica A,1999,272:173-187.
58.Newman M.“Assortative mixing in networks,” Phys.Rev.Lett.2002,89:208701
59.Xulvi-Brunet R.and Sokolov I.M.“Changing correlations in networks:Assortativity and dissortativity,” Acta Physica Polonica B,2005,36:1431-1455.
60.Xulvi-Brunet R.and Sokolov I.M.“Reshuffling scale-free networks:From random to assortative,” Phys Rev.E.2004,70:066102.
61.Erdos P.and Renyi A.“On Random Graphs,” Publ.Math.(Debrecen),1959,6:290-297.
62.Erdos P.and Renyi A.“On the strength of connectedness of a random graph,”Acta Mathematica Scientia Hungary,1961,12:261-267.
63.Newman M.and Watts D.“Renormalization Group Analysis of the Small-World Network Model,” Phys.Lett.A,1999,263:341-346.
64.McGraw P.N.and Menzinger M.“Clustering and the synchronization of oscillator networks,”Phys.Rev.E,2005,72:015101.
65.Motter A.E.Zhou C.S.and Kurths J.“Enhancing complex-networks synchronization,”Europhys Lett.2005,69:334-340.
66.Motter A.E.Zhou C.S.and Kurths J.“Network synchronization,diffusion and the paradox of heterogeneity,”Phys.Rev.E,2005,71:016116.
67.Wang X.F.“Complex networks:Topology,dynamics and synchronization,”InternationalJournal of Bifurcation and Chaos,2002,12:885-916.
68.Barahona M.and Pecora L.M.“Synchronization in small-world systems,”Phys.Rev.Lett.2002,89.
69.Nishikawa T.Motter A.E.and Lai Y.C.Hoppensteadt F.C.“Heterogeneity in oscillator networks:Are smaller worlds easier to synchronize?”Phys.Rev.Lett.2003,91:014101.
70.Chavez M.Hwang D.U.and Amann A.Hentschel H.G.E.and Boccaletti S.“Synchronization is enhanced in weighted complex networks,”Phys.Rev.Lett.2005,94:218701.
71.Zhao M.Zhou T.Wang B.H.Ou Q.and Ren J.“Better synchronizability predicted by a new coupling method,”European Physical Journal B,2006,53:375-379.
72.Arenas A.Diaz-Guilera A.Kurths J.Moreno Y.and Zhou C.S.“Synchronization in complex networks,”arXiv:cond-mat/0805.2976.
73.Nishikawa T.and Motter A.E.“Synchronization is optimal in nondiagonalizable networks,”Phys Rev.E,2006,73:065106.
74.Pecora L.M.and Carroll T.L.“Master stability functions for synchronized coupled systems,”Phys.Rev.Lett.1998,80:2109-2112.
75.Heagy J.F.Carroll T.L.and Pecora L.M.“Synchronous chaos in coupled oscillator systems,”Phys.Rev.E,1994,50:1874-1885.
76.Gade P.M.Cerdeira H.and Ramaswamy R.“Coupled maps on trees,”Phys.Rev.E,1995,52:2478-2485.
77.Gade P.M.“Synchronization of oscillators with random nonlocal connectivity,”Phys.Rev.E,1996,54:64-70.
78.Hu G.Yang J.and Liu W.Phys.Rev.E,1998,58:4440-4453.
79.Fink K.S.Johnson G.Carroll T.Mar D.and Pecora L.“Three coupled oscillators as a universal probe of synchronization stability in coupled oscillator arrays,”Phys.Rev.E,2000,61:5080-5090.
80.Jost J.and Joy M.P.Phys.Rev.E,2001,65:016201.
81.Wang X.F.and Chen G.“Synchronization in scale-free dynamical networks:Robustness and fragility,” IEEE Transactions on Circuits and Systems I,2002,49:54-62.
82.L(u|¨) J.and Chen G.,“A time-varying complex dynamical network model and its controlled synchronization criteria,” IEEE Transactions on Automatic Control,2005,50:841-846.
83.L(u|¨) J.Chen G.Cheng D.and Celikovsky S.“Bridge the gap between the Lorenz system and the Chen system,” International Journal of Bifurcation and Chaos,2002,12:2917-2926.
84.L(u|¨) J.Han F.Yu X.and Chen G.“Generating 3-D multi-scroll chaotic attractors:A hysteresis series switching method,” Automatica,2004,40(10):1677-1687.
85.L(u|¨) J.Yu X.and Chen G.“Chaos synchronization of general complex dynamical networks,” Phys.A,2004,334:281-302.
86.Barahona M.and Pecora L.M.“Synchronization in small-world systems,” Phys.Rev.Lett.2004,89:054101.
87.Chen Y.Rangarajan G.and Ding M.“General stability analysis of synchronized dynamics in coupled systems,” Phys.Rev.E,2003,67:026209.
88.Wu C.W.and Chua L.O.“Synchronization in an array of linearly coupled dynamical systems,” IEEE Transactions on Circuits and Systems I,1995,42:430-447.
89.Wu C.W.“Synchronization in networks of nonlinear dynamical systems coupled via a directed graph,” Nonlinearity,2005,18(3):1057.
90.Wu C.W.“On Rayleigh-Ritz ratios of a generalized Laplacian matrix of directed graphs,” Linear Algebra and its Applications,2005,402:207.
91.Wu C.W.“Synchronization and convergence of linear dynamics in random directed networks,” IEEE Transactions on Automatic Control,2006,51(7):1207-1210.
92.Zhou J.Lu J.A.and L(u|¨) J.“Adaptive synchronization of an uncertain complex dynamical network,” IEEE Transactions on Automatic Control,2006,51(4):652-656.
