几个新连续混沌系统的分析与控制
|
摘要
本文针对一类特殊的Loenz型自治微分方程,提出了一个基于Lyapunov理论的辅助搜索方法。主要思想是通过构造满足一定条件的Lyapunov函数的导数,进而反推出系统方程的形式。和已有的混沌化方法比较,新方法简便可行,有一定的理论依据和系统性,大大缩小了在复杂数学模型中搜寻混沌行为的范围,并容易推广到高维系统。利用这个方法,发现了3个新的Lorenz型混沌系统。新系统都是三维二次自治耗散系统,其线性部分系数矩阵的主对角元都是负数,且有三个或四个非线性项。因此和已有的Lorenz型混沌系统比如Lorenz系统、Ro¨ssler系统、Chen系统、Lu¨系统和LC系统类似,但有明显的区别。利用平衡点的局部和全局稳定性分析、系统相轨迹、Lyapunov指数、Poincare′映象、分岔图等理论和数值方法详细分析了新混沌系统的动力学行为和特征。结果显示,第一个新系统具有不动点与极限环、不动点与混沌吸引子以及极限环与混沌吸引子共存的现象,第二个系统有一个三螺旋吸引子,而第三个系统具有复杂的多层锥形结构。因此新混沌系统的吸引子拓扑结构较为复杂。对于第一个和第二个系统,给出了全局指数吸引集,并进行了严格的数学证明,这个结论在一般混沌系统中是很难得到的。
研究了新混沌系统和LC系统不稳定平衡点的镇定。讨论了基于线性和非线性反馈实现系统不稳定平衡点的指数镇定,得到了一系列简便的代数充分条件。然后针对参数不确定情形,研究了基于自适应方法的混沌控制问题,应用Lyapunov稳定性理论,给出了针对一般混沌系统平衡点自适应控制的渐近稳定的条件。并且研究了脉冲方法镇定系统不稳定平衡点的控制器设计策略,给出了系统稳定时脉冲间隔与控制器增益的关系。
研究了新混沌系统和LC系统的同步。设计了非线性反馈控制器,保证两个具有不同初值的相同系统实现全局指数同步,得到了一系列简便的代数同步判据。当系统参数未知时,基于自适应控制策略,得到了系统全局渐近同步的代数充分条件和控制器设计方法;考虑了如果系统参数未知且是时变的,基于自适应控制和滑模控制思想以及参数变化有界但界未知的假设,设计了新的自适应滑模型控制器,实现了系统的鲁棒同步。
讨论了几个时滞混沌系统动力学特性。利用超越特征方程讨论了时滞logistic模型平衡点与时滞相关和与时滞无关的局部稳定性判据,并针对时滞参数用中心流形定理和规范形理论确定了Hopf分岔的方向和首个Hopf分岔点处分岔周期解的稳定性。提出了一个新的四神经元时滞混沌神经网络新模型,基于Lyapunov稳定性理论和数值动力学分析方法分析了系统的动力学行为。最后介绍了一个简单中立型混沌神经网络,就我们所掌握的材料,直至目前尚未有中立型混沌系统的研究报道,由于理论分析和数值分析都有很大困难,这个系统还有待进一步深入研究。
Based on Lyapunov theory, an new auxiliary search method is put forward for aspecial kind of Lorenz-type autonomous differential equations. The basic idea is toobtain the expected form of system equations, by means of constructing the derivativeof Lyapunov function which satisfies some determinate conditions. The new methodis very simple and convenient. The searching range is extremely reduced for findingchaotic behavior in complex mathematical model. Utilizing the new method, three newLorenz-type chaotical system is found. They are three dimensional quadratic dissipa-tive system with three or four quadratic terms and negative main diagonal elements ofcoefficient matrix in the linear part of the system. Therefore, they are similar to theexisting Lorenz-type systems, such as Lorenz system, Ro¨ssler system, Chen system, lu¨system and LC system, but explicitly different from them. The dynamical behaviors areinvestigated in terms of local and global stability analysis, phase trajectories, lyapunovexponents, Poincare′mapping and bifurcation diagrams. The results show there is co-existence phenomena of stable equilibrium, limit loop and chaotic attractor in the firstsystem, a 3-scroll chaotical attractor in the second, and a multilayer taper attractor in thethird. However, there are more complex chaotic attractors in the new systems. Further-more, the global attractive sets are found for the first and the second system, and strictmathematical proofs are given.
Stabilization for unstable equilibria of the new systems and LC system is studied.Globally exponential stabilization of euilibria are discussed based on linear or nonlin-ear state feedback, and a series of simple algebraic criteria are obtained. The adaptivecontrol technique of the systems with unknown parameters is considered. Furthermore,by means of Lyapunov stability theory, an asymptotically stable criterion for a generalchaotical system is obtained. Finally, approach of controller design for chaos controlvia impulsive control technique is discussed, and the relation of impulse interval andcontroller gain is given if the controlled system is stable.
The chaos synchronization of the new systems and LC system is investigated. Non-linear feedback controllers are presented to achieve globally exponential synchronization for two same chaotic systems with different initial conditions. Several algebraic suffi-cient conditions are developed. Adaptive synchronization strategy is introduced whenthe parameters of systems are all unknown. An adaptive controller is designed, whichcan guarantee globally asymptotic synchronization. If the unknown parameters varywith time, assuming that the varying parameters are bounded but the boundedness is un-known, the adaptive sliding mode type controller is designed, and globally asymptoticrobust synchronization is also realized.
Dynamical behaviors of several time-delayed chaotical system is studied. Basedon transcendant characteristic equation associated with the Halanay inequality, the localstability criteria, delay independent or delay dependent, is obtained for the equilibria ofdelay logistic equation. The stability of bifurcation periodic solutions and the directionof Hopf bifurcation are determined by applying the normal form theory and the centermanifold theorem. Chaotic behavior of the parameterized Logistic differential systemswith a single delay is detected by numerical examples. A new chaotical time-delayrecurrent neural network with four neurons is put forward. Its dynamical propertyis investigated via both Lyapunov theory and numerical simulation method. Finally,a simple neutral chaotical system is presented. To our best knowledge, no neutralchaotical system is reported by far.
研究了新混沌系统和LC系统不稳定平衡点的镇定。讨论了基于线性和非线性反馈实现系统不稳定平衡点的指数镇定,得到了一系列简便的代数充分条件。然后针对参数不确定情形,研究了基于自适应方法的混沌控制问题,应用Lyapunov稳定性理论,给出了针对一般混沌系统平衡点自适应控制的渐近稳定的条件。并且研究了脉冲方法镇定系统不稳定平衡点的控制器设计策略,给出了系统稳定时脉冲间隔与控制器增益的关系。
研究了新混沌系统和LC系统的同步。设计了非线性反馈控制器,保证两个具有不同初值的相同系统实现全局指数同步,得到了一系列简便的代数同步判据。当系统参数未知时,基于自适应控制策略,得到了系统全局渐近同步的代数充分条件和控制器设计方法;考虑了如果系统参数未知且是时变的,基于自适应控制和滑模控制思想以及参数变化有界但界未知的假设,设计了新的自适应滑模型控制器,实现了系统的鲁棒同步。
讨论了几个时滞混沌系统动力学特性。利用超越特征方程讨论了时滞logistic模型平衡点与时滞相关和与时滞无关的局部稳定性判据,并针对时滞参数用中心流形定理和规范形理论确定了Hopf分岔的方向和首个Hopf分岔点处分岔周期解的稳定性。提出了一个新的四神经元时滞混沌神经网络新模型,基于Lyapunov稳定性理论和数值动力学分析方法分析了系统的动力学行为。最后介绍了一个简单中立型混沌神经网络,就我们所掌握的材料,直至目前尚未有中立型混沌系统的研究报道,由于理论分析和数值分析都有很大困难,这个系统还有待进一步深入研究。
Based on Lyapunov theory, an new auxiliary search method is put forward for aspecial kind of Lorenz-type autonomous differential equations. The basic idea is toobtain the expected form of system equations, by means of constructing the derivativeof Lyapunov function which satisfies some determinate conditions. The new methodis very simple and convenient. The searching range is extremely reduced for findingchaotic behavior in complex mathematical model. Utilizing the new method, three newLorenz-type chaotical system is found. They are three dimensional quadratic dissipa-tive system with three or four quadratic terms and negative main diagonal elements ofcoefficient matrix in the linear part of the system. Therefore, they are similar to theexisting Lorenz-type systems, such as Lorenz system, Ro¨ssler system, Chen system, lu¨system and LC system, but explicitly different from them. The dynamical behaviors areinvestigated in terms of local and global stability analysis, phase trajectories, lyapunovexponents, Poincare′mapping and bifurcation diagrams. The results show there is co-existence phenomena of stable equilibrium, limit loop and chaotic attractor in the firstsystem, a 3-scroll chaotical attractor in the second, and a multilayer taper attractor in thethird. However, there are more complex chaotic attractors in the new systems. Further-more, the global attractive sets are found for the first and the second system, and strictmathematical proofs are given.
Stabilization for unstable equilibria of the new systems and LC system is studied.Globally exponential stabilization of euilibria are discussed based on linear or nonlin-ear state feedback, and a series of simple algebraic criteria are obtained. The adaptivecontrol technique of the systems with unknown parameters is considered. Furthermore,by means of Lyapunov stability theory, an asymptotically stable criterion for a generalchaotical system is obtained. Finally, approach of controller design for chaos controlvia impulsive control technique is discussed, and the relation of impulse interval andcontroller gain is given if the controlled system is stable.
The chaos synchronization of the new systems and LC system is investigated. Non-linear feedback controllers are presented to achieve globally exponential synchronization for two same chaotic systems with different initial conditions. Several algebraic suffi-cient conditions are developed. Adaptive synchronization strategy is introduced whenthe parameters of systems are all unknown. An adaptive controller is designed, whichcan guarantee globally asymptotic synchronization. If the unknown parameters varywith time, assuming that the varying parameters are bounded but the boundedness is un-known, the adaptive sliding mode type controller is designed, and globally asymptoticrobust synchronization is also realized.
Dynamical behaviors of several time-delayed chaotical system is studied. Basedon transcendant characteristic equation associated with the Halanay inequality, the localstability criteria, delay independent or delay dependent, is obtained for the equilibria ofdelay logistic equation. The stability of bifurcation periodic solutions and the directionof Hopf bifurcation are determined by applying the normal form theory and the centermanifold theorem. Chaotic behavior of the parameterized Logistic differential systemswith a single delay is detected by numerical examples. A new chaotical time-delayrecurrent neural network with four neurons is put forward. Its dynamical propertyis investigated via both Lyapunov theory and numerical simulation method. Finally,a simple neutral chaotical system is presented. To our best knowledge, no neutralchaotical system is reported by far.
