混沌投影同步研究及其应用
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摘要
由于其在不同科学领域,特别是在混沌保密通信领域的广泛潜在应用,混沌同步吸引了越来越多的科学家和工程技术人员的广泛重视。本文主要研究和混沌投影同步相关的一些问题,包括控制方法,在混沌保密通信系统、系统参数辨识等方面的应用,主要工作和成果如下:
简单介绍了混沌的四个发展阶段,描述了常用的混沌同步方法,简要说明了混沌同步技术在混沌保密通信中的关键性。
线性自适应反馈控制技术用来实现预期投影因子的投影同步。自适应反馈控制技术首次用来控制具有部分线性结构的耦合混沌系统的投影因子,从而达到控制投影同步最终状态的目的。应用这种控制方法实现的投影同步技术来设计混沌保密通信系统,对所提方案作了理论上的严格证明。
脉冲控制技术用来实现两个混沌系统的完全同步和投影同步。首先,应用已有的脉冲微分方程理论结果,对脉冲控制系统进行分析和设计,提出了一个新的混沌同步方法并应用到Lorenz混沌系统。然后,应用脉冲微分方程理论里的比较系统方法,首次用来研究部分线性系统的投影同步问题。
提出了驱动-响应复杂动力学网络模型,并研究了其上的网络投影同步。理论上给出了实现网络投影同步的充分条件,并且应用牵制控制方法实现对网络投影同步的控制。
提出了网络投影聚类同步的概念。在驱动-响应复杂动力学网络模型里研究了实现网络投影聚类同步的充分条件,并且应用牵制控制方法实现对网络投影聚类同步的控制。
提出了全状态混合投影同步的概念。利用Lyapunov第二方法,理论上给出了连续型混沌系统之间的全状态混合投影同步实现的判别条件、离散混沌映射之间的全状态混合投影同步控制器设计的一般形式。值得一提的是全状态混合投影同步的概念包含了完全同步、反同步,全状态投影同步,部分同步,错位同步作为其特例。
应用全状态混合投影同步方法辨识混沌系统未知参数。理论上给出了驱动系统参数未知情况下两个混沌系统实现全状态混合投影同步的自适应控制器,该控制器同时保证可以正确辨识出未知参数,并具有一定的抗噪能力。
研究了两个混沌系统参数皆未知的情况下全状态混合投影同步问题。两个混沌系统的维数可以相同,也可以不同。对于不同维数的两个混沌系统,提出了降阶全状态混合投影同步、增阶全状态混合投影同步的概念。理论上给出了一般的自适应控制器的表达式。
数值研究证实了混合投影同步的存在性。针对一个具体的具有部分线性结构的混沌系统,数值实验发现了混合投影同步现象是存在的,从而支持了全状态混合投影同步概念的合理性。
研究了更一般的Q-S同步问题。利用矩阵的广义逆理论和Lyapunov稳定性理论,给出了实现更一般的Q-S混沌同步的控制器设计方法,所涉及到的驱动-响应系统可以具有不同的结构,甚至可以具有不同的维数。该方法的特点在于可以实现事先给定的系统状态可观测变量之间的同步。
以上所有的理论方法都做了相应的数值实验,给出了相应的数值结果,验证了理论方法的正确性和有效性。
Due to its wide-scope potential applications in various scientific fields, especially in chaotic secure communication, the problem of chaos synchronization has attracted increased attention from scientists and engineers. This dissertation is devoted to study on the related problem of projective synchronization, including the control scheme and some application in chaotic security communication and parameters identification. The author's main research work and contributions are as follows:
In this dissertation, the four stages of development in chaos theory are briefly introduced, some existing chaos control approaches are surveyed and the key role of chaos synchronization played in chaotic communication scheme is described.
Linear adaptive feedback controller is first designed to direct the scaling factor, characterized the synchronized dynamics of projective synchronization in partially linear chaotic systems, onto a desired value. Based on the adaptive projective synchronization, a secure communication scheme is proposed and proved rigorously.
Impulsive control approach is adopted to realize the chaossynchronization and projective synchronization, respectively. First, we derive some conditions for (asymptotic) stability of impulsive control systems with impulses at fixed times based on a existing theory result of impulsive differential equations, and the results are used to design impulsive synchronization for Lorenz systems. Then, we first use the impulsive control approach to control the scaling factor of projective synchronization onto any desired scale.
A drive-response dynamical networks model is presented and its projective synchronization is investigated. The sufficient condition of projective synchronization in drive-response network is derived. Because the scaling factor is difficult to be estimated, pinning control techniques are used to direct the scaling factor onto the desired value.
The definition of projective cluster synchronization in drive-response dynamical networks is introduced and the sufficient condition on this novel synchronization scheme is derived. For the same reason, we use pinning control methods to control the scaling factors onto the desired values.
The definition of full state hybrid projective synchronization is first presented. Based on the Lyapunov stability theory, a general controller is designed for full state hybrid projective synchronization of continuous-time chaotic systems and discrete-time chaotic maps, respectively. It is worthy noting that our full state hybrid projective synchronization includes complete synchronization, anti-synchronization, full state projective synchronization, partial synchronization and mismatched synchronization as special cases.
Estimating the model parameters of the drive system based on the full state hybrid projective synchronization scheme is presented. The adaptive controller is designed to realize the full state projective synchronization and to identify the unknown parameters, simultaneously. In addition, the proposed adaptive controller is quite robust against the effect of noise.
The full state hybrid projective synchronization of chaotic and hyper-chaotic systems with fully unknown parameters is further investigated. A unified adaptive controller and parameters update law is designed for achieving full state hybrid projective synchronization between two chaotic systems with the same and different order based on the Lyapunov stability theory. Especially, for the case of two chaotic systems with different order, two new concepts, reduced order full state hybrid projective synchronization and increased order full state hybrid projective synchronization, are also studied.
The existence of hybrid projective synchronization in a five dimensional chaotic systems is experimentally found, which shows that the definition of full state hybrid projective synchronization presented before is reasonable.
The general problem of Q-S synchronization between chaotic and/or hyper-chaotic systems is studied. We propose a general controller for Q-S synchronization of chaotic and/or hyper-chaotic systems based on the Lyapunov stability theorem and the theory of generalized inverse matrix. The drive and response systems studied in this part can be strictly different dynamical systems (including different dimensional systems). In addition, the proposed controller can realize the Q-S synchronization with respect to the freely selected observable variables Qi and S_i.
For each proposed theory result, we give numerical simulations to illustrate the effectiveness in this dissertation.
简单介绍了混沌的四个发展阶段,描述了常用的混沌同步方法,简要说明了混沌同步技术在混沌保密通信中的关键性。
线性自适应反馈控制技术用来实现预期投影因子的投影同步。自适应反馈控制技术首次用来控制具有部分线性结构的耦合混沌系统的投影因子,从而达到控制投影同步最终状态的目的。应用这种控制方法实现的投影同步技术来设计混沌保密通信系统,对所提方案作了理论上的严格证明。
脉冲控制技术用来实现两个混沌系统的完全同步和投影同步。首先,应用已有的脉冲微分方程理论结果,对脉冲控制系统进行分析和设计,提出了一个新的混沌同步方法并应用到Lorenz混沌系统。然后,应用脉冲微分方程理论里的比较系统方法,首次用来研究部分线性系统的投影同步问题。
提出了驱动-响应复杂动力学网络模型,并研究了其上的网络投影同步。理论上给出了实现网络投影同步的充分条件,并且应用牵制控制方法实现对网络投影同步的控制。
提出了网络投影聚类同步的概念。在驱动-响应复杂动力学网络模型里研究了实现网络投影聚类同步的充分条件,并且应用牵制控制方法实现对网络投影聚类同步的控制。
提出了全状态混合投影同步的概念。利用Lyapunov第二方法,理论上给出了连续型混沌系统之间的全状态混合投影同步实现的判别条件、离散混沌映射之间的全状态混合投影同步控制器设计的一般形式。值得一提的是全状态混合投影同步的概念包含了完全同步、反同步,全状态投影同步,部分同步,错位同步作为其特例。
应用全状态混合投影同步方法辨识混沌系统未知参数。理论上给出了驱动系统参数未知情况下两个混沌系统实现全状态混合投影同步的自适应控制器,该控制器同时保证可以正确辨识出未知参数,并具有一定的抗噪能力。
研究了两个混沌系统参数皆未知的情况下全状态混合投影同步问题。两个混沌系统的维数可以相同,也可以不同。对于不同维数的两个混沌系统,提出了降阶全状态混合投影同步、增阶全状态混合投影同步的概念。理论上给出了一般的自适应控制器的表达式。
数值研究证实了混合投影同步的存在性。针对一个具体的具有部分线性结构的混沌系统,数值实验发现了混合投影同步现象是存在的,从而支持了全状态混合投影同步概念的合理性。
研究了更一般的Q-S同步问题。利用矩阵的广义逆理论和Lyapunov稳定性理论,给出了实现更一般的Q-S混沌同步的控制器设计方法,所涉及到的驱动-响应系统可以具有不同的结构,甚至可以具有不同的维数。该方法的特点在于可以实现事先给定的系统状态可观测变量之间的同步。
以上所有的理论方法都做了相应的数值实验,给出了相应的数值结果,验证了理论方法的正确性和有效性。
Due to its wide-scope potential applications in various scientific fields, especially in chaotic secure communication, the problem of chaos synchronization has attracted increased attention from scientists and engineers. This dissertation is devoted to study on the related problem of projective synchronization, including the control scheme and some application in chaotic security communication and parameters identification. The author's main research work and contributions are as follows:
In this dissertation, the four stages of development in chaos theory are briefly introduced, some existing chaos control approaches are surveyed and the key role of chaos synchronization played in chaotic communication scheme is described.
Linear adaptive feedback controller is first designed to direct the scaling factor, characterized the synchronized dynamics of projective synchronization in partially linear chaotic systems, onto a desired value. Based on the adaptive projective synchronization, a secure communication scheme is proposed and proved rigorously.
Impulsive control approach is adopted to realize the chaossynchronization and projective synchronization, respectively. First, we derive some conditions for (asymptotic) stability of impulsive control systems with impulses at fixed times based on a existing theory result of impulsive differential equations, and the results are used to design impulsive synchronization for Lorenz systems. Then, we first use the impulsive control approach to control the scaling factor of projective synchronization onto any desired scale.
A drive-response dynamical networks model is presented and its projective synchronization is investigated. The sufficient condition of projective synchronization in drive-response network is derived. Because the scaling factor is difficult to be estimated, pinning control techniques are used to direct the scaling factor onto the desired value.