93.Sorrentino F.Bernardo M.D.Cuellar G.H.and Boccaletti S.“Synchronization in weighted scale-free networks with degree-degree correlation,” Physica D,2006,224:123-129
94.Bernardo M.D.Garofalo F.and Sorrentino F.“Effects of degree correlation on the synchronization of networks of oscillators,” International Journal of Bifurcation and Chaos,2007,17:3499-3506
95.汪小帆,李翔,陈关荣,复杂网络理论及其应用,清华大学出版社,2006.
96.Wang X.F.and Chen G.“Pinning control of scale-free dynamical networks,”Phys.A,2002,310:521-531.
97.Xiang L.Y.Liu Z.X.Chen Z.Q.Chen F.and Yuan Z.Z.“Pinning control of complex dynamical networks with general topology,” Phys.A,2007,379:298-306.
98.Parekha N.and Sinhab S.“Controllability of spatiotemporal systems using constant pinning,” Phys.A,2003,318:200-212.
99.Wu C.W.“Localization of effective pinning control in complex networks of dynamical systems,” ISCAS 2008.
100.Hu G.and Qu Z.L.“Controlling spatiotemporal chaos in coupled map lattice systems,” Phys.Rev.Lett.1994,72:68-71.
101.Li X.Wang X.F.and Chen G.“Pinning a complex dynamical network to its equilibrium,” IEEE Transactions on Circuits and Systems I,2004,51:2074-2087.
102.Xiang L.Y.Chen Z.Q.Liu Z.X.Chen F and Yuan Z.Z.“Stabilizing weighted complex networks,” J.Phys.A:Math.Theor.2007,40:14369-14382.
103.Xiang L.Y.Liu Z.X.Chen Z.Q.and Yuan Z.Z.“Pinning weighted complex networks with heterogeneous delays by a small number of feedback controllers,”Science in China Series F:Information Sciences,2008,51:511-523.
104.Sorrentino F.Bernardo M.D.Garofalo F.and Chen G.“Controllability of complex networks via pinning,” Phys Rev.E,2007,75:046103.
105.Sorrentino F.“Effects of the network structural properties on its controllability,”Chaos,2007,17:033101.
106.Chen T.P.Liu X.W.and Lu W.L.“Pinning complex networks by a single controller,” IEEE Transactions on Circuits and Systems I,2007,54:1317-1326.
107.Li Z.,Feng G..and David H.“Controlling complex dynamical networks with coupling delays to a desired orbit,” Phys.Lett.A,2006,359:42-46
108.Wu X.F.Chen G.and Cai J.P.“Chaos synchronization of the master-slave generalized Lorenz systems via linear state error feedback control,” Phys.D,2007,229:52-80.
109.Yu W.W.and Cao J.D.“Adaptive synchronization and lag synchronization of uncertain dynamical system with time delay based on parameter identification,” Phys.A,2007,375:467-482.
110.Jia Q.“Projective synchronization of a new hyperchaotic Lorenz system,”Phys.Lett.A,2007,370:40-45.
111.Li Rui-hong,Xu Wei and Li Shuang,“Adaptive generalized projective synchronization in different chaotic systems based on parameter identification,”Phys.Lett.A,2007,367:199-206.
112.Jia Zhen,“Linear generalized synchronization of chaotic systems with uncertain parameters,”Journal of Systems Engineering and Electronics,2008,19(4):779-784
113.Li R.H.,Xu W.and Li S.,“Adaptive generalized projective synchronization in different chaotic systems based on parameter identification,”Phys.Lett.A,2007,367:199-206.
114.Li G.H.,Zhou S.P.and Yang K.,“Generalized projective synchronization between two different chaotic systems using active back stepping control,”Phys.Lett.A,2006,355:326-330.
115.Tang Y.and Fang J.A.“General methods for modified projective synchronization of hyperchaotic systems with known or unknown parameters,”Phys.Lett.A,2008,372:1816-1826.
116.Hu A.H.and Xu Z.Y.“Stochastic linear generalized synchronization of chaotic systems via robust control,”Phys.Lett.A,2008,372:3814-3818.
117.Luo R.Z.“Adaptive function project synchronization of RCssler hyperchaotic system with uncertain parameters,”Phys.Lett.A,2008,372:3667-3671.
118.Wang B.Wen G.J.and Xie K.“On the synchronization of a hyperchaotic system based on adaptive method,”Phys.Lett.A,2008,372:3015-3020.
119.Hu M.F.Xu Z.Y.and Zhang R.“Full state hybrid projective synchronization in continuous-time chaotic(hyperchaotic)systems,”Nonlinear Sci.Numer.Simul.2008,13:456-464.
120.Li G.H.“Modified projective synchronization of chaotic system,”Chaos,Solitons and Fractals,2007,32:1786-1790.
121.Hu M.F.Xu Z.Y.and Zhang R.“Adaptive full state hybrid projective synchronization of chaotic systems with the same and different order,”Phys.Lett.A,2007,365:315.
122.Hu M.F.Xu Z.Y.and Zhang R.“Parameters identification and adaptive full state hybrid projective synchronization of chaotic(hyper-chaotic)systems,”Phys.Lett.4 2007,361:23
123.Cao J.D.and Lu J.Q.“Adaptive complete synchronization of two identical or different chaotic(hyperchaotic)systems with fully unknown parameters,”Chaos,2005,15:043901.
124.Femat R.and SlisPerales G.“Synchronization of chaotic systems with different order,”Phys Rev.E,2002,65:036226.
125.Bowong S.and McClintock P.V.E.“Adaptive synchronization between chaotic dynamical systems of different order,”Phys.Lett.A,2006,358:134-141.