引文
[1] Lorenz E N. Deterministic nonperiodic ?ow . Journal of the Atmospheric Sciences,1963, 20(3):130–141
[2] Li T, Yorke J A. Period three implies chaos . American Mathematical Monthly, 1975,8(10):985–992
[3] May R M. Simple mathematical models with very complicated dynamics . Nature,1976, 261(5560):459–467
[4] May R M, Oster G F. Bifurcations and Dynamic Complexity in Simple EcologicalModels . The American Naturalist, 1976, 110(974):573–599
[5] Feigenbaum M J. Quantitative universality for a class of non-linear transformations .Journal of Statistical Physics, 1992, 19(1):25–52
[6] Mandelbrot B B. Fractal Geometry of Nature . California: W. H. Freeman Company,1982
[7] Matsumoto T. A chaotic attractor from Chua’s circuit . IEEE Transactions on Circuitsand Systems, 1984, 31:1055–1058
[8] Ott E, Grebogi C, Yorke J A. Controlling chaos . Physical Review Letters, 1990,64(11):1196–1199
[9] Pecora L M, Carroll T L. Synchronization in chaotic systems . Physical ReviewLetters, 1990, 64(8):821–824
[10] Ditto W L, Showalter K. Introduction: Control and synchronization of chaos . Chaos,1997, 7(4):509–511
[11] Hubler A, Lusher E. Resonant stimulation and control of nonlinear oscillators . Natur-wissenschaft, 1989, 76(1):67–72
[12] M. P. Controlling chaos through parametric exitations . in: Lima R, Streit I, Mendes RV, editors, Proceedings of Dynamics and Stochastic Processes, New York: Springer-Verlag, 1988, 242-250
[13] Fradkov A L, Evans R J. Control of chaos: Methods and applications in engineering .Annual Reviews in Control, 2005, 29(1):33–56
[14]陈关荣,吕金虎. Lorenz系统族的动力学分析、控制与同步.北京:科学出版社,2002
[15] Pyragas. Generalized synchronization of chaos . Physics Letters A, 1993, 4(2):210–221
[16] Genesio R, Tesi A. Harmonic balance methods for the analysis of chaotic dynamicsin nonlinear systems . Automatica, 1992, 28(3):531–548
[17] Hsu R R, Su H T, Chern J L, et al. Conditions to control chaotic dynamics by weakperiodic perturbation . Physics Review Letters, 1997, 78:2936–2939
[18] Jackson E A, Hubler A W. Periodic entrainment of chaotic logistic map dynamics .Physica D, 1990, 44(3):407–420
[19] Jackson E A. The entrainment and migration contols of multiple attractor systems .Physics Letters A, 1990, 151:478–484
[20] Jackson E A, Kodogeorgiou A. Entrainment and migration controls of two dimen-sional maps . Physica D, 1991, 54(3):253–265
[21] Frison T W. Controlling chaos with a neural network. Neural Networks, 1992.IJCNN., International Joint Conference on , 1992, 2(7-11):75–80
[22] Lin C, Jou C. Controlling chaos by GA-based reinforcement learning neural network .Neural Networks, IEEE Transactions on, 1999, 10(4):846–859
[23] Hunt E R. Stabilizing high-period orbits in a chaotic system-The diode resonator .Physics Review Letters, 1991, 67:1953–1955
[24] Ushio T. Limitation of delayed feedback control in nonlinear discrete-timesystems. Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactionson , 1996, 43(9):815–816
[25] Kittel A, Parisi J, Pyragas K. Delay feedback control of chaos by self-adaptive delaytime . Physics Letters A, 1995, 198:433–436
[26] Pyragas K. Control of chaos via extended delay feedback . Physics Letters A, 1995,206(5):323–330
[27] Ahlborn A, Parlitz U. Stabilizing unstable steady states using multiple delay feedbackcontrol . Physical Review Letters, 2004, 93(26):264101
[28] Nakajima H. On analytical properties of delayed feedback control of chaos . PhysicsLetters A, 1997, 232:207–210
[29] Lai Y C, Grebogi C. Converting transient chaos into sustained chaos by feedbackcontrol . Physical Review E, 1994, 49(2):1094–1098
[30] Brandt M E, Chen G. Feedback control of a quadratic map model of cardiac chaos .International Journal of Bifurcation and Chaos, 1996, 6:715–723
[31] Rulkov N, Tsimring L, Abarbanel H. Tracking unstable orbits in chaos using dissi-pative feedback control . Physical Review E, 1994, 50(1):314–324
[32] Hu G, Qu Z, He K. Feedback control of chaos in spatiotemporal systems . Interna-tional Journal of Bifurcation and Chaos, 1995, 5(4):901–936
[33] Cai C, Xu Z, Xu W. Converting chaos into periodic motion by state feedback control.Automatica, 2002, 38(11):1927–1933
[34] Yassen M. Controlling chaos and synchronization for new chaotic system using linearfeedback control . Chaos, Solitons and Fractals, 2005, 26:913–920
[35] Yagasaki K, Yamashita S. Controlling chaos using nonlinear approximations for apendulum with feedforward and feedback control . Int. J. Bifurcation and Chaos,1999, 9(1):233–241
[36] Jiang G, Chen G, Tang K. A new criterion for chaos synchronization using linear statefeedback control . International Journal of Bifurcation and Chaos, 2003, 13(8):2343–2351
[37] Fang J Q, Ali M K. Nonlinear feedback control of spatiotemporal chaos in coupledmap lattices . Discrete Dynamics in Nature and Society, 1998, 1(2):283–305
[38] Andrievskii B R, Fradkov A L. Control of chaos: Methods and applications. I. Meth-ods . Automation and Remote Control, 2003, 64(5):673–713
[39] Araujo A D, Singh S N. Output feedback adaptive variable structure control of chaosin Lorenz system . International Journal of Bifurcation and Chaos, 2002, 12(3):571–582
[40] Arecchi F T, Boccaletti S, Ciofini M, et al. The control of chaos: Theoretical schemesand experimental realizations . International Journal of Bifurcation and Chaos, 1998,8:1643–1655
[41] Boccaletti S, Arecchi F T. Adaptive control of chaos . Europhysics letters, 1995,31(3):127–132
[42] Boccaletti S, Grebogi C, Lai Y C, et al. The control of chaos: theory and applications .Physics Reports, 2000, 329(3):103–197
[43] Ge S S. Adaptive control of uncertain lorenz system using decoupled backstepping .International Journal of Bifurcation and Chaos, 2004, 14(4):1439–1445
[44] Ge S S, Wang C. Adaptive control of uncertain Chua’s circuits . Circuits and SystemsI: Fundamental Theory and Applications, IEEE Transactions on [see also Circuits andSystems I: Regular Papers, IEEE Transactions on], 2000, 47(9):1397–1402
[45] Hogg T, Huberman B A. Controlling chaos in distributed systems . Systems, Manand Cybernetics, IEEE Transactions on, 1991, 21(6):1325–1332
[46] Hua C, Guan X. Adaptive control for chaotic systems . Chaos, Solitons and Fractals,2004, 22(1):103–110
[47] Huang D. Stabilizing Near-Nonhyperbolic Chaotic Systems with Applications . Phys-ical Review Letters, 2004, 93(21):214101
[48] Huang D. Synchronization-based estimation of all parameters of chaotic systemsfrom time series . Physical Review E, 2004, 69(6):67201
[49] Huang D. Simple adaptive-feedback controller for identical chaos synchronization .Physical Review E, 2005, 71(3):37203
[50] Huang D, Guo R. Identifying parameter by identical synchronization between dif-ferent systems . Chaos: An Interdisciplinary Journal of Nonlinear Science, 2003,14:152
[51] Li Z, Chen G, Shi S, et al. Robust adaptive tracking control for a class of uncertainchaotic systems . Physics Letters A, 2003, 310(1):40–43
[52] Lindner J F, Ditto W L. Removal, suppression, and control of chaos by nonlineardesign . Applied Mechanics Review, 1995, 48(12):795–807
[53] Mascolo S, Grassi G. Controlling chaos via backstepping design . Physical ReviewE, 1997, 56(5):6166–6169
[54] Paskota M, Mees A I, Teo K L. On control of chaos: Higher periodic orbits . Dynam-ics and Control, 1995, 5(4):365–387
[55] Petrov V, Michael F C, Showalter K. An adaptive control algorithm for tracking unsta-ble periodic orbits . International Journal of Bifurcation and Chaos, 1994, 4(5):1311–1317
[56] Pyragas K, Pyragas V, Kiss I Z, et al. Adaptive control of unknown unstable steadystates of dynamical systems . Physical Review E, 2004, 70(2):26215
[57] Tian Y, Furong G. Adaptive control of chaotic continuous-time systems with delay .Physica D, 1998, 117(1):1–12
[58] Tian Y P, Yu X. Adaptive control of chaotic dynamical systems using invariant mani-fold approach . Circuits and Systems I: Fundamental Theory and Applications, IEEETransactions on], 2000, 47(10):1537–1542
[59] Vassiliadis D. Parametric adaptive control of an NMR laser model . InternationalJournal of Bifurcation and Chaos, 1993, 3(3):793–796
[60] Wang C, Ge S S. Adaptive backstepping control of uncertain Lorenz system . Inter-national Journal of Bifurcation and Chaos, 2001, 11(4):1115–1119
[61] Wu T, Chen M S. Chaos control of the modified Chua’s circuit system . Physica D,2002, 164(1-2):53–58
[62] Yang T, Yang C M, Yang L B. A Detailed Study of Adaptive Control of ChaoticSystems with Unknown Parameters . Dynamics and Control, 1998, 8(3):255–267
[63] Yassen M T. Adaptive control and synchronization of a modified Chua’s circuit sys-tem . Applied Mathematics and Computation, 2003, 135(1):113–128
[64] Cao Y J, Zhang H X. An adaptive strategy for controlling chaotic system . JOURNALOF ZHEJIANG UNIVERSITY (SCIENCE), 2003, 4(3):258–263
[65] Yu X. Tracking inherent periodic orbits in chaotic dynamic systems viaadaptive vari-able structure time-delayed self control . Circuits and Systems I: Fundamental Theoryand Applications, IEEE Transactions on [see also Circuits and Systems I: Regular Pa-pers, IEEE Transactions on], 1999, 46(11):1408–1411