The definition of projective cluster synchronization in drive-response dynamical networks is introduced and the sufficient condition on this novel synchronization scheme is derived. For the same reason, we use pinning control methods to control the scaling factors onto the desired values.
The definition of full state hybrid projective synchronization is first presented. Based on the Lyapunov stability theory, a general controller is designed for full state hybrid projective synchronization of continuous-time chaotic systems and discrete-time chaotic maps, respectively. It is worthy noting that our full state hybrid projective synchronization includes complete synchronization, anti-synchronization, full state projective synchronization, partial synchronization and mismatched synchronization as special cases.
Estimating the model parameters of the drive system based on the full state hybrid projective synchronization scheme is presented. The adaptive controller is designed to realize the full state projective synchronization and to identify the unknown parameters, simultaneously. In addition, the proposed adaptive controller is quite robust against the effect of noise.
The full state hybrid projective synchronization of chaotic and hyper-chaotic systems with fully unknown parameters is further investigated. A unified adaptive controller and parameters update law is designed for achieving full state hybrid projective synchronization between two chaotic systems with the same and different order based on the Lyapunov stability theory. Especially, for the case of two chaotic systems with different order, two new concepts, reduced order full state hybrid projective synchronization and increased order full state hybrid projective synchronization, are also studied.
The existence of hybrid projective synchronization in a five dimensional chaotic systems is experimentally found, which shows that the definition of full state hybrid projective synchronization presented before is reasonable.
The general problem of Q-S synchronization between chaotic and/or hyper-chaotic systems is studied. We propose a general controller for Q-S synchronization of chaotic and/or hyper-chaotic systems based on the Lyapunov stability theorem and the theory of generalized inverse matrix. The drive and response systems studied in this part can be strictly different dynamical systems (including different dimensional systems). In addition, the proposed controller can realize the Q-S synchronization with respect to the freely selected observable variables Qi and S_i.
For each proposed theory result, we give numerical simulations to illustrate the effectiveness in this dissertation.
引文
[1] Lorenz E.N. Deterministic non-periodic flows. J.Atmos.Sci, 1963,20:130-141
[2] Li T Y, Yorke J A. Period three implies chaos. Amer.Math.Monthly, 1975, 82:985-992
[3]陈关荣,汪小帆.动力系统的混沌化——理论、方法与应用[M].上海:上海交通大学出版社,2006.1-9
[4]闫振亚.复杂非线性波的构造性理论及其应用[M].北京:科学出版社,2007.14-21,233-302
[5] Devaney R L. An Introduction to Chaotic Dynamical Systems. 2nd edition. New York: Addison-Wesley, 2003.
[6] Ott E, Grebogi C, Yorke J A. Controlling chaos. Phys.Rev.Lett, 1990,64:1196-1199
[7] Boccaletti S, Grebogi C, Lai Y-C et al. The control of chaos: theory and application. Phys.Rep, 2000,329:103-197
[8]邹恩,李祥飞,陈建国.混沌控制及其优化应用[M].长沙:国防科技大学出版社,2002
[9]黄润生,黄浩.混沌及其应用[M].第二版.武汉:武汉大学出版社,2005
[10] Schiff S J, Jerger K, Duong D H, et al. Controlling chaos in the brain. Nature, 1994, 363: 411-417
[11]刘秉正,彭建华.非线性动力学[M].北京:高等教育出版社,2004
[12] Huygenii G. Horoloquim oscilatorium. France: Parisiis,1673.
[13]郑志刚.耦合非线性系统的时空动力学与合作行为[M].北京:高等教育出版,2004
[14] Rayleigh J. The theory of sound. New York: Dover Publishers,1945
[15] Winfree A T. Geometry of biological time. New York: Springer-Verlag, 1990
[16] Glass L, Mackay M C.From clocks to chaos: the rhythms of life. Princeton NJ: Princeton University Press, 1988
[17] Pecora L M, Carroll T L. Synchronization in chaotic systems. Phys. Rev. Lett 1990,64:821-825
[18] Lu J Q, Cao J D. Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters. Chaos, 2005, 15:043901
[19] Zhan M, Wang X G, Gong X F et al. Complete synchronization and generalizedsynchronization of one-way coupled time-delay systems.Phys.Rev.E, 2003,68:036208
[20] Rosenblum M G, Pikovsky A S, Kurths J. Phase Synchronization of Chaotic Oscillators. Phys. Rev. Lett.1996, 76:1804-1807
[21] Blasius B, Huppert A, Stone L. Complex dynamics and phase synchronization in spatially extended ecological systems. Nature. 1999, 399(6734): 354-359
[22] Parlitz U, Junge L, Lauterborn W. Experimental observation of phase synchronization. Phys. Rev. E. 1996,54:2115-2117
[23] Rulkov N F, Sushchik M M, Tsimring L S et al. Generalized synchronization of chaos in directionally coupled chaotic systems. Phys. Rev. E. 1995,51:980-994
[24] Kocarev L, Parlitz U. Generalized Synchronization, Predictability, and Equivalence of Unidirectionally Coupled Dynamical Systems. Phys. Rev. Lett. 1996,76:1816 - 1819
[25] Abarbanel H D I, Rulkov N F, Sushchik M M. Generalized synchronization of chaos: The auxiliary system approach. Phys. Rev. E.1996, 53:4528-4535
[26] Mainieri R, Rehacek J. Projective synchronization in three-dimensional chaotic systems. Phys. Rev. Lett 1999,82:3042-3045
[27] Wen G L, Xu D. Nonlinear observer control for full-state projective synchronization in chaotic continuous-time systems, Chaos, Solitons and Fractals, 2005,26:71-77
[28] Hu M F, Xu Z Y, Zhang R. Full state hybrid projective synchronization in continuous-time chaotic (hyperchaotic) systems. Communications in Nonlinear Science and Numerical Simulation.2008,13(2):456-464
[29] Hu M F, Xu Z Y, Zhang R. Full state hybrid projective synchronization of a general class of chaotic maps. Communications in Nonlinear Science and Numerical Simulation. (In press)
[30] Hu M F, Xu Z Y, Zhang R et al. Parameters identification and adaptive full state hybrid projective synchronization of chaotic (hyper-chaotic) systems. Phys Lett.A.2007, 361: 231-237
[31] Hu M F, Xu Z Y, Zhang R et al. Adaptive full state hybrid projective synchronization of chaotic systems with the same and different order. Phys Lett.A.2007, 365: 315-327
[32] Yan Z Y. Q-S (lag or anticipated) synchronization backstepping scheme in a class of continuous-time hyperchaotic systems—A symbolic-numeric computation approach. Chaos, 2005,15: 023902
[33] Yan Z Y. Chaos Q-S synchronization between R?ssler system and a new unified chaotic system. Phys. Lett. A, 2005,334:406-412
[34] Shuai J W, Wong K W, Cheng L M. Synchronization of spatiotemporal chaos with positive conditional Lyapunov exponents. Phys.Rev.E.1997, 56:2272-2275
[35] Zhou C, Lai C H. Analysis of spurious synchronization with positive conditional Lyapunov exponents in computer simulations. Physica D.2000, 135:1-23
[36] Guan S., Li K., Lai C.-H. Chaotic synchronization through coupling strategies. Chaos, 2007,16: 023107
[37]汪小帆,李翔,陈关荣.复杂网络理论及其应用[M].北京:清华大学出版社,2006
[38] Strogatz S H. Exploring complex networks. Nature. 2001,410:268-276
[39] Tutte W T. Graph theory [M].北京:机械工业出版社,2004
[40] Bollobas B. Random Graphs. 2nd edition. New York: Academic Press, 2001
[41] Watts D J, Strogatz S H. Collective dynamics of“small-world”networks. Nature. 1998, 393:440-442
[42] Barabasi A L, Albert R. Emergence of scaling in random networks. Science. 1999, 286: 509-512
[43] Neda Z, Ravasz E, Vicsek T, et al. The sound of many hands clapping. Nature, 2000,403: 849-850
[44] Kaneko k. Coupled map lattices. Singapore: World Scientific, 1992
[45] Chao L O. CNN: A paradigm for complexity. Singapore: World Scientific, 1993
[46] Motter A E, Zhou C S, Kurths J. Network synchronization, diffusion, and the paradox of heterogeneity. Phys. Rev. E .2005,71:016116
[47] Motter A E, Zhou C S, Kurths J. Enhancing complex-network synchronization. Europhys. Lett.2005, 69:334-340
[48] Zhou C S, Motter A E, Kurths J. Universality in the Synchronization of Weighted Random Networks. Phys. Rev. Lett. 2006,96:034101
[49] Zhao M, Zhou T, Wang B H et al. Enhanced synchronizability by structural perturbations. Phys. Rev. E. 2005,72:057102
[50] Komiss G. Synchronization in weighted uncorrelated complex networks in a noisy environment: Optimization and connections with transport efficiency. Phys. Rev. E. 2007,75:051121
[51] Chavez M, Hwang D.-U, Amann A et al. Synchronizing weighted complex networks. Chaos. 2006,16: 015106
[52] Barahona M, Pecora L M. Synchronization in small-world systems. Phys. Rev. Lett. 2002,89:054101
[53] Wang X F, Chen G R. Synchronization in small-world dynamical networks. International Journal of Bifurcation and Chaos.2002, 12: 187-192
[54] Li C G, Chen G R. Phase synchronization in small-world networks of chaotic oscillators. Physica A.2004, 341:73-79
[55] Pecora L M, Carroll T L. Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 1998,80:2109-2112
[56] Chen G R, Ueta T. Yet another chaotic attractor. International Journal of Bifurcation and Chaos.1999, 9(7): 1465-1466
[57] LüJ H, Chen G R, Chen D Z et al. Bridge the gap between the Lorenz system and the Chen system. International Journal of Bifurcation and Chaos.2002, 12:2917-2926