126.Ho M.C.Hung Y.C.Liu Z.Y.and Jiang I.M.“Reduced-order synchronization of chaotic systems with parameters unknown,”Phys.Lett.A,2006,348:251.
127.Moukam F.M.Bowong S.and Tchawoua C.“Reduced-order synchronization of uncertain chaotic systems via adaptive control,”Nonlinear Sci.Numer.Simul.2006,11:810.
128.Lazzouni S.A.Bowong S.Moukam F.M.and Cherki B.“An adaptive feedback control for chaos synchronization of nonlinear systems with different order,”Phys.Lett.A,2007,12:568.
129.Terman D.Koppel N.and Bose A.“Dynamics of two mutually coupled slow inhibitory neurons,”Phys.D,1998,117:241-275.
130.Gao T.G.Chen Z.Q.and Yuan Z.Z.Chin.Phys.2005,14:2421.
131.Liu C.X.“A novel chaotic attractor,”Chaos,Solitons and Fractals,in press.
132.Wang Z.D.Lauria S.Fang J.A.and Liu X.H.“Exponential stability of uncertain stochastic neural networks with mixed time-delays,”Chaos,Solitons and Fractals,2007,32:62-72.
133.Wang Z.D.Shu H.S.Fang J.A.and Liu X.H.“Robust stability for stochastic Hopfield neural networks with time delays,”Nonlinear Anal.2006,7:1119-1128.
134.Tang X.H.Lu J.A.and Zhang W.W.“The function projective synchronization of chaotic system using backstepping design,”Journal of dynamics and control,2007,5(3):216-219.
135.Wang Z.D.Liu Y.R.Fraser K.and Liu X.H.“Stochastic stability of uncertain Hopfield neural networks with discrete and distributed delays,”Phys.Lett.A,2006,354:288-297.
136.Lin W.and He Y.“Complete synchronization of the noise-perturbed Chua's circuits,”Chaos,2005,15:023705.
137.Sun Y.H.and Cao J.“Adaptive lag-synchronization of unknown chaotic delayed neural networks with noise perturbation,”Phys.Lett.A,2007,364:277-285.
138.Yu W.W.and Cao J.D.“Synchronization control of stochastic delayed neural networks,” Phys.A,2007,373:252-260.
139.Zhang C.Lin W.and Zhou J.“Complete and generalized synchronization in a class of noise perturbed chaotic systems,” Chaos,2007,17:023106.
140.Friedman,“Stochastic Differential Equations and Applications,” Academic Press,New York,1976.
141.Mianovic V.and Zaghloul M.E.“Improved masking algorithm chaotic communications systems,” Electronic Lett.1996,1:11-12.
142.Wc C.W.and Chua L.O.“A simple way to synchronize chaotic systems with applications to secure communication systems,” International Journal of Bifurcation and Chaos,1993,3:1619-1627.
143.Short K.M.“Steps toward unmasking secure communications,” International Journal of Bifurcation and Chaos,1994,4:957-977.
144.Diffie W.and Hellman M.F.“New directions in cryptograghy,” IEEE Transactions in Information Theory,1976,644-654,
145.Xulvi-Brunet R.Pietsch W.and Sokolov I.M.“Correlations in scale-free networks:Tomography and percolation,” Phys.Rev.E,2003,68:036119.
146.Yang T.and Chua L.O.“Impulsive stabilition for control and synchronization of chaotic system:Theory and application to secure communications,” IEEE Transactions on Circuits and Systems I,1997,44:976-988.
147.Celikovsky S.and Chen G.“Secure synchronization of a class of chaotic systems from a nonlinear observer approach,” IEEE Transactions on Automatic Control 2005,50:76-82.
148.Liao T.L.and Huang N.S.“An observer-based approach for chaotic synchronization with application to secure communication,” IEEE Transactions on Circuits and Systems I,1999,46:1144-1150.
149.Kaulakys B.ivanauskas F.and meskauskas T.“Synchronization of chaotic systems driven by identical noise,” Int J Bifurcation and Chaos,1997,9:533-539.
150.Boccaletti S.Latora V.Moreno Y.Chavez M.and Hwang D.U.“Complex networks:Structure and dynamics,” Phys.Rep.2006,424:175-308.
151.Hong H.Kim B.J.Choi M.Y.and Park H.“Factors that predict better synchronizability on complex networks,” Phys Rev.E,2004,69:067105.
152.Zhou C.and Kurths J.“Dynamical weights and enhanced synchronization in adaptive complex networks,” Phys.Rev.Lett.2006,96:164102.
153.Zhao M.Zhou T.Wang B.H.and Wang W.X.“Enhance synchronizability by structural perturbations,” Phy.Rev.E,2005,72:057102.
154.Zhou T.Zhao M.and Wang B.H.“Better synchronizability predicted by crossed double cycle,” Phy.Rev.E,2006,73:037101.
155.赵明,汪秉宏,蒋品群,周涛,“复杂网络上动力系统同步的研究进展,”物理学进展,2005,25:273-295.
156.Zhao M.Zhou T.Wang B.H.Yan G.Yang H.J.and Bai W.J.“Relations between average distance,heterogeneity and network synchronizability,” Phys.A,2006,371:773-780.
157.Zhao M.Zhou T.Wang B.H.Ou Q.and Ren J.“Better synchronizbility predicted by a new coupling method,” Eur.Phys.J.B.2006,53:375.
158.Wu X.Wang B.H.Zhou T.Wang W.X.Zhao M.and Yang H.J.“Synchronizability of highly clustered scale-free networks,” Chin.Phys.Lett.2006,23:1046.
159.Gu Z.M.Zhao M.Zhou T.Zhu C.P.and Wang B.H.“Phase Synchronization of non-Abelian oscillators on small-world networks,”Phys.Lett.A,2007,362:115.