[66] Zeng Y, Singh S N. Adaptive Control of Chaos in Lorenz System . Dynamics andControl, 1997, 7(2):143–154
[67] Zhang H, Qin H, Chen G. Adaptive control of chaotic systems with uncertainties .International Journal of Bifurcation and Chaos, 1998, 8(10):2041–2046
[68] Huberman B A, Lumer E. Dynamics of adaptive systems . Circuits and Systems I:Fundamental Theory and Applications, IEEE Transactions on, 1990, 37(4):547–550
[69] Pecora L M, Carroll T L. Driving systems with chaotic signals . Physical Review A,1991, 44(4):2374–2383
[70] He R, Vaidya P. Analysis and synthesis of synchronous periodic and chaotic systems. Physical Review A, 1992, 46(12):7387–7392
[71]陈志盛,孙克辉,张泰山. Liu混沌系统的非线性反馈同步控制.物理学报, 2005,54(6):2580–2583
[72]陈保颖.线性反馈实现Liu系统的混沌同步.动力学与控制学报, 2006, 4(1):1–4
[73] Jiang G P, Tang W, Chen G R. A simple global synchronization criterion for coupledchaotic systems . Chaos, Solitons and Fractals, 2003, 15(5):925–935
[74] Chen S, Lu J. Parameters identification and synchronization of chaotic systems basedupon adaptive control . Physics Letters A, 2002, 299(4):353–358
[75] Dai D, Ma X K. Chaos synchronization by using intermittent parametric adaptivecontrol method . Physics Letters A, 2001, 288(1):23–28
[76] Elabbasy E M, Agiza H N, El-Dessoky M M. Adaptive synchronization of a hy-perchaotic system with uncertain parameter . Chaos, Solitons & Fractals, 2006,30(5):1133–1142
[77] Elabbasy E M, Agiza H N, El-Dessoky M M. Adaptive synchronization of Lu systemwith uncertain parameters . Chaos, Solitons & Fractals, 2004, 21(3):657–667
[78] Han X, Lu J A, Wu X. Adaptive feedback synchronization of Lu system . Chaos,Solitons & Fractals, 2004, 22(1):221–227
[79] Hu J, Chen S, Chen L. Adaptive control for anti-synchronization of Chua’s chaoticsystem . Physics Letters A, 2005, 339(6):455–460
[80] Li Z, Shi S. Robust adaptive synchronization of Rossler and Chen chaotic systemsvia slide technique . Physics Letters A, 2003, 311(4-5):389–395
[81] Moukam Kakmeni F M, Bowong S, Tchawoua C. Nonlinear adaptive synchronizationof a class of chaotic systems . Physics Letters A, 2006, 355(1):47–54
[82] Park J H. Adaptive synchronization of hyperchaotic Chen system with uncertainparameters . Chaos, Solitons & Fractals, 2005, 26(3):959–964
[83] Ren Q, Zhao J. Impulsive synchronization of coupled chaotic systems via adaptive-feedback approach . Physics Letters A, 2006, 355(4-5):342–347
[84] Yan Z, Yu P. Linear feedback control, adaptive feedback control and their combi-nation for chaos (lag) synchronization of LC chaotic systems . Chaos, Solitons &Fractals, 2007, 33(2):419–435
[85] Yassen M T. Adaptive chaos control and synchronization for uncertain new chaoticdynamical system . Physics Letters A, 2006, 350(1-2):36–43
[86] Yassen M T. Feedback and adaptive synchronization of chaotic Lu system . Chaos,Solitons & Fractals, 2005, 25(2):379–386
[87] Yu W, Cao J. Adaptive synchronization and lag synchronization of uncertain dynam-ical system with time delay based on parameter identification . Physica A: StatisticalMechanics and its Applications, 2007, 375(2):467–482
[88] Yu Y, Zhang S. Adaptive backstepping synchronization of uncertain chaotic system .Chaos, Solitons & Fractals, 2004, 21(3):643–649
[89] Zhang H, Huang W, Wang Z, et al. Adaptive synchronization between two differentchaotic systems with unknown parameters . Physics Letters A, 2006, 350(5-6):363–366
[90] Yang X S, Chen G. Some observer-based criteria for discrete-time generalized chaossynchronization . Chaos, Solitons and Fractals, 2002, 13(6):1303–1308
[91] Li K, Yang L X, He Z Y, et al. Sporadic driving chaos synchronization based onnonlinear observer. Acta Automatica Sinica, 2001, 27(2):280–283
[92] Li K, Yang L, He Z Y. Stability analysis of nonlinear observer with application tochaos synchronization . Science in China (Information Sciences), 2001, 44(6):430–447
[93] Li G H, Zhou S P, Xu D M. Chaos synchronization based on intermittent state ob-server . Chinese Physics, 2004, 13(2):168–172
[94] Celikovsky S. Observer form of the hyperbolic-type generalized Lorenz system andits use for chaos synchronization . Kybernetika, 2004, 40(6):649–664
[95] Fotsin H, Kakmeni F, Bowong S. An adaptive observer for chaos synchronization ofa nonlinear electronic circuit . International Journal of Bifurcation and Chaos, 2006,16(9):2671–2679
[96] Sanchez E, Perez J, Ricalde L. Neural network design for chaos synchronization.Lecture Notes in Control and Information Sciences, 2004, 292:137–158
[97] Barajas-Ramirez J, Chen G, Shieh L. Fuzzy chaos synchronization via sampled driv-ing signals . International Journal of Bifurcation and Chaos, 2004, 14(8):2721–2733
[98] Yu G R. Fuzzy synchronization of chaos using gray prediction for secure commu-nications . Systems, Man and Cybernetics, 2003. IEEE International Conference on,2003, 4(5-8):3104–3109
[99] Liao X X, Chen G R. Chaos synchronization of general Lurie systems via time-delayfeedback control . Int. J. Bifurcation and Chaos, 2003, 13:207–213
[100]吴立刚,王常虹,曾庆双.滑模自适应控制实现一类不确定混沌系统的同步.控制与决策, 2006, 21(2):229–232
[101] Yau H T. Design of adaptive sliding mode controller for chaos synchronization withuncertainties . Chaos,Solitons and Fractals, 2004, 22(2):341–347
[102] Fan C, Jiang C, Fan C, et al. Backstepping based synchronization control for unifiedchaos systems . Harbin Gongye Daxue Xuebao, 2004, 36(11):1571–1574
[103] Hao Z, Xi-Kui M, Yu Y, et al. Generalized synchronization of hyperchaos and chaosusing active backstepping design . Chinese Physics, 2005, 14(1):86–94
[104] Quan M, Rong C, Dan Z, et al. Approach to Generalized Synchronization with Ap-plication to Chaos–Based Secure Communication . Communication in TheoreticalPhysics, 2004, 42(4):632–640
[105] Rulkov N F, Afraimovich V S, Lewis C T, et al. Multivalued mappings in generalizedchaos synchronization . Physical Review E, 2001, 64(1):16217
[106] Afraimovich V, Cordonet A, Rulkov N F. Generalized synchronization of chaos innoninvertible maps . Physical Review E, 2002, 66(1):16208
[107] Chen G, Liu S. On generalized synchronization of spatial chaos . Chaos, Solitonsand Fractals, 2003, 15(2):311–318
[108] Zhang X, Min L. A generalized chaos synchronization based encryption algorithmfor sound signal communication . Circuits, Systems, and Signal Processing, 2005,24(5):535–548
[109] Kittel A, Parisi J, Pyragas K. Generalized synchronization of chaos in electroniccircuit experiments . Physica D, 1998, 112(3-4):459–471
[110] Morgu¨l O¨. Necessary Condition for Observer-Based Chaos Synchronization. Physi-cal Review Letters, 1999, 82(1):77–80
[111] Yalcinkaya T, Lai Y C. Phase synchronization of chaos . Physical Review Letters,1997, 79:3885–3888
[112] Blasius B, Stone L. Chaos and phase synchronization in ecological systems . Int. J.Bifurcation Chaos, 2000, 10:2361–2380
[113] Santoboni G, Pogromsky A Y, Nijmeijer H. An observer for phase synchronizationof chaos . Physics Letters A, 2001, 291(4):265–273
[114] Zheng Z G, Hu G, Zhou C S, et al. Phase synchronization in coupled chaotic sys-tems - Transitions from high-to low-dimensional chaos . Acta Physica Sinica, 2000,49(12):2320–2327
[115] Feng X Q, Shen K. Phase synchronization and anti-phase synchronization of chaosfor degenerate optical parametric oscillator . Chinese Physics, 2005, 14(8):1526
[116] Rosa Jr E, Ott E, Hess M H. Transition to Phase Synchronization of Chaos . PhysicalReview Letters, 1998, 80(8):1642–1645
[117] Rosenblum M G, Pikovsky A S, Kurths J. From Phase to Lag Synchronization inCoupled Chaotic Oscillators . Physical Review Letters, 1997, 78(22):4193–4196
[118] Zhu S, Wu L. Anticipating and lag synchronization in chaotic laser system . Interna-tional Journal of Modern Physics B, 2004, 18(17-19):2547–2551
[119] Barsella A, Lepers C. Chaotic lag synchronization and pulse-induced transient chaosin lasers coupled by saturable absorber . Optics Communications, 2002, 205(4):397–403
[120] Boccaletti S, Valladares D L. Characterization of intermittent lag synchronization .Physical Review E, 2000, 62(5):7497–7500
[121] Taherion S, Lai Y C. Observability of lag synchronization of coupled chaotic oscil-lators . Physical Review E, 1999, 59(6):6247–6250
[122] Shahverdiev E M, Sivaprakasam S, Shore K A. Lag synchronization in time-delayedsystems . Phys. Lett. A, 2002, 292(6):320–324
[123] Zhan M, Wei G W, Lai C H. Transition from intermittency to periodicity in lag syn-chronization in coupled Ro¨ssler oscillators . Physical Review E, 2002, 65(3):36202
[124] Sosnovtseva O V, Balanov A G, Vadivasova T E, et al. Loss of lag synchronization incoupled chaotic systems . Physical Review E, 1999, 60(6):6560–6565
[125] Pazo D, Zaks M, Kurths J. Role of unstable periodic orbits in phase and lag synchro-nization between coupled chaotic oscillators . Chaos, 2003, 13(1):309–318
[126] Li C, Liao X, Wong K W. Chaotic lag synchronization of coupled time-delayedsystems and its applications in secure communication . Physica D: Nonlinear Phe-nomena, 2004, 194(3-4):187–202
[127] Taherion S, Lai Y C. Experimental observation of lag synchronization in coupledchaotic systems . Int. J. Bifurcation and chaos, 2000, 10:2587–2594