[58] LüJ H, Chen G R. A new chaotic attractor coined. International Journal of Bifurcation and Chaos.2002, 12: 659-661
[59] Genesio R, Tesi A. Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica. 1992, 28 : 531-548
[60] R?ssler O E. An equation for continuous chaos. Phys.Lett.A.1976, 57:397-398
[61] Stenflo L. Generalized Lorenz equations for acoustic-gravity waves in the atmosphere. Phys. Scr. 1996, 53: 83-84
[62] Li Y, Tang WKS, Chen G. Generating hyperchaos via state feedback control. International Journal of Bifurcation and Chaos. 2005, 10:3367-3375
[63] Chen A , Lu J , LǚJ , Yu S. Generating hyperchaotic Lǚattractor via state feedback control. Phys.A. 2006, 364:103-110
[64] Stefanski K. Modelling chaos and hyperchaos with 3D maps. Chaos, Solitons and Fractals.1998, 9:83-93
[65] Grassi G, Miller D A. Theory and experimental realization of observer-based discrete-time hyperchaos synchronization. IEEE Trans.Circ.Syst.I.2002, 49:373-378
[66] Murali K,Lakshmanan M. Drive-response scenario of chaos synchronization in identical nonlinear systems. Phys. Rev. E. 1994,49:4882 - 4885
[67] Yassen M T. Controlling chaos and synchronization for new chaotic system using linear feedback control. Chaos, Solitons and Fractals.2005, 26(3): 913-920
[68] Wang F Q, Liu C X. A new criterion for chaos and hyperchaos synchronization using linear feedback control. Phys.Lett.A.2006, 360:274-278
[69]王发强,刘崇新. Liu混沌系统的线性反馈同步控制及电路实验的研究[J].物理学报,2006,55(10): 5055-5060
[70] Wu X F, Chen G, Cai J P. Chaos synchronization of the master–slave generalized Lorenz systems via linear state error feedback control. Physica D.2007, 229:52-80
[71] Lu J A, Wu X Q, Han X P et al. Adaptive feedback synchronization of a unified chaotic system. Phys.Lett.A.2004, 329:327-333
[72] Lei Y M, Xu W, Shen J W. Robust synchronization of chaotic non-autonomous systems using adaptive-feedback control. Chaos, Solitons and Fractal, 2007, 31:371-379
[73] Yassen M T. Feedback and adaptive synchronization of chaotic Lüsystem. Chaos, Solitons and Fractal, 2005, 25:379-386
[74] Tao C H, Liu X F. Feedback and adaptive control and synchronization of a set of chaotic and hyperchaotic systems. Chaos, Solitons and Fractal, 2007,32:1572-1581
[75] Yan Z Y, Yu P. Linear feedback control, adaptive feedback control and their combination for chaos (lag) synchronization of LC chaotic systems. Chaos, Solitons and Fractal, 2007, 33: 419-435
[76] Ali M K, Fang J Q. Synchronization of chaos and hyperchaos using linear and nonlinear feedback functions. Phys. Rev. E.1997, 55:5285-5290
[77]陈志盛,孙克辉,张泰山. Liu混沌系统的非线性反馈同步控制[J].物理学报,2005,54(6): 2580-2583
[78]刘扬正,费树岷. Genesio-Tesi和Coullet混沌系统之间的非线性反馈同步[J].物理学报,2005,54(8): 3486-3490
[79]武相军,王兴元.基于非线性控制的超混沌Chen系统混沌同步[J].物理学报, 2006,55(12):6261-6266
[80] Park J H. On synchronization of unified chaotic systems via nonlinear Control. Chaos,Solitons and Fractal, 2005, 25: 699-704
[81] Hu G, Qu Z L. Controlling spatiotemporal chaos in coupled map lattice systems. Phys. Rev. Lett. 1994,72:68-71
[82] Kocarev L, Parlitz U. General Approach for Chaotic Synchronization with Applications to Communication. Phys. Rev. Lett. 1995,74: 5028-5031
[83] Fischer I, Liu Y, Davis P. Synchronization of chaotic semiconductor laser dynamics on subnanosecond time scales and its potential for chaos communication. Phys. Rev. A . 2000,62:011801
[84] Jovic B, Berber S, Unsworth C P. A novel mathematical analysis for predicting master–slave synchronization for the simplest quadratic chaotic flow and Ueda chaotic system with application to communications. Physica D.2006, 213:31-50
[85] Alvarez G, Hernández L, Mu?oz Jaime et al. Security analysis of communication system based on the synchronization of different order chaotic systems. Phys.Lett.A. 2005, 345 :245-250
[86] Hyun C-H, Kim J-H, Kim E et al. Adaptive fuzzy observer based synchronization design and secure communications of chaotic systems. Chaos, Solitons and Fractals. 2006,27:930-940
[87] Yang T. A survey of chaotic secure communication systems. International Journal of Computational Cognition.2004, 2:81-130
[88]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[M].北京:国防工业出版社,2002.
[89]赵耿,方锦清,现代信息安全与混沌保密通信应用研究的进展[J].物理学进展,2003,23(2):212-255
[90] Boccaletti S, Kurths J, Osipov G et al. The synchronization of chaotic systems. Phys.Rep. 2002,366:1-101
[91] Li Z G, Xu D. stability criterion for projective synchronization in three-dimensional chaotic systems. Phys Lett A. 2001,282:175-179
[92] Xu D, Leng Ong W, Li Z G. Criteria for the occurrence of projective synchronization in chaotic systems of arbitrary dimension. Phys Lett A. 2002,305:167-172
[93] Xu D, Li Z, Bishop R. Manipulating the scaling factor of projective synchronization in three-dimensional chaotic systems. Chaos, 2001,11(3): 439-442
[94] Xu D, Control of projective synchronization in chaotic systems, Phys Rev E 2001,63(2):027201
[95] Xu D., Chee C Y, Controlling the ultimate state of projective synchronization in chaotic systems of arbitary dimension. Phys. Rev. E 2002,66(4):046218
[96]刘杰,陈士华,陆君安.统一混沌系统的投影同步与控制[J].物理学报,2003,52(7):1595-1599
[97] Li Z G, Xu D. A secure communication scheme using projective chaos synchronization. Chaos, Solitons and Fractals. 2004,22:477-481
[98] Chee C Y, Xu D. Secure digital communication using controlled projective synchronization of chaos. Chaos, Solitons and Fractals. 2005,23:1063-1070
[99] Alvarez G, Li S J, Montoya F et al. Breaking projective chaos synchronization secure communication using filtering and generalized synchronization. Chaos, Solitons and Fractals. 2005,24:775-783
[100] Xu D, Li Z G. Controlled projective synchronization in nonpartially-linear chaotic systems. Int.J.Bifurcation and chaos.2002, 12(6):1395-1402
[101] Zou Y L, Zhu J. Controlling projective synchronization in coupled chaotic systems. Chinese physics. 2006,15(9):1965-1970
[102] Wen G L, Xu D. Observer-based control for full-state projective synchronization of a general class of chaotic maps in any dimension. Phys Lett A. 2004,333:420-425
[103] Wen G L, Xu D. Nonlinear observe control for full-state projective synchronization in chaotic continuous-time systems. Chaos,Solitons and Fractals. 2005,26:71-77
[104] Feng C F, Zhang Y, Wang Y H. Projective synchronization in time-delayed chaotic systems. Chinese Phys. Lett. 2006,23:1418-1421
[105] Li C G. Projective synchronization in fractional order chaotic systems and its control. Progr.Theo.Phys.2006, 115: 661-666
[106]严建平,李常品.关于混沌洛伦兹系统和陈系统中的广义投影同步(英文) [J].上海大学学报(自然科学版)(英文版),2006,10(4):299-304
[107] Yan J P, Li C P.Generalized projective synchronization of a unified chaotic.Chaos,Solitons and Fractals,2005,26:1119-1124
[108] Li G H, Zhou S P , Yang K. Generalized projective synchronization between two differentchaotic systems using active backstepping control. Phys. Lett. A .2006,355:326-330
[109] Hung Y C, Hwang C C, Liao T L.Generalized projective synchronization of chaotic systems with unknown dead-zone input:observer-based approach. Chaos, 2006,16:033125
[110] Li G.H., Modified projective synchronization of chaotic system. Chaos, Solitons and Fractals, 2007, 32(5):1786-1790
[111] Park J H. Adaptive controller design for modified projective synchronization of Genesio-Tesi chaotic system with uncertain parameters. Chaos, Solitons and Fractals, 2007, 34(4): 1154-1159
[112] Hong T M, Nakagawa M. Projective lag synchronization of coupled multidelay feedback systems.J.Phys.Soci.Japan.2006, 75(9):094801
[113] Li C P, Yan J P. Generalized projective synchronization of chaos: the casade synchronization approach. Chaos, Solitons and Fractals. 2006,30:140-146
[114] Li G H, Xiong C A, Sun X N. Projective synchronization based on suitable separation. Chaos, Solitons and Fractals.2007, 32:561-565
[115] Yu H J, Peng J H, Liu Y Z. Projective synchronization of unidentical chaotic systems based on stability criterion. International Journal of Bifurcation and Chaos.2006,16(4):1049-1056
[116] Zhen J, Lu J A, Deng G M et al. Generalized projective synchronization of a class of chaotic (hyperchaotic) systems with uncertain parameters. Chinese physics.2007,16(5):1246-1251
[117] Wang B H, Bu S L. Controlling the Ultimate State of projective synchronization in Chaos: application to chaotic encryption. Int.J.Mod.Phys.B. 2004,18 (17-19):2415-2421
[118]王兴,王勇.基于线性分离的自治混沌系统的投影同步[J].物理学报,2007,56(5):2498-2503
[119]李国辉,孙晓南.基于观测器的混沌投影同步[J].量子电子学报,2006,23(5):677-680
[120]梁峰,相敬林,陈韶华,石杰.统一混沌系统的投影同步性质与控制研究[J].西北工业大学学报,2006,24(1):10-14
[121]肖前勇,邹艳丽,罗晓曙.混沌系统的投影同步及其控制研究[J].广西师范大学学报(自然科学版),2005,23(2):5-8
[122]姜德平,罗晓曙,唐国宁,翁甲强.超混沌Chua电路的比例同步及其研究[J].广西师范大学学报(自然科学版),2004,22(3):10-14
[123]刘杰,陈士华.四维混沌(超混沌)系统投影同步的稳定性判据[J].石河子大学学报(自然科学版),2002,6(4):311-314
[124] Huang D., Stabilizing near-nonhyperbolic chaotic systems with applications. Phys.Rev.Lett 2004,93:214101
[125] Huang D. Simple adaptive-feedback controller for identical chaos synchronization. Phys.Rev.E 2005, 71:037203.
[126] Huang D. Adaptive-feedback control algorithm. Phys.Rev.E 2006, 73:066204.