160.Zhou T.Zhao M.Chen G.Yan G.and Wang B.H.“Phase synchronization on scale-free networks with community structure,”Phys.lett.A,2007,368:431-434.
161.Pandit S.A.and Amritkar R.E.”Characterization and control of small-world networks,” Physical Rev.E,1999,60(2):1119-1122.
162.Parekh N.Parthasarathy S.and Sinha S.“Global and local control of spatiotemporal chaos in coupled map lattices,” Phys.Rev.Lett.1998,81(7):1401-1404.
163.Zhou J.Xiang L.and Liu Z.R.“Global synchronization in general complex delayed dynamical networks and its applications,” Phys.A,2007,385:729-742.
164.Liu X.Wang J.Z.and Huang L.“Stabilization of a class of dynamical complex networks based on decentralized control,” Physi.A,2007,383:733-744.
165.Zhou J.Lua J.A.and Lu J.H.“Pinning adaptive synchronization of a general complex dynamical network,”Automatica,2008,44:996-1003.
166.Xulvi-Brunet R.Pietsch W.and Sokolov I.M.“Correlations in scale-free networks:Tomography and percolation,” Phys.Rev.E,2003,68:036119.
167.Rong Z.H.Li X.and Wang X.F.“Roles of mixing patterns in cooperation on a scale-free networked game,” Phys.Rev.E,2007,76:027101.
2.Li T.Y.and Yorke J.A.“Period three implies chaos,”Amer.Math.Mont.1975,82:985-992.
3.Ott E.Grebogi C.and Yorke J.A.“Controlling Chaos,”Phys.Rev.Lett.1990,64:1196-1199.
4.Qammar H.K.and Mossagchi F.“Adaptive Control Chaos,”Phys.Rev.Lett.1993,178:279.
5.Sinha S.and Biswas D.“Adaptive Dynamics on a Chaotic Lattice,”Phys.Rev.Lett.1993,71:2010.
6.Vassiliadis D.and Parametric,“Adaptive Control and Parameter Identification of Lows Dimensional Chaotic System,”Phys.D,1994,71:319.
7.Ni W.S.and Qin T.F.“Controlling Chaos by Method of Adaptive,”Chin.Phys.Lett.1994,11:325.
8.Tian Y.C.and Gao F.R.“Adaptive Control of Chaotic Continuous-Time Systems with Delay,”Phys.D,1998,117:1.
9.Wu C.W.Yang T.and Chua L.O.“On adaptive synchronization and control of nonlinear dynamical systems,”International Journal of Bifurcation and Chaos 1996,6:455-471.
10.Chen M.and Han Z.“Controlling and synchronizing chaotic Genesio system via nonlinear feedback control,”Chaos,Solitons and Fractals,2003,17:709-716.
11.Perora L.M.and Carroll T.L.“Synchronization in chaotic systems,”Phys.Rev.Lett.1990,64:821-825.
12.Boccaletti S.Bragard J.Arecchi F.T.and Mancini H.“Synchronization in nonidentical extended systems,”Phys.Rev.Lett.1999,83:536-539.
13.Boccaletti S.Valladares D.L.Kurths J.Maza D.and Mancini H.“Synchronization of chaotic structurally nonequivalent systems,”Phys.Rev.E,2000,61:3712-3715
14.Boccalettia S.Kurths J.Osipov G.Valladaresb D.L.and Zhou C.S.“The synchronization of chaotic systems,”Phys.Rep.2002,366:1-101.
15.Nijmeijer H.“A dynamical control view on synchronization,”Phys.D,2001,154:219-228.
16.Gonzalez M.Iranda J.M.Bistable.“Generalized synchronization of chaotic systems,” Computer Physics Communications,1999,121:429-431.
17.Maritan A.and Banavar J.“Chaos,noise and synchronization,” Phys.Rev.Lett.1994,72:1451.
18.Zhang H.G.Huang W.Wang Z.L.and Chai T.Y.“Adaptive synchronization between two different chaotic systems with unknown parameters,” Phys.Lett.A,2006,350:363-366.
19.Wu X.Y.Guan Z.H.Wu Z.P.et al,“Chaos synchronization between Chen system and Genesio system,” Phys.Lett.A,2007,364:484-487.
20.Rosenblum M.G.Pikovsky A.S.and Kurths J.“Phase synchronization of chaotic oscillators,” Phys.Rev.Lett.1996,76:1804
21.Rosenblum M.G.Pikovsky A.S.and Kurths J.“From phase to lag synchronization in coupled chaotic oscillators,” Phys.Rev.Lett.1997,78:4193.
22.Kocarev L.and Parlitz U.“From phase to lag synchronization in coupled chaotic oscillators,” Phys.Rev.Lett.1996,76:1816.
23.Mainieri R.and Rehacek J.“Projective Snchronization in Three-demensional Chaotic System,” Phys.Rev.Lett.1999,82:3042.
24.Kolcarev L.and Parlitz U.,“General Approach for Chaotic Synchronization with Applications to Communication,” Phys.Rev.Lett.1995,74(25):5028.
25.Park Ju H.“Chaos synchronization between two different chaotic dynamical systems,” Chaos,Solitons and Fractals,2006,27:549-554.
26.Agiza H.N.and Yassen M.T.“Synchronization of Rossler and Chen chaotic dynamical systems using active control,” Phys.Lett.A,2001,278:191-197.
27.Bai E.W.and Lonngren K.E.“Sequential synchronization of two Lorenz systems using active control,” Chaos,Solitons and Fractals,2000,11:1041-1044.