[128] Wu X F, Zhao Y, HUANG X H. Some new criteria for lag synchronization of chaoticLur’e systems by replacing variables control .控制理论与应用(英文版), 2004,2(3):259–266
[129] Zhao D Q, Liu Z R. Lag Synchronization in Nonlinear Systems Based on AdaptiveControl . Journal of Shanghai University (English Edition), 2004, 8(1):24–27
[130] Vieira M D E S, Khoury P, Lichtenberg A j, et al. Numerical and experimental studiesof self-synchronization and synchronized chaos . International Journal of Bifurcationand Chaos, 1992, 2(3):645–657
[131] Cuomo K, Oppenheim A. Circuit implementation of synchronization chaos withapplications to communications . Physical Review Letters, 1993, 71(1):65–68
[132] Belskiy Y U L, Dmitriyev A S. Transmission of information by means of determin-istic chaos . Journal of communications technology & electronics, 1993, 38(15):1–5
[133] Cruz J M, Chua L O. An IC chip of Chua’s circuit . Circuits and Systems II: Analogand Digital Signal Processing, IEEE Transactions on, 1993, 40(10):614–625
[134] Kolumban G, Kennedy M, Chua L. The role of synchronization in digital communica-tions using chaos. I. Fundamentals of digital communications . Circuits and SystemsI: Fundamental Theory and Applications, IEEE Transactions on [see also Circuits andSystems I: Regular Papers, IEEE Transactions on], 1997, 44(10):927–936
[135] Kolumban G, Kennedy M, Chua L. The role of synchronization in digital commu-nications using chaos. II. Chaotic modulation and chaotic synchronization . Circuitsand Systems I: Fundamental Theory and Applications, IEEE Transactions on, 1998,45(11):1129–1140
[136] Yang T, Chua L O. Secure communication via chaotic parameter modulation . Cir-cuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on,1996, 43(9):817–819
[137] Yang T, Yang L B, Yang C M. Breaking chaotic switching using generalized synchro-nization: examples . Circuits and Systems I: Fundamental Theory and Applications,IEEE Transactions on [see also Circuits and Systems I: Regular Papers, IEEE Trans-actions on], 1998, 45(10):1062–1067
[138] Alvarez G, Montoya F, Romera M, et al. Breaking two secure communication systemsbased on chaotic masking . Circuits and Systems II: Express Briefs, IEEE Transac-tions on, 2004, 51(10):505–506
[139]方锦清,罗晓曙.混沌保密通信应用研究的进展.广西师范大学学报:自然科学版, 2002, 20(1):6–18
[140]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(1):21–30
[141]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.物理学进展,2003, 23(2):212–255
[142] Devaney R L. Introduction to Chaotic Dynamical Systems . Massachusetts: Addivon-Weslay, 1989
[143]关新平,范正平,陈彩莲,等.混沌控制及其在保密通讯中的应用.北京:国防工业出版社, 2002
[144]杜星福.混沌定义的研究进展.宁波职业技术学院学报, 2003, 3(2):85–87
[145] Chen G, Dong X. From Chaos to Order: Methodologies, Perspectives and Applica-tions . Singapore: World Scientific Publishing Company, 2002
[146] He′non M. A two dimensional mapping with strange attractor . Communication ofMathematics Physics, 1976, 50(1):69–77
[147] Sparrow C. The Lorenz Equations: Bifurcations, Chaos and Strange Attractors . Lon-don: Springer-Verlag, 1982
[148] Chen G, Ueta T. Yet another chaotic attractor . International Journal of Bifurcationand Chaos, 1999, 9(7):1465–1466
[149] Ueta T, Chen G R. Bifurcation analysis of Chen’s attractor . International Journal ofBifurcation and Chaos, 2000, 10:1917–1931
[150] Lu¨J H, Chen G R. A new chaotic attractor coined . International Journal of Bifurca-tion and Chaos, 2002, 12(3):659–661
[151] Lu¨J H, Chen G R, Zhang S C. The compound structure of a new chaotic attractor .Chaos, Solitons & Fractals, 2002, 12:669–672
[152] Vane?c?ek A, C?elikovsky′S. Control Systems: From Linear Analysis to Synthesis ofChaos . London: Prentice-Hall, 1996
[153] Lu¨J H, Chen G R, Zhan C D, et al. Bridge the gap between the Lorenz system and theChen system . International Journal of Bifurcation and Chaos, 2002, 12(12):2917–2926
[154] Ro¨ssler O E. An equation for continuous chaos . Physics Letters A, 1976, 57(5):397–398
[155] Ro¨ssler O E. Chaotic behavior in simple reaction systems . Zeitschrift fu¨r Natur-forschung A, 1976, 31:259–264
[156] Ro¨ssler O E. Different types of chaos in two simple differential equations . Zeitschriftfu¨r Naturforschung A, 1976, 31:1664–1670
[157] Ro¨ssler O E. Chaos in abstract kinetics: Two prototypes . Bulletin of MathematicalBiology, 1977, 39:275–289
[158] Ro¨ssler O E, Wegmann K. Continuous chaos - four prototype equations . Annals ofNew York Academy of Sciences, 1979, 316:376–392
[159] Ro¨ssler O E. An equation for hyperchaos . Physics Letters A, 1979, 71(2-3):155–157
[160] Chua L O, Komuro M, Matsumato T. The double scroll family . IEEE Transactionson Circuits and Systems, 1986, 33(11):1073–1118
[161] Sprott J C. Some simple chaotic ?ows . Physical Review E, 1994, 50(2):647–650
[162] Yang X S, Li Q D. Chaotic attractor in a hybrid system . International Journal ofChaos Theories and Applications , 2003, 12(10):2255–2256
[163] Yang X S, Li Q D. Generate n-scroll attractor in linear system by scalar outputfeedback . Chaos, Solitons and Fractals, 2003, 18(1):25–29
[164] Silva C P. S?ilnikov theorem: a tutorial . IEEE Transactions on Circuits and SystemsI, 1993, 40(10):675–993
[165] S?ilnikov L P. A contribution of the problem of the structure of an extended neigh-borhood of rough equilibrium state of saddle- focus type [translated by Cezus F.A. ] .Math USSR-Shornik, 1965, 10(1):91–102
[166] Zhou T S, Chen G R, Yang Q G. Constructing a new chaotic system based on theS?ilnikov criterion . Chaos, Solitons and Fractals, 2004, 19(4):985–993
[167] Liao X X, Fu Y L, Xie S L. on the new results of global attractive set and positiveinvariant set of the Lorenz chaotic system and the applications to chaos control andsynchronization . Science in China, Series F Information Sciences, 2005, 48(3):304–321
[168]李元杰.摆的规则随机及混花运动的研究.大学物理, 1998, 17(9):6–8
[169]徐云,宋向东,濮岚澜.电学中的混沌.长春:东北师范大学出版社, 1999
[170]王珂,田真.非线性电路混沌现象实验装置的研究.实验室研究与探索, 1999,18(4):43–45
[171] Sprott J C. Simple chaotic systems and circuits . Physical Letters A, 2000, 266(1):19–23
[172]刘孝贤,谭雪琴.一个四阶非自治混池电路的同步实现及其保密通信应用.山东大学学报(工学版), 2002, 32(5):497–500
[173] Ro¨ssler O E, Wegmann K. Chaos in the Zhabotinskii reaction . Nature, 1978,271(5460):89–90
[174]李艳妮,陈兰,蔡遵生,等. Belousov-Zhabotisky化学体系中超混沌及其同步的研究.化学学报, 2002, 60(7):1173–1178
[175] Agladze K I, Krinsky V I, Pertsov A M. Chaos in the non-stirred Belousov-Zhabotinsky reaction is induced by interaction of waves and stationary dissipativestructures . Nature, 1984, 308:834–835
[176] Tomita K, Tsuda I. Chaos in the Belousov-Zhabotinsky reaction in a ?ow system .Physics Letters A, 1979, 71(5-6):489–492
[177] Costantino R F, Desharnais R A, Cushing J M, et al. Chaotic dynamics in an insectpopulation . Science, 1997, 275:389–391
[178] Henson S M, Costantino R F, Cushing J, et al. Multiple attractors saddles and pop-ulation dynamics in periodic habitats . Bulletin of Mathematical Biology, 1999,61(6):1121–1149
[179] Cushing J, Henson S M, Desharnais R A, et al. A chaotic attractor in ecology: theoryand experimental data . Chaos, Solitons and Fractals, 2001, 12(2):219–234
[180] Mackey M C, Glass L. Oscillation and Chaos in Physiological Control Systems .Science, 1977, 197:287–289
[181] Malasoma J M. What is the simplest dissipative chaotic jerk equation which is parityinvariant . Physics Letters A, 2000, 264(5):383–389
[182] Liu W B, Chen G R. A new chaotic system and its generation . International Journalof Bifurcation and Chaos, 2003, 13(1):261–267
[183] Liu W B, Chen G R. Can a three-dimensional smooth autonomous quadratic chaoticsystem generate a single four-scroll attractor . International Journal of Bifurcationand Chaos, 2004, 13(1):261–267
[184] Lu¨J H, Chen G R, Chen D Z. A new chaotic system and beyond: The gener-alized Lorenz-like system . International Journal of Bifurcation and Chaos, 2004,14(5):1507–1537
[185] Liu C X, Liu T, Liu L, et al. A new chaotic attractor . Chaos,Solitons and Fractals,2002, 22(5):1031–1038
[186] Qi G Y, Du S Z, Chen G, et al. On a four-dimensional chaotic system . Chaos, Solitonsand Fractals, 2005, 23(7):1671–1682
[187] Stewart I. Mathematics: The Lorenz attractor exists . Nature, 2000, 406(6799):948–949
[188] Tucker W. A rigorous ODE solver and Smale’s 14th problem . Foundations of Com-putational Mathematics, 2002, 2(1):53–117
[189] Leonov G A, Reitmann V. Attraktoreingrenzuny fur Nichtlineare System . Leipzing:Teubner-Verlag, 1987
[190] Leonov G, Bunin A, Koksch N. Attractor Localization of the Lorenz System .ZMAA-Journal of Applied Mathematics and Mechanics, 1987, 67(12):649–656
[191] Liao X X, Wang J. Global disspativity of continuous-time recurrent neural networks .Physical Review E, 2003, 68(1):0161181–0161187
[192] Shilnikov A, Nicolis G, Nicolis C. Bifurcation and predictability analysis of a low-order atmospheric circulation model . International Journal of Bifurcation and Chaos,1995, 5:1701–1711
[193] Park T S, Chua L O. Practical Numerical Algorithms for Chaotic Systems . NewYork: Springer-Verlag, 1982
[194] Wolf A, Swift J B, Swinney H L, et al. Determining Lyapunov exponent from a timeseries . Physica D, 1985, 16(3):285–317