[127] Cao J D, Lu J. Adaptive synchronization of neural networks with or without time-varying delay. Chaos. 2006,16:013133
[128] Chen M, Zhou D. Synchronization in uncertain complex networks. Chaos.2006,16: 013101
[129] Tao C H. Dislocated feedback synchronization of Lorenz chaotic system. Phys.Lett.A.2006, 348:201-209
[130] Lu J A, Wu X Q, LǚJ H. Synchronization of unified chaotic system and the application in secure communication. Phys. Lett. A. 2002,305:365-370
[131] Hu M F, Xu Z Y. Adaptive feedback controller for projective synchronization. Nonlinear Analysis B: Real World Applications.2007, In press
[132] Hu M F, Xu Z Y. Adaptive projective synchronization of unified chaotic systems and its application to secure communication. Chinese Physics. 2007, In Press
[133] Yang T. Impulsive control. IEEE Trans.Automat.Contr.1999, 44:1081-1083
[134] Chen D L, Sun J T, Wu Q D. Impulsive control and its application to Lü's chaotic system. Chaos, Solitons and Fractals.2004, 21:1135-1142
[135] Yang T. Impulsive control theory. Berlin: Springer.2001
[136] Yang T, Chua L O. Impulsive control and synchronization of nonlinear dynamical systems and application to secure communication. International Journal of Bifurcation and Chaos.1997, 7 (3): 645-664
[137] Yang T, Yang L B, Yang C M. Impulsive synchronization of Lorenz systems. Phys. Lett.A.1997, 226:349-354
[138] Itoh M, Yang T, Chua L O. Conditions for impulsive synchronization of chaotic and hyperchaotic systems. Int. J. Bifurcation and Chaos. 2001,11:551-560
[139] Sun J T, Zhang Y P, Qiao Fet al. Some impulsive synchronization criterions for coupled chaotic systems via unidirectional linear error feedback approach.. Chaos, Solitons andFractals.2004, 20:751-758
[140] Li C D, Liao X F, Zhang X Y. Impulsive synchronization of chaotic systems. Chaos. 2005, 15:023104
[141] Bainov D D, Simeonov P S. Systems with impulsive effect: stability, theory and applications.. New York: Halsted Press, 1989.
[142] Lakshmikantham V, Bainov D D, Simeonov P S. Theory of impulsive differential equations. Singapore: World scientific, 1989.
[143]郭大均.非线性分析中的半序方法[M].济南:山东科技出版社.1997.
[144]傅希林,闫宝强,刘衍胜.脉冲微分系统引论[M].北京:科学出版社.2005.
[145] Sun J T, Zhang Y P, Wu Q D. Less conservative conditions for asymptotic stability of impulsive control systems. Automatic Control, IEEE Transactions on. 2003,48:829-831
[146] Sun J T. Impulsive control of a new chaotic system. Mathematics and Computers in Simulation.2004, 64:669-677
[147] Sun J T, Zhang Y P. Impulsive control of a nuclear spin generator. Journal of Computational and Applied Mathematics.2003, 157:235-242
[148] Yan J, Shen J. Impulsive stabilization of functional differential equations by Lyapunov Razumikhin functions. Nonlinear Analysis.1999, 37:245-255
[149] Akhmetov M U, Zafer A. Stability of the Zero Solution of Impulsive Differential Equations by the Lyapunov Second Method. Journal of Mathematical Analysis and Applications. 2000,248:69-82
[150] Zhao H Q, Feng E M. Stability of impulsive system by perturbing Lyapunov functions. Applied Mathematics Letters.2007, 20:194-198
[151] Liu X Z, Wang Q. On stability in terms of two measures for impulsive systems of functional differential equations. J. Math. Anal. Appl. 2007,326:252-265
[152] Lakshmikantham V, Liu X Z. Stability criteria for impulsive differential equations in terms of two measures, J. Math. Anal. Appl. 1989,137:591–604
[153] Wang P G, Liu X. Practical stability of impulsive hybrid differential systems in terms of two measures on time scales. Nonlinear Analysis.2006, 65:2035-2042
[154] Li Z G, Wen C Y, Soh Y C. Analysis and design of impulsive control systems. IEEE Trans.Automat.Contr.2001, 46:894-897
[155] Liu X Z. Stability results for impulsive differential systems with applications to population growth models. Dynam. Stability Systems.1994, 9:163-174
[156] Hu M F, Xu Z Y. Analysis and design of impulsive control system. DCDIS proceedings. 2005, 3: 571-578. Wuxi, China, October 28-30, 2005.
[157] Liu Z X, Chen Z Q, Yuan Z Z. Pinning control of scale-free dynamical networks. Physica A.2007,375:345-354
[158] Wang X F, Chen G R. Synchronization in scale-free dynamical networks: robustness and fragility. IEEE Trans.Circuits Syst.I.2002, 49:54-62
[159] Li X, Chen G R.synchronization and desynchronization of complex dynamical networks: an engineering viewpoint. IEEE Trans.Circuits Syst.I.2003, 50: 1381-1390
[160] Belykh V N, Belykh I V, Hasler M. Connection graph stability method for synchronized coupled chaotic systems. Physica D.2004, 159-187
[161] Belykh I V, Belykh V N, Hasler M.Synchronization in asymmetrically coupled networks with node balance. Chaos.2006, 16:015102
[162] LüJ H, Yu X H, Chen G R et al. Characterizing the synchronizability of small-world dynamical networks. IEEE Trans.Circuits Syst.I.2004, 51:787-796
[163] Grigoriev R O, Cross M C, Schuster H G. Pinning Control of Spatiotemporal Chaos. Phys. Rev. Lett. 1997,79:2795-2798
[164] Wang X F, Chen G R. Pinning control of scale-free dynamical networks. Physica A.2002, 310:521-531
[165] Xiang L Y, Liu Z X, Chen Z Q et al. Pinning control of complex dynamical networks with general topology. Physica A.2007, 379:298-306
[166] Li X, Wang X F, Chen G R. Pinning a complex dynamical network to its equilibrium. IEEE Trans.Circuits Syst.I.2004, 51:2074-2087
[167] Hu G, Xiao J H, Gao J H. Analytical study of spatiotemporal chaos control by applying local injections. Phys. Rev. E.2000, 62: 3043 -3046
[168] Hu M F, Yang Y Q, Xu Z Y et al. Projective synchronization in drive-response dynamical networks. Physica A. 2007,381:457-466
[169] Ma Z J, Liu Z R, Zhang G. A new method to realize cluster synchronization in connected chaotic networks. Chaos.2006, 16:023103
[170]马忠军.混沌动力学网络上的几类同步现象研究[D]:[博士学位论文].上海:上海大学,2006
[171]匡继昌.常用不等式[M].(第三版)济南:山东科学技术出版社.2004,447-458
[172] Song Q K, Cao J D. Synchronization and anti-synchronization for chaotic systems. Chaos, Solitons and Fractals.2007, 33:929-939
[173] Li C D, Liao X F. Anti-synchronization of a class of coupled chaotic systems via linear feedback control. International Journal of Bifurcation and Chaos.2006, 16:1041-1047
[174] Song Q K, Cao J D. Synchronization and anti-synchronization for chaotic systems. Chaos, Solitons and Fractals.2007, 33:929-939
[175] Kreuz T, Mormann F, Andrzejak R G et al. Measuring synchronization in coupled model systems: A comparison of different approaches. Physica D.2007, 225:29-42
[176] Li G H, Zhou S P. Anti-synchronization in different chaotic systems. Chaos, Solitons and Fractals.2007, 32:516-520
[177]胡满峰,徐振源,Lorenz混沌系统的非线性反馈错位同步控制[J].系统工程与电子技术,2007,29(8):1346-1348
[178]陈骏,刘曾荣.不同系统之间的部分同步[J].应用数学和力学,2005,26(9): 1033-1037
[179] Lu J, Xi Y. Linear generalized synchronization of continuous-time chaotic systems. Chaos, Solitons and Fractals.2003, 17(5):825-831
[180] Yang T, Chua L O. Generalized synchronization of chaos via linear transformations. International Journal of Bifurcation and Chaos.1999, 9(1): 215-219
[181]胡爱花,徐振源,李芳.基于自适应控制的混沌系统的线性广义同步化[J].系统仿真学报,2006,18(4): 975-977,981
[182] Rubezic V, Lutovac B, Ostojic R. Linear generalized synchronization of two chaotic Colpitts oscillators. Electronics, Circuits and Systems, 2002. 9th International Conference on. 2002,1:223-225
[183]刘曾荣.关于同步的几个理论问题[J].自然,2004, 26(5): 298-300
[184] Yassen M T. Chaos synchronization between two different chaotic systems using active control. Chaos, Solitons and Fractals.2005, 23:131-140
[185] Femat R, Alvarez-Ramírez J. Synchronization of a class of strictly different chaotic oscillators.Phys.Lett.A.1997, 236:307-313
[186] Sun F Y. Global chaos synchronization between two new different chaotic systems via active control. Chinese Physics Letters.2006, 23:32-34
[187] Li R H, Xu W, Li S. Adaptive generalized projective synchronization in different chaotic systems based on parameter identification.Phys.Lett.A.2007, 367:199-206
[188] Femat R, Solís-Perales G. Synchronization of chaotic systems with different order. Phys. Rev. E.200265:036226
[189] Lazzouni S A, Bowong S, Kakmeni F M et al. An adaptive feedback control for chaos synchronization of nonlinear systems with different order. Communications in Nonlinear Science and Numerical Simulation.2007, 12(4):568-583
[190] Bowong S. Stability analysis for the synchronization of chaotic systems with different order: application to secure communications.Phys.Lett.A. 2004, 326:102-113
[191] Ge Z M, Chang C M, Chen Y S. Anti-control of chaos of single time scale brushless dc motors and chaos synchronization of different order systems. Chaos, Solitons and Fractals. 2006, 27:1298-1315
[192] Chen J Y, Wong K W, Chen Z X et al. Phase synchronization in discrete chaotic systems. Phys. Rev. E .2000,61:2559-2562
[193] Rosenblum M G, Pikovsky A S, Kurths J. Comment on“Phase synchronization in discrete chaotic systems”. Phys. Rev. E.2001, 63:058201
[194] Zheng Y A, Nian Y B, Liu Z R. Impulsive Synchronization of Discrete Chaotic Systems. Chinese Physics Letters.2003, 20:199-201
[195] Zhang H B, Li C G, Liao X F. Observer-based synchronization for piecewise-linear discrete chaotic systems based on piecewise quadratic Lyapunov functions. International Journal of Bifurcation and Chaos.2005,19:4389-4397
[196] Lian K Y, Liu P, Chiu C S et al. Robust dead-beat synchronization and communication for discrete-time chaotic systems. International Journal of Bifurcation and Chaos. 2002, 12: 835-846
[197]陈菊芳,程丽,刘颖等.延迟变量反馈法控制离散混沌系统的电路实验[J].物理学报, 2003, 52(1):18-24
[198]陈菊芳,张入元,彭建华.脉冲驱动离散混沌系统同步的实验与理论研究[J].物理学报, 2003, 52(7):1589-1594
[199]刘丁,钱富才,任海鹏等.离散混沌系统的最小能量控制[J].物理学报,2004, 53 (7) : 2074-2079
[200] Atay F M, Jost J, Wende A. Delays, connection topology, and synchronization of coupled chaotic maps. Phys. Rev. Lett. 2004,92:144101
[201] Minai A A, Anand T. Synchronization of chaotic maps through a noisy coupling channel with application to digital communication. Phys. Rev. E.1999, 59:312-320
[202] De Oliveira A G, Jones A J. Synchronization of Chaotic Maps by Feedback Control and Application to Secure Communications Using Chaotic Neural Networks. International Journal of Bifurcation and Chaos.1998,8: 2225-2237
[203] T?rres L A B. Discrete-time dynamic systems synchronization: Information transmission and model matching, Physica D. 2007,228:31-39
[204] Brzozowska-Rup K, Or?owski A. Application of bootstrap to detecting chaos in financial time series. Physica A. 2004, 344:317-321
[205]吕金虎,陆君安,陈士华.混沌时间序列分析及其应用[M].武汉:武汉大学出版社.2002.