28.Li Z.Han C.Z.and Shi S.J.“Modification for synchronization of Rossler and Chen chaotic systems,” Phys.Lett.A,2002,301.
29.Li D.M.Wang Z.D.etc.“A note on chaotic synchronization of time-delay secure communication systems,”Chaos,Solitons and Fractals,2007,27:549-554.
30.于娜,丁群等,“异结构系统混沌同步及其在保密通信中的应用,”通信学报,2007,28(10):73-78.
31.Corron N.J.and Hahs D.W.“A new approach to communication using chaotic signals,” IEEE Trans.Circuits and Systems-I,1997,44:373-382.
32.Chua L.O.and Kocarev L.“Transmission of digital signals by chaotic synchronization,” International Journal of Bifurcation and Chaos,1992,2:973-977.
33.Wang X.F.Wang Z.Q.and Chen G.“A new criterion for synchronization of coupled chaotic oscillators with application to Chua's circuits,” International Journal of Bifurcation and Chaos,1999,1169-1174.
34.赵耿,方锦清.“混沌通信分类及其保密通信的研究,”自然杂志,2003,25:21-30.
35.关新平,范正平,陈彩莲,华长春,“混沌控制及其在保密通信中的应用,”国防工业出版社,2002.
36.Haier M.“Synchronization of Chaotic and Transmission of Information,”International Journa of Bifureation and Chaos,1998,8(4):647-659.
37.Milanovic V.and zagloul M.E.“Improved Masking Algorithm for Chaotic Communications systems,” Electronics Letters,1996,32(1):11-12.
38.Yan T.and Chua L.O.“Secure communication via chaotic parameter modulation,” IEEE Transactions on Circuits and Systems I,1996,43:817-819.
39.Fang J.A.and Miao Q.Y.“Chaotic parameter modulation and secure communication,” Proceedings of the 4th World Congress on Intelligent Control and Automation,2002.
40.关新平,范正平,彭海鹏,李丽香,“一种新的多级混沌系统在保密通信中的应用,”燕山大学学报,2001,25(4):287-289.
41.赵耿,方锦清,“现代信息安全与混沌保密通信应用研究与进展,”物理学进展,2003,23(2):212-255.
42.Milanovic V.and zagloul M.E.“Improved Masking Algorithm for Chaotic Communications systems,” Electronics Letters,1996,32(1):11-12.
43.Albert R.and Barab(?)si A.L.“Statistical mechanics of complex networks,” Rev.Mod.Phys.2002,74:47-97.
44.Newman M.“The structure and function of complex networks,” SIAM Rev.2003,45:167-256.
45.Wang X.F.“Complex networks:Topology,dynamics and synchronization,”International Journa of Bifureation and Chaos,2002,12:885-916.
46.Dorogovtsev S.N.and Mendes J.F.“Evolution of networks,” Adv.Phys.2002,51:1079-1187.
47.吴金闪,狄增如,“从统计物理学看复杂网络研究,”物理学进展,2004,24:18-46.
48.周涛,柏文洁,汪秉宏,刘之景,严钢,“复杂网络研究概述,”物理,2005,34:31-36.
49.Watts D.J.and Strogatz S.H.“Collective dynamics of small-world networks,”Nature,1998,393:440-442.
50.Faloutsos M.Faloutsos P.and Faloutsos C.“On power-law relationships of the internet topology,” Computer Communications Review,1999,29:251-262.
51.Liljeros F.Rdling C.R.Amaral L.A.N.Stanley H.E.and Aberg Y.“Random Walks and Scale-Free Networks,” Nature,2001,411:907-908.
52.Ebel H.Mielsch L.I.and Borbholdt S.“Scale-free topology of E-mail networks,”Phys.Rev.E,2002,66,035103.
53.Li W.and Cai X.“Statistical analysis of airport network of China,” Phys.Rev.E,2004,69:046106.
54.Watts D.J.“Networks,dynamics,and the small world phenomenon,” Am.J.Sociol.1999,105:493-592.
55.Watts D.J.“Small Worlds,” Princeton University Press,Princeton,1999.
56.Barab(?)si A.L.and Albert R.“Emergence of scaling in random networks,” Science,1999,286:509-512.
57.Barab(?)si A.L.Albert R.and Jeong H.“Mean-field theory for scale-free random networks,” Physica A,1999,272:173-187.
58.Newman M.“Assortative mixing in networks,” Phys.Rev.Lett.2002,89:208701
59.Xulvi-Brunet R.and Sokolov I.M.“Changing correlations in networks:Assortativity and dissortativity,” Acta Physica Polonica B,2005,36:1431-1455.
60.Xulvi-Brunet R.and Sokolov I.M.“Reshuffling scale-free networks:From random to assortative,” Phys Rev.E.2004,70:066102.
61.Erdos P.and Renyi A.“On Random Graphs,” Publ.Math.(Debrecen),1959,6:290-297.
62.Erdos P.and Renyi A.“On the strength of connectedness of a random graph,”Acta Mathematica Scientia Hungary,1961,12:261-267.
63.Newman M.and Watts D.“Renormalization Group Analysis of the Small-World Network Model,” Phys.Lett.A,1999,263:341-346.
64.McGraw P.N.and Menzinger M.“Clustering and the synchronization of oscillator networks,”Phys.Rev.E,2005,72:015101.
65.Motter A.E.Zhou C.S.and Kurths J.“Enhancing complex-networks synchronization,”Europhys Lett.2005,69:334-340.
66.Motter A.E.Zhou C.S.and Kurths J.“Network synchronization,diffusion and the paradox of heterogeneity,”Phys.Rev.E,2005,71:016116.
67.Wang X.F.“Complex networks:Topology,dynamics and synchronization,”InternationalJournal of Bifurcation and Chaos,2002,12:885-916.