[195] Sinha S. Unidirectional adaptive dynamics . Physical Review E, 1994, 49(6):4832–4842
[196]廖晓昕.动力系统的稳定性理论和应用,第1版.北京:国防工业出版社, 1999
[197]廖晓昕.稳定性的数学理论与应用,第2版.武汉:华中师范大学出版社, 2001
[198] Hua C C, Guan X P. Adaptive control for chaotic systems . Chaos, Solitons andFractals, 2004, 22(1):55–60
[199] Huang D B. Adaptive-feedback control algorithm . Physical Review E, 2006,73:066204
[200] Lu W L. Comment on“Adaptive-feedback control algorithm”. Physical Review E,2007, 75:018201
[201] Bainov D D, Simeonov P S. Systems with Impulse Effect: Stability, Theory, andApplications . Chichester: Ellis Homood Limited, 1989
[202] Lakshmikantham V. Theory of Impulsive Differential Equations . Singapore: WorldScientific, 1989
[203] Yang T. Impulsive Control Theory . Berlin Heidelberg: Springer-Verlag, 1989
[204] Yang T, Yang L B, Yang C M. Impulsive control of Lorenz system . Physica D, 1997,110(1-2):18–24
[205] Yang T, Chua L O. Impulsive stabilization for control and synchronization of chaotic-systems: theory and application to secure communication . Circuits and Systems I:Fundamental Theory and Applications, IEEE Transactions on [see also Circuits andSystems I: Regular Papers, IEEE Transactions on], 1997, 44(10):976–988
[206] Yang T, Yang L B, Yang C M. Impulsive synchronization of Lorenz systems . PhysicsLetters A, 1997, 226(6):349–354
[207] Xie W X, Wen C Y, Li Z G. Impulsive control for the stabilization and synchroniza-tion of Lorenz systems . Physics Letters A, 2000, 275:67–72
[208] Li Z G, Wen C Y, Soh Y C. Analysis and design of impulsive control systems .Automatic Control, IEEE Transactions on, 2001, 46(6):894–897
[209] Guan Z H, Liao R Q, Zhou F, et al. On impulsive control and its application to Chen’schaotic system . Int. J. of Bifurcation and Chaos, 2002, 12(5):1191–1197
[210] Sun J T, Wu Q D. Impulsive Control for the Stabilization and Synchronizationof Lurie Systems . Applied Mathematics and Mechanics (English Edition), 2004,25(3):322–328
[211] Sun J T. Impulsive control of a new chaotic system . Mathematics and Computers inSimulation, 2004, 64(6):669–677
[212] Sun J T. Impulsive control of Lurie systems . International Journal of Systems Sci-ence, 2005, 36(13):809–813
[213] Li C, Liao X, Yang X, et al. Impulsive stabilization and synchronization of a classof chaotic delay systems . Chaos: An Interdisciplinary Journal of Nonlinear Science,2005, 15:043103
[214] Khadra A L X, Shen X. Impulsive control and synchronization of spatiotemporalchaos . Chaos, Solitons & Fractals, 2005, 26:615–636
[215] Haeri M, Dehghani M. Impulsive synchronization of Chen’s hyperchaotic system .Physics Letters A, 2006, 356(3):226–230
[216] Farmer J. Chaotic attractors of an infinite-dimensional dynamical system . PhysicaD, 1982, 4(2):366–393
[217] Lu H, He Z. Chaotic behavior in first-order autonomous continuous-time systemswith delay . IEEE transactions on circuits and systems. I, Fundamental theory andapplications, 1996, 43(8):700–702
[218] Liao X F L, Wong K W, Leung C S, et al. Hopf bifurcation and chaos in a singledelayed neuron equation with non-monotonic activation function . Chaos, Solitonsand Fractals, 2001, 12(8):1535–1547
[219] Zhou S, Liao X, Yu J, et al. Chaos and its synchronization in two-neuron systemswith discrete delays . Chaos, Solitons and Fractals, 2004, 21(1):133–142
[220] Levins R. Some demographic and genetic consequences of environmental hetero-geneity for biological control . Bulletin of the Entomology Society of America, 1969,15:237–240
[221] Hosting A. Population Biology . New York: Springer-Verlag, 1997
[222] Li C D, Liao X F, Yu J B. Hopf bifurcation in a prototype delayed system . Solitionand Fractals, 2004, 19:779–787
[223] Sadkowski A. On the application of the logistic differential equation in electrochem-ical dynamics . Journal of Electroanalytical Chemistry, 2000, 486(1):92–94
[224] Kowalczyk R, Forys U. Qualitative analysis on the initial value problem to the logisticequation with delay . Mathematical and computer Modelling, 2002, 35(1):1–13
[225] Hale J, Verduyn S, Lunel I. Introduction to Functional Differential Equations . NewYork: Springer-Verlag, 1993
[226] Schechter M. Principles of functional analysis . New York: Academic Press, 1971
[227] Skarda C, Freeman W. How brains make chaos in order to make sense of the world .Behavioral and Brain Sciences, 1987, 10(1):161–195
[228] Chen L N, Kazuyuki A. Chaotic simulated annealing by a neural network model withtransient chaos . Neural Network, 1995, 8(6):915–930
[229]张学义,胡仕诚,谢荣生.一种混沌神经网络及其在优化计算中的应用.系统工程与电子技术, 2000, 22(7):69–71
[230]毛亚林,张国忠,周明,等.一种改进混沌神经网络及其在组合优化问题中的应用.山东大学学报, 2005, 35(2):72–76
[231] Lewis J E, Glass L. Steady states, limit cycles, and chaos in models of complexbiological networks . International Journal of Bifurcation and Chaos, 1991, 1:477–483
[232] Renals S, Rohwer R. A study of network dynamics . Journal of Statistical Physics,1990, 58:825–849
[233] Das P K, Schieve W C, Zeng Z. Chaos in an effective four-neuron neural network .Physics Letters A, 1991, 161(1):60–66
[234] Das P K, Schieve W C. A bifurcation analysis of the four dimensional generalizedHopfield neural network . Physica D: Nonlinear Phenomena, 1995, 88(1):14–28
[235] Das A, Pritha D, Roy A. Chaos in three dimensional neural network. Applications inMathematical Modelling, 2000, 24:511–522
[236] Das A, Das P, Roy A B. Chaos in a threedimensional general model of neural network.International Journal of Bifurcation and Chaos, 2002, 12(10):2271–2281
[237] Yang X S, Yuan Q. Chaos and transient chaos in simple Hopfield neural networks .Neurocomputing, 2005, 69(1-3):232–241
[238] Yang X S, Huang Y. Complex dynamics in simple Hopfield neural networks . Chaos,2006, 16:033114
[239] Li Q D, Yang X S, Yang H Y. Hyperchaos in Hopfield-type neural networks . Neuro-computing, 2005, 67:275–280
[240] Park J H, Won S. Stability analysis for neutral delay-differential systems . Journal ofThe Franklin Institute, 2000, 337(1):1–9
[241] Hu G D, Mitsui T. Stability analysis of numerical methods for systems of neutraldelay-differential equations . BIT Numerical Mathematics, 1995, 35(4):504–515
[242] Niculescu S I. On delay-dependent stability under model transformations of someneutral linear systems . International Journal of Control, 2001, 74(6):609–617
[243] Marcus C M, Westervelt R M. Stability of analog neutral networks with delay . Phys-ical Review A-General Physics, 3 rd Series, 1989, 39:347–359
[244] Han Q. Robust stability of uncertain delay-differential systems of neutral type . Au-tomatica, 2002, 38(4):719–723
[245] Thompson S, Shampine L F. A friendly Fortran DDE solver. Applied NumericalMathematics, 2006, 56(3-4):503–516
[2] Li T, Yorke J A. Period three implies chaos . American Mathematical Monthly, 1975,8(10):985–992
[3] May R M. Simple mathematical models with very complicated dynamics . Nature,1976, 261(5560):459–467
[4] May R M, Oster G F. Bifurcations and Dynamic Complexity in Simple EcologicalModels . The American Naturalist, 1976, 110(974):573–599
[5] Feigenbaum M J. Quantitative universality for a class of non-linear transformations .Journal of Statistical Physics, 1992, 19(1):25–52
[6] Mandelbrot B B. Fractal Geometry of Nature . California: W. H. Freeman Company,1982
[7] Matsumoto T. A chaotic attractor from Chua’s circuit . IEEE Transactions on Circuitsand Systems, 1984, 31:1055–1058
[8] Ott E, Grebogi C, Yorke J A. Controlling chaos . Physical Review Letters, 1990,64(11):1196–1199
[9] Pecora L M, Carroll T L. Synchronization in chaotic systems . Physical ReviewLetters, 1990, 64(8):821–824
[10] Ditto W L, Showalter K. Introduction: Control and synchronization of chaos . Chaos,1997, 7(4):509–511
[11] Hubler A, Lusher E. Resonant stimulation and control of nonlinear oscillators . Natur-wissenschaft, 1989, 76(1):67–72
[12] M. P. Controlling chaos through parametric exitations . in: Lima R, Streit I, Mendes RV, editors, Proceedings of Dynamics and Stochastic Processes, New York: Springer-Verlag, 1988, 242-250
[13] Fradkov A L, Evans R J. Control of chaos: Methods and applications in engineering .Annual Reviews in Control, 2005, 29(1):33–56
[14]陈关荣,吕金虎. Lorenz系统族的动力学分析、控制与同步.北京:科学出版社,2002
[15] Pyragas. Generalized synchronization of chaos . Physics Letters A, 1993, 4(2):210–221
[16] Genesio R, Tesi A. Harmonic balance methods for the analysis of chaotic dynamicsin nonlinear systems . Automatica, 1992, 28(3):531–548
[17] Hsu R R, Su H T, Chern J L, et al. Conditions to control chaotic dynamics by weakperiodic perturbation . Physics Review Letters, 1997, 78:2936–2939
[18] Jackson E A, Hubler A W. Periodic entrainment of chaotic logistic map dynamics .Physica D, 1990, 44(3):407–420
[19] Jackson E A. The entrainment and migration contols of multiple attractor systems .Physics Letters A, 1990, 151:478–484
[20] Jackson E A, Kodogeorgiou A. Entrainment and migration controls of two dimen-sional maps . Physica D, 1991, 54(3):253–265
[21] Frison T W. Controlling chaos with a neural network. Neural Networks, 1992.IJCNN., International Joint Conference on , 1992, 2(7-11):75–80