[206] Huang D B. Synchronization-based estimation of all parameters of chaotic systems from time series. Phys. Rev. E.2004, 69:067201
[207] Huang D B, Guo R W. Identifying parameter by identical synchronization between different systems. Chaos.2004, 14:152-159
[208] Konnur R. Synchronization-based approach for estimating all model parameters of chaotic systems. Phys. Rev. E. 2003,67: 027204
[209] Li S, Xu W, Li R et al. A general method for chaos synchronization and parameters estimation between different systems. Journal of Sound and Vibration.2007,302:777-788
[210] Grosu I. Estimation of model parameters from time series by OPCL auto-synchronization. Journal of Bifurcation and Chaos. 2004,14:2133-2141
[211] Sun F Y, Zhao Y, Zhou T S. Identify fully uncertain parameters and design controllers based on synchronization. Chaos, Solitons and Fractals.2007,34:1677-1682
[212] Yang X L, Xu W, Sun Z K. Estimating model parameters in nonautonomous chaotic systems using synchronization. Phys.Lett.A.2007,364: 378-388
[213] Rakshit B, Chowdhury A. R, Saha P. Parameter estimation of a delay dynamical system usingsynchronization in presence of noise. Chaos, Solitons and Fractals.2007,32: 1278-1284
[214] Ge Z M, Lee J K. Chaos synchronization and parameter identification for gyroscope system. Applied Mathematics and Computation. 2005, 163:667-682
[215] Lin W, Chen G R. Using white noise to enhance synchronization of coupled chaotic systems. Chaos. 2006,16:013134
[216] Kiss I Z, Zhai Y M, Hudson J L et al. Noise enhanced phase synchronization and coherence resonance in sets of chaotic oscillators with weak global coupling.Chaos.2003,13:267-278
[217] Park K, Lai Y C, Krishnamoorthy S et al. Effect of common noise on phase synchronization in coupled chaotic oscillators. Chaos. 2007,17:013105
[218] Sánchez E, Matías M A, Pérez-Mu?uzuri V. Analysis of synchronization of chaotic systems by noise: An experimental study. Phys. Rev. E. 1997,56:4068-4071
[219]胡爱花,徐振源.利用白噪声实现混沌系统线性广义同步的研究[J].物理学报,2007, 56(6): 3132-3136
[220] Bowong S, McClintock P V E. Adaptive synchronization between chaotic dynamical systems of different order. Phys.Lett.A. 2006,358:134-141
[221]王发强,刘崇新.不同阶混沌系统的自适应同步控制[J].华北电力学学报,2005, 32:11-14
[222] Zhang G, Liu Z R, Ma Z J. Generalized synchronization of different dimensional chaotic dynamical systems. Chaos, Solitons and Fractals.2007,32: 773-779
[223] Gao T G, Chen Z Q, Yuan Z Z. Generalized reduced-order synchronization of chaotic system based on fast slide mode. Chinese Physics. 2005, 14(12):2421-2427
[224] Li R H, Xu W, Li S. Chaos control and reduced-order generalized synchronization for the Chen--Liao system. Chinese Physics. 2007, 16(6):1591-1596
[225] Kakmeni F M M, Bowong S, Tchawoua C. Reduced-order synchronization of uncertain chaotic systems via adaptive control. Communications in Nonlinear Science and Numerical Simulation.2006,11:810-830
[226] Ho M C, Hung Y C, Liu Z Y et al. Reduced-order synchronization of chaotic systems with parameters unknown. Phys.Lett.A. 2006, 348:251-259
[227] Mahmoud G M, Aly S A, Al-Kashif M A. Dynamical properties and chaos synchronization of a new chaotic complex nonlinear system. Nonlinear Dyn. 2007, (In press) doi:10.1007/s11071-007-9200-y
[228] Yang X S. Concepts of synchronization in dynamical systems. Phys. Lett. A. 1999,260: 340-344
[229] Yan Z Y. Q-S synchronization in 3D Henon-like map and generalized Henon map via a scalar controller. Phys.Lett.A.2005,342:309-317
[230] Yan Z Y. Q-S (complete or anticipated) synchronization backstepping scheme in a class of discrete-time chaotic (hyperchaotic) systems: A symbolic-numeric computation approach. Chaos, 2005,16: 013199
[231]方保镕,周继东,李医民.矩阵论[M].北京:清华大学出版社.2004.
[232] Yu W W, Cao J D. Adaptive Q-S (lag, anticipated, and complete) time-varying synchronization and parameters identification of uncertain delayed neural networks. Chaos. 2006, 16: 023119
[233] Wang Q, Chen Y. Generalized Q-S (lag, anticipated, and complete) synchronization in modified Chua’s circuit and Hindmarsh-Rose systems. Applied Mathematics and Computation. 2006, 181:48-56
[234] Hu M F. Xu Z Y. A general scheme for Q-S synchronization of chaotic systems. Nonlinear Analysis. 2007, In press
[2] Li T Y, Yorke J A. Period three implies chaos. Amer.Math.Monthly, 1975, 82:985-992
[3]陈关荣,汪小帆.动力系统的混沌化——理论、方法与应用[M].上海:上海交通大学出版社,2006.1-9
[4]闫振亚.复杂非线性波的构造性理论及其应用[M].北京:科学出版社,2007.14-21,233-302
[5] Devaney R L. An Introduction to Chaotic Dynamical Systems. 2nd edition. New York: Addison-Wesley, 2003.
[6] Ott E, Grebogi C, Yorke J A. Controlling chaos. Phys.Rev.Lett, 1990,64:1196-1199
[7] Boccaletti S, Grebogi C, Lai Y-C et al. The control of chaos: theory and application. Phys.Rep, 2000,329:103-197
[8]邹恩,李祥飞,陈建国.混沌控制及其优化应用[M].长沙:国防科技大学出版社,2002
[9]黄润生,黄浩.混沌及其应用[M].第二版.武汉:武汉大学出版社,2005
[10] Schiff S J, Jerger K, Duong D H, et al. Controlling chaos in the brain. Nature, 1994, 363: 411-417
[11]刘秉正,彭建华.非线性动力学[M].北京:高等教育出版社,2004
[12] Huygenii G. Horoloquim oscilatorium. France: Parisiis,1673.