68.Barahona M.and Pecora L.M.“Synchronization in small-world systems,”Phys.Rev.Lett.2002,89.
69.Nishikawa T.Motter A.E.and Lai Y.C.Hoppensteadt F.C.“Heterogeneity in oscillator networks:Are smaller worlds easier to synchronize?”Phys.Rev.Lett.2003,91:014101.
70.Chavez M.Hwang D.U.and Amann A.Hentschel H.G.E.and Boccaletti S.“Synchronization is enhanced in weighted complex networks,”Phys.Rev.Lett.2005,94:218701.
71.Zhao M.Zhou T.Wang B.H.Ou Q.and Ren J.“Better synchronizability predicted by a new coupling method,”European Physical Journal B,2006,53:375-379.
72.Arenas A.Diaz-Guilera A.Kurths J.Moreno Y.and Zhou C.S.“Synchronization in complex networks,”arXiv:cond-mat/0805.2976.
73.Nishikawa T.and Motter A.E.“Synchronization is optimal in nondiagonalizable networks,”Phys Rev.E,2006,73:065106.
74.Pecora L.M.and Carroll T.L.“Master stability functions for synchronized coupled systems,”Phys.Rev.Lett.1998,80:2109-2112.
75.Heagy J.F.Carroll T.L.and Pecora L.M.“Synchronous chaos in coupled oscillator systems,”Phys.Rev.E,1994,50:1874-1885.
76.Gade P.M.Cerdeira H.and Ramaswamy R.“Coupled maps on trees,”Phys.Rev.E,1995,52:2478-2485.
77.Gade P.M.“Synchronization of oscillators with random nonlocal connectivity,”Phys.Rev.E,1996,54:64-70.
78.Hu G.Yang J.and Liu W.Phys.Rev.E,1998,58:4440-4453.
79.Fink K.S.Johnson G.Carroll T.Mar D.and Pecora L.“Three coupled oscillators as a universal probe of synchronization stability in coupled oscillator arrays,”Phys.Rev.E,2000,61:5080-5090.
80.Jost J.and Joy M.P.Phys.Rev.E,2001,65:016201.
81.Wang X.F.and Chen G.“Synchronization in scale-free dynamical networks:Robustness and fragility,” IEEE Transactions on Circuits and Systems I,2002,49:54-62.
82.L(u|¨) J.and Chen G.,“A time-varying complex dynamical network model and its controlled synchronization criteria,” IEEE Transactions on Automatic Control,2005,50:841-846.
83.L(u|¨) J.Chen G.Cheng D.and Celikovsky S.“Bridge the gap between the Lorenz system and the Chen system,” International Journal of Bifurcation and Chaos,2002,12:2917-2926.
84.L(u|¨) J.Han F.Yu X.and Chen G.“Generating 3-D multi-scroll chaotic attractors:A hysteresis series switching method,” Automatica,2004,40(10):1677-1687.
85.L(u|¨) J.Yu X.and Chen G.“Chaos synchronization of general complex dynamical networks,” Phys.A,2004,334:281-302.
86.Barahona M.and Pecora L.M.“Synchronization in small-world systems,” Phys.Rev.Lett.2004,89:054101.
87.Chen Y.Rangarajan G.and Ding M.“General stability analysis of synchronized dynamics in coupled systems,” Phys.Rev.E,2003,67:026209.
88.Wu C.W.and Chua L.O.“Synchronization in an array of linearly coupled dynamical systems,” IEEE Transactions on Circuits and Systems I,1995,42:430-447.
89.Wu C.W.“Synchronization in networks of nonlinear dynamical systems coupled via a directed graph,” Nonlinearity,2005,18(3):1057.
90.Wu C.W.“On Rayleigh-Ritz ratios of a generalized Laplacian matrix of directed graphs,” Linear Algebra and its Applications,2005,402:207.
91.Wu C.W.“Synchronization and convergence of linear dynamics in random directed networks,” IEEE Transactions on Automatic Control,2006,51(7):1207-1210.
92.Zhou J.Lu J.A.and L(u|¨) J.“Adaptive synchronization of an uncertain complex dynamical network,” IEEE Transactions on Automatic Control,2006,51(4):652-656.
93.Sorrentino F.Bernardo M.D.Cuellar G.H.and Boccaletti S.“Synchronization in weighted scale-free networks with degree-degree correlation,” Physica D,2006,224:123-129
94.Bernardo M.D.Garofalo F.and Sorrentino F.“Effects of degree correlation on the synchronization of networks of oscillators,” International Journal of Bifurcation and Chaos,2007,17:3499-3506
95.汪小帆,李翔,陈关荣,复杂网络理论及其应用,清华大学出版社,2006.
96.Wang X.F.and Chen G.“Pinning control of scale-free dynamical networks,”Phys.A,2002,310:521-531.
97.Xiang L.Y.Liu Z.X.Chen Z.Q.Chen F.and Yuan Z.Z.“Pinning control of complex dynamical networks with general topology,” Phys.A,2007,379:298-306.
98.Parekha N.and Sinhab S.“Controllability of spatiotemporal systems using constant pinning,” Phys.A,2003,318:200-212.
99.Wu C.W.“Localization of effective pinning control in complex networks of dynamical systems,” ISCAS 2008.
100.Hu G.and Qu Z.L.“Controlling spatiotemporal chaos in coupled map lattice systems,” Phys.Rev.Lett.1994,72:68-71.
101.Li X.Wang X.F.and Chen G.“Pinning a complex dynamical network to its equilibrium,” IEEE Transactions on Circuits and Systems I,2004,51:2074-2087.