[22] Lin C, Jou C. Controlling chaos by GA-based reinforcement learning neural network .Neural Networks, IEEE Transactions on, 1999, 10(4):846–859
[23] Hunt E R. Stabilizing high-period orbits in a chaotic system-The diode resonator .Physics Review Letters, 1991, 67:1953–1955
[24] Ushio T. Limitation of delayed feedback control in nonlinear discrete-timesystems. Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactionson , 1996, 43(9):815–816
[25] Kittel A, Parisi J, Pyragas K. Delay feedback control of chaos by self-adaptive delaytime . Physics Letters A, 1995, 198:433–436
[26] Pyragas K. Control of chaos via extended delay feedback . Physics Letters A, 1995,206(5):323–330
[27] Ahlborn A, Parlitz U. Stabilizing unstable steady states using multiple delay feedbackcontrol . Physical Review Letters, 2004, 93(26):264101
[28] Nakajima H. On analytical properties of delayed feedback control of chaos . PhysicsLetters A, 1997, 232:207–210
[29] Lai Y C, Grebogi C. Converting transient chaos into sustained chaos by feedbackcontrol . Physical Review E, 1994, 49(2):1094–1098
[30] Brandt M E, Chen G. Feedback control of a quadratic map model of cardiac chaos .International Journal of Bifurcation and Chaos, 1996, 6:715–723
[31] Rulkov N, Tsimring L, Abarbanel H. Tracking unstable orbits in chaos using dissi-pative feedback control . Physical Review E, 1994, 50(1):314–324
[32] Hu G, Qu Z, He K. Feedback control of chaos in spatiotemporal systems . Interna-tional Journal of Bifurcation and Chaos, 1995, 5(4):901–936
[33] Cai C, Xu Z, Xu W. Converting chaos into periodic motion by state feedback control.Automatica, 2002, 38(11):1927–1933
[34] Yassen M. Controlling chaos and synchronization for new chaotic system using linearfeedback control . Chaos, Solitons and Fractals, 2005, 26:913–920
[35] Yagasaki K, Yamashita S. Controlling chaos using nonlinear approximations for apendulum with feedforward and feedback control . Int. J. Bifurcation and Chaos,1999, 9(1):233–241
[36] Jiang G, Chen G, Tang K. A new criterion for chaos synchronization using linear statefeedback control . International Journal of Bifurcation and Chaos, 2003, 13(8):2343–2351
[37] Fang J Q, Ali M K. Nonlinear feedback control of spatiotemporal chaos in coupledmap lattices . Discrete Dynamics in Nature and Society, 1998, 1(2):283–305
[38] Andrievskii B R, Fradkov A L. Control of chaos: Methods and applications. I. Meth-ods . Automation and Remote Control, 2003, 64(5):673–713
[39] Araujo A D, Singh S N. Output feedback adaptive variable structure control of chaosin Lorenz system . International Journal of Bifurcation and Chaos, 2002, 12(3):571–582
[40] Arecchi F T, Boccaletti S, Ciofini M, et al. The control of chaos: Theoretical schemesand experimental realizations . International Journal of Bifurcation and Chaos, 1998,8:1643–1655
[41] Boccaletti S, Arecchi F T. Adaptive control of chaos . Europhysics letters, 1995,31(3):127–132
[42] Boccaletti S, Grebogi C, Lai Y C, et al. The control of chaos: theory and applications .Physics Reports, 2000, 329(3):103–197
[43] Ge S S. Adaptive control of uncertain lorenz system using decoupled backstepping .International Journal of Bifurcation and Chaos, 2004, 14(4):1439–1445
[44] Ge S S, Wang C. Adaptive control of uncertain Chua’s circuits . Circuits and SystemsI: Fundamental Theory and Applications, IEEE Transactions on [see also Circuits andSystems I: Regular Papers, IEEE Transactions on], 2000, 47(9):1397–1402
[45] Hogg T, Huberman B A. Controlling chaos in distributed systems . Systems, Manand Cybernetics, IEEE Transactions on, 1991, 21(6):1325–1332
[46] Hua C, Guan X. Adaptive control for chaotic systems . Chaos, Solitons and Fractals,2004, 22(1):103–110
[47] Huang D. Stabilizing Near-Nonhyperbolic Chaotic Systems with Applications . Phys-ical Review Letters, 2004, 93(21):214101
[48] Huang D. Synchronization-based estimation of all parameters of chaotic systemsfrom time series . Physical Review E, 2004, 69(6):67201
[49] Huang D. Simple adaptive-feedback controller for identical chaos synchronization .Physical Review E, 2005, 71(3):37203
[50] Huang D, Guo R. Identifying parameter by identical synchronization between dif-ferent systems . Chaos: An Interdisciplinary Journal of Nonlinear Science, 2003,14:152
[51] Li Z, Chen G, Shi S, et al. Robust adaptive tracking control for a class of uncertainchaotic systems . Physics Letters A, 2003, 310(1):40–43
[52] Lindner J F, Ditto W L. Removal, suppression, and control of chaos by nonlineardesign . Applied Mechanics Review, 1995, 48(12):795–807
[53] Mascolo S, Grassi G. Controlling chaos via backstepping design . Physical ReviewE, 1997, 56(5):6166–6169
[54] Paskota M, Mees A I, Teo K L. On control of chaos: Higher periodic orbits . Dynam-ics and Control, 1995, 5(4):365–387
[55] Petrov V, Michael F C, Showalter K. An adaptive control algorithm for tracking unsta-ble periodic orbits . International Journal of Bifurcation and Chaos, 1994, 4(5):1311–1317
[56] Pyragas K, Pyragas V, Kiss I Z, et al. Adaptive control of unknown unstable steadystates of dynamical systems . Physical Review E, 2004, 70(2):26215
[57] Tian Y, Furong G. Adaptive control of chaotic continuous-time systems with delay .Physica D, 1998, 117(1):1–12
[58] Tian Y P, Yu X. Adaptive control of chaotic dynamical systems using invariant mani-fold approach . Circuits and Systems I: Fundamental Theory and Applications, IEEETransactions on], 2000, 47(10):1537–1542
[59] Vassiliadis D. Parametric adaptive control of an NMR laser model . InternationalJournal of Bifurcation and Chaos, 1993, 3(3):793–796
[60] Wang C, Ge S S. Adaptive backstepping control of uncertain Lorenz system . Inter-national Journal of Bifurcation and Chaos, 2001, 11(4):1115–1119
[61] Wu T, Chen M S. Chaos control of the modified Chua’s circuit system . Physica D,2002, 164(1-2):53–58
[62] Yang T, Yang C M, Yang L B. A Detailed Study of Adaptive Control of ChaoticSystems with Unknown Parameters . Dynamics and Control, 1998, 8(3):255–267
[63] Yassen M T. Adaptive control and synchronization of a modified Chua’s circuit sys-tem . Applied Mathematics and Computation, 2003, 135(1):113–128
[64] Cao Y J, Zhang H X. An adaptive strategy for controlling chaotic system . JOURNALOF ZHEJIANG UNIVERSITY (SCIENCE), 2003, 4(3):258–263
[65] Yu X. Tracking inherent periodic orbits in chaotic dynamic systems viaadaptive vari-able structure time-delayed self control . Circuits and Systems I: Fundamental Theoryand Applications, IEEE Transactions on [see also Circuits and Systems I: Regular Pa-pers, IEEE Transactions on], 1999, 46(11):1408–1411
[66] Zeng Y, Singh S N. Adaptive Control of Chaos in Lorenz System . Dynamics andControl, 1997, 7(2):143–154
[67] Zhang H, Qin H, Chen G. Adaptive control of chaotic systems with uncertainties .International Journal of Bifurcation and Chaos, 1998, 8(10):2041–2046
[68] Huberman B A, Lumer E. Dynamics of adaptive systems . Circuits and Systems I:Fundamental Theory and Applications, IEEE Transactions on, 1990, 37(4):547–550
[69] Pecora L M, Carroll T L. Driving systems with chaotic signals . Physical Review A,1991, 44(4):2374–2383
[70] He R, Vaidya P. Analysis and synthesis of synchronous periodic and chaotic systems. Physical Review A, 1992, 46(12):7387–7392
[71]陈志盛,孙克辉,张泰山. Liu混沌系统的非线性反馈同步控制.物理学报, 2005,54(6):2580–2583
[72]陈保颖.线性反馈实现Liu系统的混沌同步.动力学与控制学报, 2006, 4(1):1–4
[73] Jiang G P, Tang W, Chen G R. A simple global synchronization criterion for coupledchaotic systems . Chaos, Solitons and Fractals, 2003, 15(5):925–935
[74] Chen S, Lu J. Parameters identification and synchronization of chaotic systems basedupon adaptive control . Physics Letters A, 2002, 299(4):353–358
[75] Dai D, Ma X K. Chaos synchronization by using intermittent parametric adaptivecontrol method . Physics Letters A, 2001, 288(1):23–28
[76] Elabbasy E M, Agiza H N, El-Dessoky M M. Adaptive synchronization of a hy-perchaotic system with uncertain parameter . Chaos, Solitons & Fractals, 2006,30(5):1133–1142
[77] Elabbasy E M, Agiza H N, El-Dessoky M M. Adaptive synchronization of Lu systemwith uncertain parameters . Chaos, Solitons & Fractals, 2004, 21(3):657–667
[78] Han X, Lu J A, Wu X. Adaptive feedback synchronization of Lu system . Chaos,Solitons & Fractals, 2004, 22(1):221–227
[79] Hu J, Chen S, Chen L. Adaptive control for anti-synchronization of Chua’s chaoticsystem . Physics Letters A, 2005, 339(6):455–460
[80] Li Z, Shi S. Robust adaptive synchronization of Rossler and Chen chaotic systemsvia slide technique . Physics Letters A, 2003, 311(4-5):389–395
[81] Moukam Kakmeni F M, Bowong S, Tchawoua C. Nonlinear adaptive synchronizationof a class of chaotic systems . Physics Letters A, 2006, 355(1):47–54
[82] Park J H. Adaptive synchronization of hyperchaotic Chen system with uncertainparameters . Chaos, Solitons & Fractals, 2005, 26(3):959–964
[83] Ren Q, Zhao J. Impulsive synchronization of coupled chaotic systems via adaptive-feedback approach . Physics Letters A, 2006, 355(4-5):342–347