[13]郑志刚.耦合非线性系统的时空动力学与合作行为[M].北京:高等教育出版,2004
[14] Rayleigh J. The theory of sound. New York: Dover Publishers,1945
[15] Winfree A T. Geometry of biological time. New York: Springer-Verlag, 1990
[16] Glass L, Mackay M C.From clocks to chaos: the rhythms of life. Princeton NJ: Princeton University Press, 1988
[17] Pecora L M, Carroll T L. Synchronization in chaotic systems. Phys. Rev. Lett 1990,64:821-825
[18] Lu J Q, Cao J D. Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters. Chaos, 2005, 15:043901
[19] Zhan M, Wang X G, Gong X F et al. Complete synchronization and generalizedsynchronization of one-way coupled time-delay systems.Phys.Rev.E, 2003,68:036208
[20] Rosenblum M G, Pikovsky A S, Kurths J. Phase Synchronization of Chaotic Oscillators. Phys. Rev. Lett.1996, 76:1804-1807
[21] Blasius B, Huppert A, Stone L. Complex dynamics and phase synchronization in spatially extended ecological systems. Nature. 1999, 399(6734): 354-359
[22] Parlitz U, Junge L, Lauterborn W. Experimental observation of phase synchronization. Phys. Rev. E. 1996,54:2115-2117
[23] Rulkov N F, Sushchik M M, Tsimring L S et al. Generalized synchronization of chaos in directionally coupled chaotic systems. Phys. Rev. E. 1995,51:980-994
[24] Kocarev L, Parlitz U. Generalized Synchronization, Predictability, and Equivalence of Unidirectionally Coupled Dynamical Systems. Phys. Rev. Lett. 1996,76:1816 - 1819
[25] Abarbanel H D I, Rulkov N F, Sushchik M M. Generalized synchronization of chaos: The auxiliary system approach. Phys. Rev. E.1996, 53:4528-4535
[26] Mainieri R, Rehacek J. Projective synchronization in three-dimensional chaotic systems. Phys. Rev. Lett 1999,82:3042-3045
[27] Wen G L, Xu D. Nonlinear observer control for full-state projective synchronization in chaotic continuous-time systems, Chaos, Solitons and Fractals, 2005,26:71-77
[28] Hu M F, Xu Z Y, Zhang R. Full state hybrid projective synchronization in continuous-time chaotic (hyperchaotic) systems. Communications in Nonlinear Science and Numerical Simulation.2008,13(2):456-464
[29] Hu M F, Xu Z Y, Zhang R. Full state hybrid projective synchronization of a general class of chaotic maps. Communications in Nonlinear Science and Numerical Simulation. (In press)
[30] Hu M F, Xu Z Y, Zhang R et al. Parameters identification and adaptive full state hybrid projective synchronization of chaotic (hyper-chaotic) systems. Phys Lett.A.2007, 361: 231-237
[31] Hu M F, Xu Z Y, Zhang R et al. Adaptive full state hybrid projective synchronization of chaotic systems with the same and different order. Phys Lett.A.2007, 365: 315-327
[32] Yan Z Y. Q-S (lag or anticipated) synchronization backstepping scheme in a class of continuous-time hyperchaotic systems—A symbolic-numeric computation approach. Chaos, 2005,15: 023902
[33] Yan Z Y. Chaos Q-S synchronization between R?ssler system and a new unified chaotic system. Phys. Lett. A, 2005,334:406-412
[34] Shuai J W, Wong K W, Cheng L M. Synchronization of spatiotemporal chaos with positive conditional Lyapunov exponents. Phys.Rev.E.1997, 56:2272-2275
[35] Zhou C, Lai C H. Analysis of spurious synchronization with positive conditional Lyapunov exponents in computer simulations. Physica D.2000, 135:1-23
[36] Guan S., Li K., Lai C.-H. Chaotic synchronization through coupling strategies. Chaos, 2007,16: 023107
[37]汪小帆,李翔,陈关荣.复杂网络理论及其应用[M].北京:清华大学出版社,2006
[38] Strogatz S H. Exploring complex networks. Nature. 2001,410:268-276
[39] Tutte W T. Graph theory [M].北京:机械工业出版社,2004
[40] Bollobas B. Random Graphs. 2nd edition. New York: Academic Press, 2001
[41] Watts D J, Strogatz S H. Collective dynamics of“small-world”networks. Nature. 1998, 393:440-442
[42] Barabasi A L, Albert R. Emergence of scaling in random networks. Science. 1999, 286: 509-512
[43] Neda Z, Ravasz E, Vicsek T, et al. The sound of many hands clapping. Nature, 2000,403: 849-850
[44] Kaneko k. Coupled map lattices. Singapore: World Scientific, 1992
[45] Chao L O. CNN: A paradigm for complexity. Singapore: World Scientific, 1993
[46] Motter A E, Zhou C S, Kurths J. Network synchronization, diffusion, and the paradox of heterogeneity. Phys. Rev. E .2005,71:016116
[47] Motter A E, Zhou C S, Kurths J. Enhancing complex-network synchronization. Europhys. Lett.2005, 69:334-340
[48] Zhou C S, Motter A E, Kurths J. Universality in the Synchronization of Weighted Random Networks. Phys. Rev. Lett. 2006,96:034101
[49] Zhao M, Zhou T, Wang B H et al. Enhanced synchronizability by structural perturbations. Phys. Rev. E. 2005,72:057102
[50] Komiss G. Synchronization in weighted uncorrelated complex networks in a noisy environment: Optimization and connections with transport efficiency. Phys. Rev. E. 2007,75:051121
[51] Chavez M, Hwang D.-U, Amann A et al. Synchronizing weighted complex networks. Chaos. 2006,16: 015106
[52] Barahona M, Pecora L M. Synchronization in small-world systems. Phys. Rev. Lett. 2002,89:054101
[53] Wang X F, Chen G R. Synchronization in small-world dynamical networks. International Journal of Bifurcation and Chaos.2002, 12: 187-192
[54] Li C G, Chen G R. Phase synchronization in small-world networks of chaotic oscillators. Physica A.2004, 341:73-79
[55] Pecora L M, Carroll T L. Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 1998,80:2109-2112
[56] Chen G R, Ueta T. Yet another chaotic attractor. International Journal of Bifurcation and Chaos.1999, 9(7): 1465-1466
[57] LüJ H, Chen G R, Chen D Z et al. Bridge the gap between the Lorenz system and the Chen system. International Journal of Bifurcation and Chaos.2002, 12:2917-2926
[58] LüJ H, Chen G R. A new chaotic attractor coined. International Journal of Bifurcation and Chaos.2002, 12: 659-661
[59] Genesio R, Tesi A. Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica. 1992, 28 : 531-548
[60] R?ssler O E. An equation for continuous chaos. Phys.Lett.A.1976, 57:397-398
[61] Stenflo L. Generalized Lorenz equations for acoustic-gravity waves in the atmosphere. Phys. Scr. 1996, 53: 83-84
[62] Li Y, Tang WKS, Chen G. Generating hyperchaos via state feedback control. International Journal of Bifurcation and Chaos. 2005, 10:3367-3375
[63] Chen A , Lu J , LǚJ , Yu S. Generating hyperchaotic Lǚattractor via state feedback control. Phys.A. 2006, 364:103-110
[64] Stefanski K. Modelling chaos and hyperchaos with 3D maps. Chaos, Solitons and Fractals.1998, 9:83-93
[65] Grassi G, Miller D A. Theory and experimental realization of observer-based discrete-time hyperchaos synchronization. IEEE Trans.Circ.Syst.I.2002, 49:373-378
[66] Murali K,Lakshmanan M. Drive-response scenario of chaos synchronization in identical nonlinear systems. Phys. Rev. E. 1994,49:4882 - 4885
[67] Yassen M T. Controlling chaos and synchronization for new chaotic system using linear feedback control. Chaos, Solitons and Fractals.2005, 26(3): 913-920
[68] Wang F Q, Liu C X. A new criterion for chaos and hyperchaos synchronization using linear feedback control. Phys.Lett.A.2006, 360:274-278
[69]王发强,刘崇新. Liu混沌系统的线性反馈同步控制及电路实验的研究[J].物理学报,2006,55(10): 5055-5060
[70] Wu X F, Chen G, Cai J P. Chaos synchronization of the master–slave generalized Lorenz systems via linear state error feedback control. Physica D.2007, 229:52-80
[71] Lu J A, Wu X Q, Han X P et al. Adaptive feedback synchronization of a unified chaotic system. Phys.Lett.A.2004, 329:327-333
[72] Lei Y M, Xu W, Shen J W. Robust synchronization of chaotic non-autonomous systems using adaptive-feedback control. Chaos, Solitons and Fractal, 2007, 31:371-379
[73] Yassen M T. Feedback and adaptive synchronization of chaotic Lüsystem. Chaos, Solitons and Fractal, 2005, 25:379-386
[74] Tao C H, Liu X F. Feedback and adaptive control and synchronization of a set of chaotic and hyperchaotic systems. Chaos, Solitons and Fractal, 2007,32:1572-1581
[75] Yan Z Y, Yu P. Linear feedback control, adaptive feedback control and their combination for chaos (lag) synchronization of LC chaotic systems. Chaos, Solitons and Fractal, 2007, 33: 419-435
[76] Ali M K, Fang J Q. Synchronization of chaos and hyperchaos using linear and nonlinear feedback functions. Phys. Rev. E.1997, 55:5285-5290
[77]陈志盛,孙克辉,张泰山. Liu混沌系统的非线性反馈同步控制[J].物理学报,2005,54(6): 2580-2583
[78]刘扬正,费树岷. Genesio-Tesi和Coullet混沌系统之间的非线性反馈同步[J].物理学报,2005,54(8): 3486-3490
[79]武相军,王兴元.基于非线性控制的超混沌Chen系统混沌同步[J].物理学报, 2006,55(12):6261-6266
[80] Park J H. On synchronization of unified chaotic systems via nonlinear Control. Chaos,Solitons and Fractal, 2005, 25: 699-704
[81] Hu G, Qu Z L. Controlling spatiotemporal chaos in coupled map lattice systems. Phys. Rev. Lett. 1994,72:68-71
[82] Kocarev L, Parlitz U. General Approach for Chaotic Synchronization with Applications to Communication. Phys. Rev. Lett. 1995,74: 5028-5031
[83] Fischer I, Liu Y, Davis P. Synchronization of chaotic semiconductor laser dynamics on subnanosecond time scales and its potential for chaos communication. Phys. Rev. A . 2000,62:011801
[84] Jovic B, Berber S, Unsworth C P. A novel mathematical analysis for predicting master–slave synchronization for the simplest quadratic chaotic flow and Ueda chaotic system with application to communications. Physica D.2006, 213:31-50
[85] Alvarez G, Hernández L, Mu?oz Jaime et al. Security analysis of communication system based on the synchronization of different order chaotic systems. Phys.Lett.A. 2005, 345 :245-250
[86] Hyun C-H, Kim J-H, Kim E et al. Adaptive fuzzy observer based synchronization design and secure communications of chaotic systems. Chaos, Solitons and Fractals. 2006,27:930-940
[87] Yang T. A survey of chaotic secure communication systems. International Journal of Computational Cognition.2004, 2:81-130
[88]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[M].北京:国防工业出版社,2002.