102.Xiang L.Y.Chen Z.Q.Liu Z.X.Chen F and Yuan Z.Z.“Stabilizing weighted complex networks,” J.Phys.A:Math.Theor.2007,40:14369-14382.
103.Xiang L.Y.Liu Z.X.Chen Z.Q.and Yuan Z.Z.“Pinning weighted complex networks with heterogeneous delays by a small number of feedback controllers,”Science in China Series F:Information Sciences,2008,51:511-523.
104.Sorrentino F.Bernardo M.D.Garofalo F.and Chen G.“Controllability of complex networks via pinning,” Phys Rev.E,2007,75:046103.
105.Sorrentino F.“Effects of the network structural properties on its controllability,”Chaos,2007,17:033101.
106.Chen T.P.Liu X.W.and Lu W.L.“Pinning complex networks by a single controller,” IEEE Transactions on Circuits and Systems I,2007,54:1317-1326.
107.Li Z.,Feng G..and David H.“Controlling complex dynamical networks with coupling delays to a desired orbit,” Phys.Lett.A,2006,359:42-46
108.Wu X.F.Chen G.and Cai J.P.“Chaos synchronization of the master-slave generalized Lorenz systems via linear state error feedback control,” Phys.D,2007,229:52-80.
109.Yu W.W.and Cao J.D.“Adaptive synchronization and lag synchronization of uncertain dynamical system with time delay based on parameter identification,” Phys.A,2007,375:467-482.
110.Jia Q.“Projective synchronization of a new hyperchaotic Lorenz system,”Phys.Lett.A,2007,370:40-45.
111.Li Rui-hong,Xu Wei and Li Shuang,“Adaptive generalized projective synchronization in different chaotic systems based on parameter identification,”Phys.Lett.A,2007,367:199-206.
112.Jia Zhen,“Linear generalized synchronization of chaotic systems with uncertain parameters,”Journal of Systems Engineering and Electronics,2008,19(4):779-784
113.Li R.H.,Xu W.and Li S.,“Adaptive generalized projective synchronization in different chaotic systems based on parameter identification,”Phys.Lett.A,2007,367:199-206.
114.Li G.H.,Zhou S.P.and Yang K.,“Generalized projective synchronization between two different chaotic systems using active back stepping control,”Phys.Lett.A,2006,355:326-330.
115.Tang Y.and Fang J.A.“General methods for modified projective synchronization of hyperchaotic systems with known or unknown parameters,”Phys.Lett.A,2008,372:1816-1826.
116.Hu A.H.and Xu Z.Y.“Stochastic linear generalized synchronization of chaotic systems via robust control,”Phys.Lett.A,2008,372:3814-3818.
117.Luo R.Z.“Adaptive function project synchronization of RCssler hyperchaotic system with uncertain parameters,”Phys.Lett.A,2008,372:3667-3671.
118.Wang B.Wen G.J.and Xie K.“On the synchronization of a hyperchaotic system based on adaptive method,”Phys.Lett.A,2008,372:3015-3020.
119.Hu M.F.Xu Z.Y.and Zhang R.“Full state hybrid projective synchronization in continuous-time chaotic(hyperchaotic)systems,”Nonlinear Sci.Numer.Simul.2008,13:456-464.
120.Li G.H.“Modified projective synchronization of chaotic system,”Chaos,Solitons and Fractals,2007,32:1786-1790.
121.Hu M.F.Xu Z.Y.and Zhang R.“Adaptive full state hybrid projective synchronization of chaotic systems with the same and different order,”Phys.Lett.A,2007,365:315.
122.Hu M.F.Xu Z.Y.and Zhang R.“Parameters identification and adaptive full state hybrid projective synchronization of chaotic(hyper-chaotic)systems,”Phys.Lett.4 2007,361:23
123.Cao J.D.and Lu J.Q.“Adaptive complete synchronization of two identical or different chaotic(hyperchaotic)systems with fully unknown parameters,”Chaos,2005,15:043901.
124.Femat R.and SlisPerales G.“Synchronization of chaotic systems with different order,”Phys Rev.E,2002,65:036226.
125.Bowong S.and McClintock P.V.E.“Adaptive synchronization between chaotic dynamical systems of different order,”Phys.Lett.A,2006,358:134-141.
126.Ho M.C.Hung Y.C.Liu Z.Y.and Jiang I.M.“Reduced-order synchronization of chaotic systems with parameters unknown,”Phys.Lett.A,2006,348:251.
127.Moukam F.M.Bowong S.and Tchawoua C.“Reduced-order synchronization of uncertain chaotic systems via adaptive control,”Nonlinear Sci.Numer.Simul.2006,11:810.
128.Lazzouni S.A.Bowong S.Moukam F.M.and Cherki B.“An adaptive feedback control for chaos synchronization of nonlinear systems with different order,”Phys.Lett.A,2007,12:568.
129.Terman D.Koppel N.and Bose A.“Dynamics of two mutually coupled slow inhibitory neurons,”Phys.D,1998,117:241-275.
130.Gao T.G.Chen Z.Q.and Yuan Z.Z.Chin.Phys.2005,14:2421.
131.Liu C.X.“A novel chaotic attractor,”Chaos,Solitons and Fractals,in press.
132.Wang Z.D.Lauria S.Fang J.A.and Liu X.H.“Exponential stability of uncertain stochastic neural networks with mixed time-delays,”Chaos,Solitons and Fractals,2007,32:62-72.
133.Wang Z.D.Shu H.S.Fang J.A.and Liu X.H.“Robust stability for stochastic Hopfield neural networks with time delays,”Nonlinear Anal.2006,7:1119-1128.