[84] Yan Z, Yu P. Linear feedback control, adaptive feedback control and their combi-nation for chaos (lag) synchronization of LC chaotic systems . Chaos, Solitons &Fractals, 2007, 33(2):419–435
[85] Yassen M T. Adaptive chaos control and synchronization for uncertain new chaoticdynamical system . Physics Letters A, 2006, 350(1-2):36–43
[86] Yassen M T. Feedback and adaptive synchronization of chaotic Lu system . Chaos,Solitons & Fractals, 2005, 25(2):379–386
[87] Yu W, Cao J. Adaptive synchronization and lag synchronization of uncertain dynam-ical system with time delay based on parameter identification . Physica A: StatisticalMechanics and its Applications, 2007, 375(2):467–482
[88] Yu Y, Zhang S. Adaptive backstepping synchronization of uncertain chaotic system .Chaos, Solitons & Fractals, 2004, 21(3):643–649
[89] Zhang H, Huang W, Wang Z, et al. Adaptive synchronization between two differentchaotic systems with unknown parameters . Physics Letters A, 2006, 350(5-6):363–366
[90] Yang X S, Chen G. Some observer-based criteria for discrete-time generalized chaossynchronization . Chaos, Solitons and Fractals, 2002, 13(6):1303–1308
[91] Li K, Yang L X, He Z Y, et al. Sporadic driving chaos synchronization based onnonlinear observer. Acta Automatica Sinica, 2001, 27(2):280–283
[92] Li K, Yang L, He Z Y. Stability analysis of nonlinear observer with application tochaos synchronization . Science in China (Information Sciences), 2001, 44(6):430–447
[93] Li G H, Zhou S P, Xu D M. Chaos synchronization based on intermittent state ob-server . Chinese Physics, 2004, 13(2):168–172
[94] Celikovsky S. Observer form of the hyperbolic-type generalized Lorenz system andits use for chaos synchronization . Kybernetika, 2004, 40(6):649–664
[95] Fotsin H, Kakmeni F, Bowong S. An adaptive observer for chaos synchronization ofa nonlinear electronic circuit . International Journal of Bifurcation and Chaos, 2006,16(9):2671–2679
[96] Sanchez E, Perez J, Ricalde L. Neural network design for chaos synchronization.Lecture Notes in Control and Information Sciences, 2004, 292:137–158
[97] Barajas-Ramirez J, Chen G, Shieh L. Fuzzy chaos synchronization via sampled driv-ing signals . International Journal of Bifurcation and Chaos, 2004, 14(8):2721–2733
[98] Yu G R. Fuzzy synchronization of chaos using gray prediction for secure commu-nications . Systems, Man and Cybernetics, 2003. IEEE International Conference on,2003, 4(5-8):3104–3109
[99] Liao X X, Chen G R. Chaos synchronization of general Lurie systems via time-delayfeedback control . Int. J. Bifurcation and Chaos, 2003, 13:207–213
[100]吴立刚,王常虹,曾庆双.滑模自适应控制实现一类不确定混沌系统的同步.控制与决策, 2006, 21(2):229–232
[101] Yau H T. Design of adaptive sliding mode controller for chaos synchronization withuncertainties . Chaos,Solitons and Fractals, 2004, 22(2):341–347
[102] Fan C, Jiang C, Fan C, et al. Backstepping based synchronization control for unifiedchaos systems . Harbin Gongye Daxue Xuebao, 2004, 36(11):1571–1574
[103] Hao Z, Xi-Kui M, Yu Y, et al. Generalized synchronization of hyperchaos and chaosusing active backstepping design . Chinese Physics, 2005, 14(1):86–94
[104] Quan M, Rong C, Dan Z, et al. Approach to Generalized Synchronization with Ap-plication to Chaos–Based Secure Communication . Communication in TheoreticalPhysics, 2004, 42(4):632–640
[105] Rulkov N F, Afraimovich V S, Lewis C T, et al. Multivalued mappings in generalizedchaos synchronization . Physical Review E, 2001, 64(1):16217
[106] Afraimovich V, Cordonet A, Rulkov N F. Generalized synchronization of chaos innoninvertible maps . Physical Review E, 2002, 66(1):16208
[107] Chen G, Liu S. On generalized synchronization of spatial chaos . Chaos, Solitonsand Fractals, 2003, 15(2):311–318
[108] Zhang X, Min L. A generalized chaos synchronization based encryption algorithmfor sound signal communication . Circuits, Systems, and Signal Processing, 2005,24(5):535–548
[109] Kittel A, Parisi J, Pyragas K. Generalized synchronization of chaos in electroniccircuit experiments . Physica D, 1998, 112(3-4):459–471
[110] Morgu¨l O¨. Necessary Condition for Observer-Based Chaos Synchronization. Physi-cal Review Letters, 1999, 82(1):77–80
[111] Yalcinkaya T, Lai Y C. Phase synchronization of chaos . Physical Review Letters,1997, 79:3885–3888
[112] Blasius B, Stone L. Chaos and phase synchronization in ecological systems . Int. J.Bifurcation Chaos, 2000, 10:2361–2380
[113] Santoboni G, Pogromsky A Y, Nijmeijer H. An observer for phase synchronizationof chaos . Physics Letters A, 2001, 291(4):265–273
[114] Zheng Z G, Hu G, Zhou C S, et al. Phase synchronization in coupled chaotic sys-tems - Transitions from high-to low-dimensional chaos . Acta Physica Sinica, 2000,49(12):2320–2327
[115] Feng X Q, Shen K. Phase synchronization and anti-phase synchronization of chaosfor degenerate optical parametric oscillator . Chinese Physics, 2005, 14(8):1526
[116] Rosa Jr E, Ott E, Hess M H. Transition to Phase Synchronization of Chaos . PhysicalReview Letters, 1998, 80(8):1642–1645
[117] Rosenblum M G, Pikovsky A S, Kurths J. From Phase to Lag Synchronization inCoupled Chaotic Oscillators . Physical Review Letters, 1997, 78(22):4193–4196
[118] Zhu S, Wu L. Anticipating and lag synchronization in chaotic laser system . Interna-tional Journal of Modern Physics B, 2004, 18(17-19):2547–2551
[119] Barsella A, Lepers C. Chaotic lag synchronization and pulse-induced transient chaosin lasers coupled by saturable absorber . Optics Communications, 2002, 205(4):397–403
[120] Boccaletti S, Valladares D L. Characterization of intermittent lag synchronization .Physical Review E, 2000, 62(5):7497–7500
[121] Taherion S, Lai Y C. Observability of lag synchronization of coupled chaotic oscil-lators . Physical Review E, 1999, 59(6):6247–6250
[122] Shahverdiev E M, Sivaprakasam S, Shore K A. Lag synchronization in time-delayedsystems . Phys. Lett. A, 2002, 292(6):320–324
[123] Zhan M, Wei G W, Lai C H. Transition from intermittency to periodicity in lag syn-chronization in coupled Ro¨ssler oscillators . Physical Review E, 2002, 65(3):36202
[124] Sosnovtseva O V, Balanov A G, Vadivasova T E, et al. Loss of lag synchronization incoupled chaotic systems . Physical Review E, 1999, 60(6):6560–6565
[125] Pazo D, Zaks M, Kurths J. Role of unstable periodic orbits in phase and lag synchro-nization between coupled chaotic oscillators . Chaos, 2003, 13(1):309–318
[126] Li C, Liao X, Wong K W. Chaotic lag synchronization of coupled time-delayedsystems and its applications in secure communication . Physica D: Nonlinear Phe-nomena, 2004, 194(3-4):187–202
[127] Taherion S, Lai Y C. Experimental observation of lag synchronization in coupledchaotic systems . Int. J. Bifurcation and chaos, 2000, 10:2587–2594
[128] Wu X F, Zhao Y, HUANG X H. Some new criteria for lag synchronization of chaoticLur’e systems by replacing variables control .控制理论与应用(英文版), 2004,2(3):259–266
[129] Zhao D Q, Liu Z R. Lag Synchronization in Nonlinear Systems Based on AdaptiveControl . Journal of Shanghai University (English Edition), 2004, 8(1):24–27
[130] Vieira M D E S, Khoury P, Lichtenberg A j, et al. Numerical and experimental studiesof self-synchronization and synchronized chaos . International Journal of Bifurcationand Chaos, 1992, 2(3):645–657
[131] Cuomo K, Oppenheim A. Circuit implementation of synchronization chaos withapplications to communications . Physical Review Letters, 1993, 71(1):65–68
[132] Belskiy Y U L, Dmitriyev A S. Transmission of information by means of determin-istic chaos . Journal of communications technology & electronics, 1993, 38(15):1–5
[133] Cruz J M, Chua L O. An IC chip of Chua’s circuit . Circuits and Systems II: Analogand Digital Signal Processing, IEEE Transactions on, 1993, 40(10):614–625
[134] Kolumban G, Kennedy M, Chua L. The role of synchronization in digital communica-tions using chaos. I. Fundamentals of digital communications . Circuits and SystemsI: Fundamental Theory and Applications, IEEE Transactions on [see also Circuits andSystems I: Regular Papers, IEEE Transactions on], 1997, 44(10):927–936
[135] Kolumban G, Kennedy M, Chua L. The role of synchronization in digital commu-nications using chaos. II. Chaotic modulation and chaotic synchronization . Circuitsand Systems I: Fundamental Theory and Applications, IEEE Transactions on, 1998,45(11):1129–1140
[136] Yang T, Chua L O. Secure communication via chaotic parameter modulation . Cir-cuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on,1996, 43(9):817–819
[137] Yang T, Yang L B, Yang C M. Breaking chaotic switching using generalized synchro-nization: examples . Circuits and Systems I: Fundamental Theory and Applications,IEEE Transactions on [see also Circuits and Systems I: Regular Papers, IEEE Trans-actions on], 1998, 45(10):1062–1067
[138] Alvarez G, Montoya F, Romera M, et al. Breaking two secure communication systemsbased on chaotic masking . Circuits and Systems II: Express Briefs, IEEE Transac-tions on, 2004, 51(10):505–506
[139]方锦清,罗晓曙.混沌保密通信应用研究的进展.广西师范大学学报:自然科学版, 2002, 20(1):6–18
[140]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(1):21–30
[141]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.物理学进展,2003, 23(2):212–255