[89]赵耿,方锦清,现代信息安全与混沌保密通信应用研究的进展[J].物理学进展,2003,23(2):212-255
[90] Boccaletti S, Kurths J, Osipov G et al. The synchronization of chaotic systems. Phys.Rep. 2002,366:1-101
[91] Li Z G, Xu D. stability criterion for projective synchronization in three-dimensional chaotic systems. Phys Lett A. 2001,282:175-179
[92] Xu D, Leng Ong W, Li Z G. Criteria for the occurrence of projective synchronization in chaotic systems of arbitrary dimension. Phys Lett A. 2002,305:167-172
[93] Xu D, Li Z, Bishop R. Manipulating the scaling factor of projective synchronization in three-dimensional chaotic systems. Chaos, 2001,11(3): 439-442
[94] Xu D, Control of projective synchronization in chaotic systems, Phys Rev E 2001,63(2):027201
[95] Xu D., Chee C Y, Controlling the ultimate state of projective synchronization in chaotic systems of arbitary dimension. Phys. Rev. E 2002,66(4):046218
[96]刘杰,陈士华,陆君安.统一混沌系统的投影同步与控制[J].物理学报,2003,52(7):1595-1599
[97] Li Z G, Xu D. A secure communication scheme using projective chaos synchronization. Chaos, Solitons and Fractals. 2004,22:477-481
[98] Chee C Y, Xu D. Secure digital communication using controlled projective synchronization of chaos. Chaos, Solitons and Fractals. 2005,23:1063-1070
[99] Alvarez G, Li S J, Montoya F et al. Breaking projective chaos synchronization secure communication using filtering and generalized synchronization. Chaos, Solitons and Fractals. 2005,24:775-783
[100] Xu D, Li Z G. Controlled projective synchronization in nonpartially-linear chaotic systems. Int.J.Bifurcation and chaos.2002, 12(6):1395-1402
[101] Zou Y L, Zhu J. Controlling projective synchronization in coupled chaotic systems. Chinese physics. 2006,15(9):1965-1970
[102] Wen G L, Xu D. Observer-based control for full-state projective synchronization of a general class of chaotic maps in any dimension. Phys Lett A. 2004,333:420-425
[103] Wen G L, Xu D. Nonlinear observe control for full-state projective synchronization in chaotic continuous-time systems. Chaos,Solitons and Fractals. 2005,26:71-77
[104] Feng C F, Zhang Y, Wang Y H. Projective synchronization in time-delayed chaotic systems. Chinese Phys. Lett. 2006,23:1418-1421
[105] Li C G. Projective synchronization in fractional order chaotic systems and its control. Progr.Theo.Phys.2006, 115: 661-666
[106]严建平,李常品.关于混沌洛伦兹系统和陈系统中的广义投影同步(英文) [J].上海大学学报(自然科学版)(英文版),2006,10(4):299-304
[107] Yan J P, Li C P.Generalized projective synchronization of a unified chaotic.Chaos,Solitons and Fractals,2005,26:1119-1124
[108] Li G H, Zhou S P , Yang K. Generalized projective synchronization between two differentchaotic systems using active backstepping control. Phys. Lett. A .2006,355:326-330
[109] Hung Y C, Hwang C C, Liao T L.Generalized projective synchronization of chaotic systems with unknown dead-zone input:observer-based approach. Chaos, 2006,16:033125
[110] Li G.H., Modified projective synchronization of chaotic system. Chaos, Solitons and Fractals, 2007, 32(5):1786-1790
[111] Park J H. Adaptive controller design for modified projective synchronization of Genesio-Tesi chaotic system with uncertain parameters. Chaos, Solitons and Fractals, 2007, 34(4): 1154-1159
[112] Hong T M, Nakagawa M. Projective lag synchronization of coupled multidelay feedback systems.J.Phys.Soci.Japan.2006, 75(9):094801
[113] Li C P, Yan J P. Generalized projective synchronization of chaos: the casade synchronization approach. Chaos, Solitons and Fractals. 2006,30:140-146
[114] Li G H, Xiong C A, Sun X N. Projective synchronization based on suitable separation. Chaos, Solitons and Fractals.2007, 32:561-565
[115] Yu H J, Peng J H, Liu Y Z. Projective synchronization of unidentical chaotic systems based on stability criterion. International Journal of Bifurcation and Chaos.2006,16(4):1049-1056
[116] Zhen J, Lu J A, Deng G M et al. Generalized projective synchronization of a class of chaotic (hyperchaotic) systems with uncertain parameters. Chinese physics.2007,16(5):1246-1251
[117] Wang B H, Bu S L. Controlling the Ultimate State of projective synchronization in Chaos: application to chaotic encryption. Int.J.Mod.Phys.B. 2004,18 (17-19):2415-2421
[118]王兴,王勇.基于线性分离的自治混沌系统的投影同步[J].物理学报,2007,56(5):2498-2503
[119]李国辉,孙晓南.基于观测器的混沌投影同步[J].量子电子学报,2006,23(5):677-680
[120]梁峰,相敬林,陈韶华,石杰.统一混沌系统的投影同步性质与控制研究[J].西北工业大学学报,2006,24(1):10-14
[121]肖前勇,邹艳丽,罗晓曙.混沌系统的投影同步及其控制研究[J].广西师范大学学报(自然科学版),2005,23(2):5-8
[122]姜德平,罗晓曙,唐国宁,翁甲强.超混沌Chua电路的比例同步及其研究[J].广西师范大学学报(自然科学版),2004,22(3):10-14
[123]刘杰,陈士华.四维混沌(超混沌)系统投影同步的稳定性判据[J].石河子大学学报(自然科学版),2002,6(4):311-314
[124] Huang D., Stabilizing near-nonhyperbolic chaotic systems with applications. Phys.Rev.Lett 2004,93:214101
[125] Huang D. Simple adaptive-feedback controller for identical chaos synchronization. Phys.Rev.E 2005, 71:037203.
[126] Huang D. Adaptive-feedback control algorithm. Phys.Rev.E 2006, 73:066204.
[127] Cao J D, Lu J. Adaptive synchronization of neural networks with or without time-varying delay. Chaos. 2006,16:013133
[128] Chen M, Zhou D. Synchronization in uncertain complex networks. Chaos.2006,16: 013101
[129] Tao C H. Dislocated feedback synchronization of Lorenz chaotic system. Phys.Lett.A.2006, 348:201-209
[130] Lu J A, Wu X Q, LǚJ H. Synchronization of unified chaotic system and the application in secure communication. Phys. Lett. A. 2002,305:365-370
[131] Hu M F, Xu Z Y. Adaptive feedback controller for projective synchronization. Nonlinear Analysis B: Real World Applications.2007, In press
[132] Hu M F, Xu Z Y. Adaptive projective synchronization of unified chaotic systems and its application to secure communication. Chinese Physics. 2007, In Press
[133] Yang T. Impulsive control. IEEE Trans.Automat.Contr.1999, 44:1081-1083
[134] Chen D L, Sun J T, Wu Q D. Impulsive control and its application to Lü's chaotic system. Chaos, Solitons and Fractals.2004, 21:1135-1142
[135] Yang T. Impulsive control theory. Berlin: Springer.2001
[136] Yang T, Chua L O. Impulsive control and synchronization of nonlinear dynamical systems and application to secure communication. International Journal of Bifurcation and Chaos.1997, 7 (3): 645-664
[137] Yang T, Yang L B, Yang C M. Impulsive synchronization of Lorenz systems. Phys. Lett.A.1997, 226:349-354
[138] Itoh M, Yang T, Chua L O. Conditions for impulsive synchronization of chaotic and hyperchaotic systems. Int. J. Bifurcation and Chaos. 2001,11:551-560
[139] Sun J T, Zhang Y P, Qiao Fet al. Some impulsive synchronization criterions for coupled chaotic systems via unidirectional linear error feedback approach.. Chaos, Solitons andFractals.2004, 20:751-758
[140] Li C D, Liao X F, Zhang X Y. Impulsive synchronization of chaotic systems. Chaos. 2005, 15:023104
[141] Bainov D D, Simeonov P S. Systems with impulsive effect: stability, theory and applications.. New York: Halsted Press, 1989.
[142] Lakshmikantham V, Bainov D D, Simeonov P S. Theory of impulsive differential equations. Singapore: World scientific, 1989.
[143]郭大均.非线性分析中的半序方法[M].济南:山东科技出版社.1997.
[144]傅希林,闫宝强,刘衍胜.脉冲微分系统引论[M].北京:科学出版社.2005.
[145] Sun J T, Zhang Y P, Wu Q D. Less conservative conditions for asymptotic stability of impulsive control systems. Automatic Control, IEEE Transactions on. 2003,48:829-831
[146] Sun J T. Impulsive control of a new chaotic system. Mathematics and Computers in Simulation.2004, 64:669-677
[147] Sun J T, Zhang Y P. Impulsive control of a nuclear spin generator. Journal of Computational and Applied Mathematics.2003, 157:235-242
[148] Yan J, Shen J. Impulsive stabilization of functional differential equations by Lyapunov Razumikhin functions. Nonlinear Analysis.1999, 37:245-255
[149] Akhmetov M U, Zafer A. Stability of the Zero Solution of Impulsive Differential Equations by the Lyapunov Second Method. Journal of Mathematical Analysis and Applications. 2000,248:69-82
[150] Zhao H Q, Feng E M. Stability of impulsive system by perturbing Lyapunov functions. Applied Mathematics Letters.2007, 20:194-198
[151] Liu X Z, Wang Q. On stability in terms of two measures for impulsive systems of functional differential equations. J. Math. Anal. Appl. 2007,326:252-265
[152] Lakshmikantham V, Liu X Z. Stability criteria for impulsive differential equations in terms of two measures, J. Math. Anal. Appl. 1989,137:591–604
[153] Wang P G, Liu X. Practical stability of impulsive hybrid differential systems in terms of two measures on time scales. Nonlinear Analysis.2006, 65:2035-2042
[154] Li Z G, Wen C Y, Soh Y C. Analysis and design of impulsive control systems. IEEE Trans.Automat.Contr.2001, 46:894-897
[155] Liu X Z. Stability results for impulsive differential systems with applications to population growth models. Dynam. Stability Systems.1994, 9:163-174
[156] Hu M F, Xu Z Y. Analysis and design of impulsive control system. DCDIS proceedings. 2005, 3: 571-578. Wuxi, China, October 28-30, 2005.