134.Tang X.H.Lu J.A.and Zhang W.W.“The function projective synchronization of chaotic system using backstepping design,”Journal of dynamics and control,2007,5(3):216-219.
135.Wang Z.D.Liu Y.R.Fraser K.and Liu X.H.“Stochastic stability of uncertain Hopfield neural networks with discrete and distributed delays,”Phys.Lett.A,2006,354:288-297.
136.Lin W.and He Y.“Complete synchronization of the noise-perturbed Chua's circuits,”Chaos,2005,15:023705.
137.Sun Y.H.and Cao J.“Adaptive lag-synchronization of unknown chaotic delayed neural networks with noise perturbation,”Phys.Lett.A,2007,364:277-285.
138.Yu W.W.and Cao J.D.“Synchronization control of stochastic delayed neural networks,” Phys.A,2007,373:252-260.
139.Zhang C.Lin W.and Zhou J.“Complete and generalized synchronization in a class of noise perturbed chaotic systems,” Chaos,2007,17:023106.
140.Friedman,“Stochastic Differential Equations and Applications,” Academic Press,New York,1976.
141.Mianovic V.and Zaghloul M.E.“Improved masking algorithm chaotic communications systems,” Electronic Lett.1996,1:11-12.
142.Wc C.W.and Chua L.O.“A simple way to synchronize chaotic systems with applications to secure communication systems,” International Journal of Bifurcation and Chaos,1993,3:1619-1627.
143.Short K.M.“Steps toward unmasking secure communications,” International Journal of Bifurcation and Chaos,1994,4:957-977.
144.Diffie W.and Hellman M.F.“New directions in cryptograghy,” IEEE Transactions in Information Theory,1976,644-654,
145.Xulvi-Brunet R.Pietsch W.and Sokolov I.M.“Correlations in scale-free networks:Tomography and percolation,” Phys.Rev.E,2003,68:036119.
146.Yang T.and Chua L.O.“Impulsive stabilition for control and synchronization of chaotic system:Theory and application to secure communications,” IEEE Transactions on Circuits and Systems I,1997,44:976-988.
147.Celikovsky S.and Chen G.“Secure synchronization of a class of chaotic systems from a nonlinear observer approach,” IEEE Transactions on Automatic Control 2005,50:76-82.
148.Liao T.L.and Huang N.S.“An observer-based approach for chaotic synchronization with application to secure communication,” IEEE Transactions on Circuits and Systems I,1999,46:1144-1150.
149.Kaulakys B.ivanauskas F.and meskauskas T.“Synchronization of chaotic systems driven by identical noise,” Int J Bifurcation and Chaos,1997,9:533-539.
150.Boccaletti S.Latora V.Moreno Y.Chavez M.and Hwang D.U.“Complex networks:Structure and dynamics,” Phys.Rep.2006,424:175-308.
151.Hong H.Kim B.J.Choi M.Y.and Park H.“Factors that predict better synchronizability on complex networks,” Phys Rev.E,2004,69:067105.
152.Zhou C.and Kurths J.“Dynamical weights and enhanced synchronization in adaptive complex networks,” Phys.Rev.Lett.2006,96:164102.
153.Zhao M.Zhou T.Wang B.H.and Wang W.X.“Enhance synchronizability by structural perturbations,” Phy.Rev.E,2005,72:057102.
154.Zhou T.Zhao M.and Wang B.H.“Better synchronizability predicted by crossed double cycle,” Phy.Rev.E,2006,73:037101.
155.赵明,汪秉宏,蒋品群,周涛,“复杂网络上动力系统同步的研究进展,”物理学进展,2005,25:273-295.
156.Zhao M.Zhou T.Wang B.H.Yan G.Yang H.J.and Bai W.J.“Relations between average distance,heterogeneity and network synchronizability,” Phys.A,2006,371:773-780.
157.Zhao M.Zhou T.Wang B.H.Ou Q.and Ren J.“Better synchronizbility predicted by a new coupling method,” Eur.Phys.J.B.2006,53:375.
158.Wu X.Wang B.H.Zhou T.Wang W.X.Zhao M.and Yang H.J.“Synchronizability of highly clustered scale-free networks,” Chin.Phys.Lett.2006,23:1046.
159.Gu Z.M.Zhao M.Zhou T.Zhu C.P.and Wang B.H.“Phase Synchronization of non-Abelian oscillators on small-world networks,”Phys.Lett.A,2007,362:115.
160.Zhou T.Zhao M.Chen G.Yan G.and Wang B.H.“Phase synchronization on scale-free networks with community structure,”Phys.lett.A,2007,368:431-434.
161.Pandit S.A.and Amritkar R.E.”Characterization and control of small-world networks,” Physical Rev.E,1999,60(2):1119-1122.
162.Parekh N.Parthasarathy S.and Sinha S.“Global and local control of spatiotemporal chaos in coupled map lattices,” Phys.Rev.Lett.1998,81(7):1401-1404.
163.Zhou J.Xiang L.and Liu Z.R.“Global synchronization in general complex delayed dynamical networks and its applications,” Phys.A,2007,385:729-742.
164.Liu X.Wang J.Z.and Huang L.“Stabilization of a class of dynamical complex networks based on decentralized control,” Physi.A,2007,383:733-744.
165.Zhou J.Lua J.A.and Lu J.H.“Pinning adaptive synchronization of a general complex dynamical network,”Automatica,2008,44:996-1003.
166.Xulvi-Brunet R.Pietsch W.and Sokolov I.M.“Correlations in scale-free networks:Tomography and percolation,” Phys.Rev.E,2003,68:036119.
167.Rong Z.H.Li X.and Wang X.F.“Roles of mixing patterns in cooperation on a scale-free networked game,” Phys.Rev.E,2007,76:027101.
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