[157] Liu Z X, Chen Z Q, Yuan Z Z. Pinning control of scale-free dynamical networks. Physica A.2007,375:345-354
[158] Wang X F, Chen G R. Synchronization in scale-free dynamical networks: robustness and fragility. IEEE Trans.Circuits Syst.I.2002, 49:54-62
[159] Li X, Chen G R.synchronization and desynchronization of complex dynamical networks: an engineering viewpoint. IEEE Trans.Circuits Syst.I.2003, 50: 1381-1390
[160] Belykh V N, Belykh I V, Hasler M. Connection graph stability method for synchronized coupled chaotic systems. Physica D.2004, 159-187
[161] Belykh I V, Belykh V N, Hasler M.Synchronization in asymmetrically coupled networks with node balance. Chaos.2006, 16:015102
[162] LüJ H, Yu X H, Chen G R et al. Characterizing the synchronizability of small-world dynamical networks. IEEE Trans.Circuits Syst.I.2004, 51:787-796
[163] Grigoriev R O, Cross M C, Schuster H G. Pinning Control of Spatiotemporal Chaos. Phys. Rev. Lett. 1997,79:2795-2798
[164] Wang X F, Chen G R. Pinning control of scale-free dynamical networks. Physica A.2002, 310:521-531
[165] Xiang L Y, Liu Z X, Chen Z Q et al. Pinning control of complex dynamical networks with general topology. Physica A.2007, 379:298-306
[166] Li X, Wang X F, Chen G R. Pinning a complex dynamical network to its equilibrium. IEEE Trans.Circuits Syst.I.2004, 51:2074-2087
[167] Hu G, Xiao J H, Gao J H. Analytical study of spatiotemporal chaos control by applying local injections. Phys. Rev. E.2000, 62: 3043 -3046
[168] Hu M F, Yang Y Q, Xu Z Y et al. Projective synchronization in drive-response dynamical networks. Physica A. 2007,381:457-466
[169] Ma Z J, Liu Z R, Zhang G. A new method to realize cluster synchronization in connected chaotic networks. Chaos.2006, 16:023103
[170]马忠军.混沌动力学网络上的几类同步现象研究[D]:[博士学位论文].上海:上海大学,2006
[171]匡继昌.常用不等式[M].(第三版)济南:山东科学技术出版社.2004,447-458
[172] Song Q K, Cao J D. Synchronization and anti-synchronization for chaotic systems. Chaos, Solitons and Fractals.2007, 33:929-939
[173] Li C D, Liao X F. Anti-synchronization of a class of coupled chaotic systems via linear feedback control. International Journal of Bifurcation and Chaos.2006, 16:1041-1047
[174] Song Q K, Cao J D. Synchronization and anti-synchronization for chaotic systems. Chaos, Solitons and Fractals.2007, 33:929-939
[175] Kreuz T, Mormann F, Andrzejak R G et al. Measuring synchronization in coupled model systems: A comparison of different approaches. Physica D.2007, 225:29-42
[176] Li G H, Zhou S P. Anti-synchronization in different chaotic systems. Chaos, Solitons and Fractals.2007, 32:516-520
[177]胡满峰,徐振源,Lorenz混沌系统的非线性反馈错位同步控制[J].系统工程与电子技术,2007,29(8):1346-1348
[178]陈骏,刘曾荣.不同系统之间的部分同步[J].应用数学和力学,2005,26(9): 1033-1037
[179] Lu J, Xi Y. Linear generalized synchronization of continuous-time chaotic systems. Chaos, Solitons and Fractals.2003, 17(5):825-831
[180] Yang T, Chua L O. Generalized synchronization of chaos via linear transformations. International Journal of Bifurcation and Chaos.1999, 9(1): 215-219
[181]胡爱花,徐振源,李芳.基于自适应控制的混沌系统的线性广义同步化[J].系统仿真学报,2006,18(4): 975-977,981
[182] Rubezic V, Lutovac B, Ostojic R. Linear generalized synchronization of two chaotic Colpitts oscillators. Electronics, Circuits and Systems, 2002. 9th International Conference on. 2002,1:223-225
[183]刘曾荣.关于同步的几个理论问题[J].自然,2004, 26(5): 298-300
[184] Yassen M T. Chaos synchronization between two different chaotic systems using active control. Chaos, Solitons and Fractals.2005, 23:131-140
[185] Femat R, Alvarez-Ramírez J. Synchronization of a class of strictly different chaotic oscillators.Phys.Lett.A.1997, 236:307-313
[186] Sun F Y. Global chaos synchronization between two new different chaotic systems via active control. Chinese Physics Letters.2006, 23:32-34
[187] Li R H, Xu W, Li S. Adaptive generalized projective synchronization in different chaotic systems based on parameter identification.Phys.Lett.A.2007, 367:199-206
[188] Femat R, Solís-Perales G. Synchronization of chaotic systems with different order. Phys. Rev. E.200265:036226
[189] Lazzouni S A, Bowong S, Kakmeni F M et al. An adaptive feedback control for chaos synchronization of nonlinear systems with different order. Communications in Nonlinear Science and Numerical Simulation.2007, 12(4):568-583
[190] Bowong S. Stability analysis for the synchronization of chaotic systems with different order: application to secure communications.Phys.Lett.A. 2004, 326:102-113
[191] Ge Z M, Chang C M, Chen Y S. Anti-control of chaos of single time scale brushless dc motors and chaos synchronization of different order systems. Chaos, Solitons and Fractals. 2006, 27:1298-1315
[192] Chen J Y, Wong K W, Chen Z X et al. Phase synchronization in discrete chaotic systems. Phys. Rev. E .2000,61:2559-2562
[193] Rosenblum M G, Pikovsky A S, Kurths J. Comment on“Phase synchronization in discrete chaotic systems”. Phys. Rev. E.2001, 63:058201
[194] Zheng Y A, Nian Y B, Liu Z R. Impulsive Synchronization of Discrete Chaotic Systems. Chinese Physics Letters.2003, 20:199-201
[195] Zhang H B, Li C G, Liao X F. Observer-based synchronization for piecewise-linear discrete chaotic systems based on piecewise quadratic Lyapunov functions. International Journal of Bifurcation and Chaos.2005,19:4389-4397
[196] Lian K Y, Liu P, Chiu C S et al. Robust dead-beat synchronization and communication for discrete-time chaotic systems. International Journal of Bifurcation and Chaos. 2002, 12: 835-846
[197]陈菊芳,程丽,刘颖等.延迟变量反馈法控制离散混沌系统的电路实验[J].物理学报, 2003, 52(1):18-24
[198]陈菊芳,张入元,彭建华.脉冲驱动离散混沌系统同步的实验与理论研究[J].物理学报, 2003, 52(7):1589-1594
[199]刘丁,钱富才,任海鹏等.离散混沌系统的最小能量控制[J].物理学报,2004, 53 (7) : 2074-2079
[200] Atay F M, Jost J, Wende A. Delays, connection topology, and synchronization of coupled chaotic maps. Phys. Rev. Lett. 2004,92:144101
[201] Minai A A, Anand T. Synchronization of chaotic maps through a noisy coupling channel with application to digital communication. Phys. Rev. E.1999, 59:312-320
[202] De Oliveira A G, Jones A J. Synchronization of Chaotic Maps by Feedback Control and Application to Secure Communications Using Chaotic Neural Networks. International Journal of Bifurcation and Chaos.1998,8: 2225-2237
[203] T?rres L A B. Discrete-time dynamic systems synchronization: Information transmission and model matching, Physica D. 2007,228:31-39
[204] Brzozowska-Rup K, Or?owski A. Application of bootstrap to detecting chaos in financial time series. Physica A. 2004, 344:317-321
[205]吕金虎,陆君安,陈士华.混沌时间序列分析及其应用[M].武汉:武汉大学出版社.2002.
[206] Huang D B. Synchronization-based estimation of all parameters of chaotic systems from time series. Phys. Rev. E.2004, 69:067201
[207] Huang D B, Guo R W. Identifying parameter by identical synchronization between different systems. Chaos.2004, 14:152-159
[208] Konnur R. Synchronization-based approach for estimating all model parameters of chaotic systems. Phys. Rev. E. 2003,67: 027204
[209] Li S, Xu W, Li R et al. A general method for chaos synchronization and parameters estimation between different systems. Journal of Sound and Vibration.2007,302:777-788
[210] Grosu I. Estimation of model parameters from time series by OPCL auto-synchronization. Journal of Bifurcation and Chaos. 2004,14:2133-2141
[211] Sun F Y, Zhao Y, Zhou T S. Identify fully uncertain parameters and design controllers based on synchronization. Chaos, Solitons and Fractals.2007,34:1677-1682
[212] Yang X L, Xu W, Sun Z K. Estimating model parameters in nonautonomous chaotic systems using synchronization. Phys.Lett.A.2007,364: 378-388
[213] Rakshit B, Chowdhury A. R, Saha P. Parameter estimation of a delay dynamical system usingsynchronization in presence of noise. Chaos, Solitons and Fractals.2007,32: 1278-1284
[214] Ge Z M, Lee J K. Chaos synchronization and parameter identification for gyroscope system. Applied Mathematics and Computation. 2005, 163:667-682
[215] Lin W, Chen G R. Using white noise to enhance synchronization of coupled chaotic systems. Chaos. 2006,16:013134
[216] Kiss I Z, Zhai Y M, Hudson J L et al. Noise enhanced phase synchronization and coherence resonance in sets of chaotic oscillators with weak global coupling.Chaos.2003,13:267-278
[217] Park K, Lai Y C, Krishnamoorthy S et al. Effect of common noise on phase synchronization in coupled chaotic oscillators. Chaos. 2007,17:013105
[218] Sánchez E, Matías M A, Pérez-Mu?uzuri V. Analysis of synchronization of chaotic systems by noise: An experimental study. Phys. Rev. E. 1997,56:4068-4071
[219]胡爱花,徐振源.利用白噪声实现混沌系统线性广义同步的研究[J].物理学报,2007, 56(6): 3132-3136
[220] Bowong S, McClintock P V E. Adaptive synchronization between chaotic dynamical systems of different order. Phys.Lett.A. 2006,358:134-141
[221]王发强,刘崇新.不同阶混沌系统的自适应同步控制[J].华北电力学学报,2005, 32:11-14
[222] Zhang G, Liu Z R, Ma Z J. Generalized synchronization of different dimensional chaotic dynamical systems. Chaos, Solitons and Fractals.2007,32: 773-779
[223] Gao T G, Chen Z Q, Yuan Z Z. Generalized reduced-order synchronization of chaotic system based on fast slide mode. Chinese Physics. 2005, 14(12):2421-2427
[224] Li R H, Xu W, Li S. Chaos control and reduced-order generalized synchronization for the Chen--Liao system. Chinese Physics. 2007, 16(6):1591-1596
[225] Kakmeni F M M, Bowong S, Tchawoua C. Reduced-order synchronization of uncertain chaotic systems via adaptive control. Communications in Nonlinear Science and Numerical Simulation.2006,11:810-830
[226] Ho M C, Hung Y C, Liu Z Y et al. Reduced-order synchronization of chaotic systems with parameters unknown. Phys.Lett.A. 2006, 348:251-259
[227] Mahmoud G M, Aly S A, Al-Kashif M A. Dynamical properties and chaos synchronization of a new chaotic complex nonlinear system. Nonlinear Dyn. 2007, (In press) doi:10.1007/s11071-007-9200-y
[228] Yang X S. Concepts of synchronization in dynamical systems. Phys. Lett. A. 1999,260: 340-344
[229] Yan Z Y. Q-S synchronization in 3D Henon-like map and generalized Henon map via a scalar controller. Phys.Lett.A.2005,342:309-317
[230] Yan Z Y. Q-S (complete or anticipated) synchronization backstepping scheme in a class of discrete-time chaotic (hyperchaotic) systems: A symbolic-numeric computation approach. Chaos, 2005,16: 013199
[231]方保镕,周继东,李医民.矩阵论[M].北京:清华大学出版社.2004.
[232] Yu W W, Cao J D. Adaptive Q-S (lag, anticipated, and complete) time-varying synchronization and parameters identification of uncertain delayed neural networks. Chaos. 2006, 16: 023119
[233] Wang Q, Chen Y. Generalized Q-S (lag, anticipated, and complete) synchronization in modified Chua’s circuit and Hindmarsh-Rose systems. Applied Mathematics and Computation. 2006, 181:48-56
[234] Hu M F. Xu Z Y. A general scheme for Q-S synchronization of chaotic systems. Nonlinear Analysis. 2007, In press
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