[142] Devaney R L. Introduction to Chaotic Dynamical Systems . Massachusetts: Addivon-Weslay, 1989
[143]关新平,范正平,陈彩莲,等.混沌控制及其在保密通讯中的应用.北京:国防工业出版社, 2002
[144]杜星福.混沌定义的研究进展.宁波职业技术学院学报, 2003, 3(2):85–87
[145] Chen G, Dong X. From Chaos to Order: Methodologies, Perspectives and Applica-tions . Singapore: World Scientific Publishing Company, 2002
[146] He′non M. A two dimensional mapping with strange attractor . Communication ofMathematics Physics, 1976, 50(1):69–77
[147] Sparrow C. The Lorenz Equations: Bifurcations, Chaos and Strange Attractors . Lon-don: Springer-Verlag, 1982
[148] Chen G, Ueta T. Yet another chaotic attractor . International Journal of Bifurcationand Chaos, 1999, 9(7):1465–1466
[149] Ueta T, Chen G R. Bifurcation analysis of Chen’s attractor . International Journal ofBifurcation and Chaos, 2000, 10:1917–1931
[150] Lu¨J H, Chen G R. A new chaotic attractor coined . International Journal of Bifurca-tion and Chaos, 2002, 12(3):659–661
[151] Lu¨J H, Chen G R, Zhang S C. The compound structure of a new chaotic attractor .Chaos, Solitons & Fractals, 2002, 12:669–672
[152] Vane?c?ek A, C?elikovsky′S. Control Systems: From Linear Analysis to Synthesis ofChaos . London: Prentice-Hall, 1996
[153] Lu¨J H, Chen G R, Zhan C D, et al. Bridge the gap between the Lorenz system and theChen system . International Journal of Bifurcation and Chaos, 2002, 12(12):2917–2926
[154] Ro¨ssler O E. An equation for continuous chaos . Physics Letters A, 1976, 57(5):397–398
[155] Ro¨ssler O E. Chaotic behavior in simple reaction systems . Zeitschrift fu¨r Natur-forschung A, 1976, 31:259–264
[156] Ro¨ssler O E. Different types of chaos in two simple differential equations . Zeitschriftfu¨r Naturforschung A, 1976, 31:1664–1670
[157] Ro¨ssler O E. Chaos in abstract kinetics: Two prototypes . Bulletin of MathematicalBiology, 1977, 39:275–289
[158] Ro¨ssler O E, Wegmann K. Continuous chaos - four prototype equations . Annals ofNew York Academy of Sciences, 1979, 316:376–392
[159] Ro¨ssler O E. An equation for hyperchaos . Physics Letters A, 1979, 71(2-3):155–157
[160] Chua L O, Komuro M, Matsumato T. The double scroll family . IEEE Transactionson Circuits and Systems, 1986, 33(11):1073–1118
[161] Sprott J C. Some simple chaotic ?ows . Physical Review E, 1994, 50(2):647–650
[162] Yang X S, Li Q D. Chaotic attractor in a hybrid system . International Journal ofChaos Theories and Applications , 2003, 12(10):2255–2256
[163] Yang X S, Li Q D. Generate n-scroll attractor in linear system by scalar outputfeedback . Chaos, Solitons and Fractals, 2003, 18(1):25–29
[164] Silva C P. S?ilnikov theorem: a tutorial . IEEE Transactions on Circuits and SystemsI, 1993, 40(10):675–993
[165] S?ilnikov L P. A contribution of the problem of the structure of an extended neigh-borhood of rough equilibrium state of saddle- focus type [translated by Cezus F.A. ] .Math USSR-Shornik, 1965, 10(1):91–102
[166] Zhou T S, Chen G R, Yang Q G. Constructing a new chaotic system based on theS?ilnikov criterion . Chaos, Solitons and Fractals, 2004, 19(4):985–993
[167] Liao X X, Fu Y L, Xie S L. on the new results of global attractive set and positiveinvariant set of the Lorenz chaotic system and the applications to chaos control andsynchronization . Science in China, Series F Information Sciences, 2005, 48(3):304–321
[168]李元杰.摆的规则随机及混花运动的研究.大学物理, 1998, 17(9):6–8
[169]徐云,宋向东,濮岚澜.电学中的混沌.长春:东北师范大学出版社, 1999
[170]王珂,田真.非线性电路混沌现象实验装置的研究.实验室研究与探索, 1999,18(4):43–45
[171] Sprott J C. Simple chaotic systems and circuits . Physical Letters A, 2000, 266(1):19–23
[172]刘孝贤,谭雪琴.一个四阶非自治混池电路的同步实现及其保密通信应用.山东大学学报(工学版), 2002, 32(5):497–500
[173] Ro¨ssler O E, Wegmann K. Chaos in the Zhabotinskii reaction . Nature, 1978,271(5460):89–90
[174]李艳妮,陈兰,蔡遵生,等. Belousov-Zhabotisky化学体系中超混沌及其同步的研究.化学学报, 2002, 60(7):1173–1178
[175] Agladze K I, Krinsky V I, Pertsov A M. Chaos in the non-stirred Belousov-Zhabotinsky reaction is induced by interaction of waves and stationary dissipativestructures . Nature, 1984, 308:834–835
[176] Tomita K, Tsuda I. Chaos in the Belousov-Zhabotinsky reaction in a ?ow system .Physics Letters A, 1979, 71(5-6):489–492
[177] Costantino R F, Desharnais R A, Cushing J M, et al. Chaotic dynamics in an insectpopulation . Science, 1997, 275:389–391
[178] Henson S M, Costantino R F, Cushing J, et al. Multiple attractors saddles and pop-ulation dynamics in periodic habitats . Bulletin of Mathematical Biology, 1999,61(6):1121–1149
[179] Cushing J, Henson S M, Desharnais R A, et al. A chaotic attractor in ecology: theoryand experimental data . Chaos, Solitons and Fractals, 2001, 12(2):219–234
[180] Mackey M C, Glass L. Oscillation and Chaos in Physiological Control Systems .Science, 1977, 197:287–289
[181] Malasoma J M. What is the simplest dissipative chaotic jerk equation which is parityinvariant . Physics Letters A, 2000, 264(5):383–389
[182] Liu W B, Chen G R. A new chaotic system and its generation . International Journalof Bifurcation and Chaos, 2003, 13(1):261–267
[183] Liu W B, Chen G R. Can a three-dimensional smooth autonomous quadratic chaoticsystem generate a single four-scroll attractor . International Journal of Bifurcationand Chaos, 2004, 13(1):261–267
[184] Lu¨J H, Chen G R, Chen D Z. A new chaotic system and beyond: The gener-alized Lorenz-like system . International Journal of Bifurcation and Chaos, 2004,14(5):1507–1537
[185] Liu C X, Liu T, Liu L, et al. A new chaotic attractor . Chaos,Solitons and Fractals,2002, 22(5):1031–1038
[186] Qi G Y, Du S Z, Chen G, et al. On a four-dimensional chaotic system . Chaos, Solitonsand Fractals, 2005, 23(7):1671–1682
[187] Stewart I. Mathematics: The Lorenz attractor exists . Nature, 2000, 406(6799):948–949
[188] Tucker W. A rigorous ODE solver and Smale’s 14th problem . Foundations of Com-putational Mathematics, 2002, 2(1):53–117
[189] Leonov G A, Reitmann V. Attraktoreingrenzuny fur Nichtlineare System . Leipzing:Teubner-Verlag, 1987
[190] Leonov G, Bunin A, Koksch N. Attractor Localization of the Lorenz System .ZMAA-Journal of Applied Mathematics and Mechanics, 1987, 67(12):649–656
[191] Liao X X, Wang J. Global disspativity of continuous-time recurrent neural networks .Physical Review E, 2003, 68(1):0161181–0161187
[192] Shilnikov A, Nicolis G, Nicolis C. Bifurcation and predictability analysis of a low-order atmospheric circulation model . International Journal of Bifurcation and Chaos,1995, 5:1701–1711
[193] Park T S, Chua L O. Practical Numerical Algorithms for Chaotic Systems . NewYork: Springer-Verlag, 1982
[194] Wolf A, Swift J B, Swinney H L, et al. Determining Lyapunov exponent from a timeseries . Physica D, 1985, 16(3):285–317
[195] Sinha S. Unidirectional adaptive dynamics . Physical Review E, 1994, 49(6):4832–4842
[196]廖晓昕.动力系统的稳定性理论和应用,第1版.北京:国防工业出版社, 1999
[197]廖晓昕.稳定性的数学理论与应用,第2版.武汉:华中师范大学出版社, 2001
[198] Hua C C, Guan X P. Adaptive control for chaotic systems . Chaos, Solitons andFractals, 2004, 22(1):55–60
[199] Huang D B. Adaptive-feedback control algorithm . Physical Review E, 2006,73:066204
[200] Lu W L. Comment on“Adaptive-feedback control algorithm”. Physical Review E,2007, 75:018201
[201] Bainov D D, Simeonov P S. Systems with Impulse Effect: Stability, Theory, andApplications . Chichester: Ellis Homood Limited, 1989
[202] Lakshmikantham V. Theory of Impulsive Differential Equations . Singapore: WorldScientific, 1989
[203] Yang T. Impulsive Control Theory . Berlin Heidelberg: Springer-Verlag, 1989
[204] Yang T, Yang L B, Yang C M. Impulsive control of Lorenz system . Physica D, 1997,110(1-2):18–24
[205] Yang T, Chua L O. Impulsive stabilization for control and synchronization of chaotic-systems: theory and application to secure communication . Circuits and Systems I:Fundamental Theory and Applications, IEEE Transactions on [see also Circuits andSystems I: Regular Papers, IEEE Transactions on], 1997, 44(10):976–988
[206] Yang T, Yang L B, Yang C M. Impulsive synchronization of Lorenz systems . PhysicsLetters A, 1997, 226(6):349–354
[207] Xie W X, Wen C Y, Li Z G. Impulsive control for the stabilization and synchroniza-tion of Lorenz systems . Physics Letters A, 2000, 275:67–72
[208] Li Z G, Wen C Y, Soh Y C. Analysis and design of impulsive control systems .Automatic Control, IEEE Transactions on, 2001, 46(6):894–897
[209] Guan Z H, Liao R Q, Zhou F, et al. On impulsive control and its application to Chen’schaotic system . Int. J. of Bifurcation and Chaos, 2002, 12(5):1191–1197
[210] Sun J T, Wu Q D. Impulsive Control for the Stabilization and Synchronizationof Lurie Systems . Applied Mathematics and Mechanics (English Edition), 2004,25(3):322–328
[211] Sun J T. Impulsive control of a new chaotic system . Mathematics and Computers inSimulation, 2004, 64(6):669–677
[212] Sun J T. Impulsive control of Lurie systems . International Journal of Systems Sci-ence, 2005, 36(13):809–813
[213] Li C, Liao X, Yang X, et al. Impulsive stabilization and synchronization of a classof chaotic delay systems . Chaos: An Interdisciplinary Journal of Nonlinear Science,2005, 15:043103
[214] Khadra A L X, Shen X. Impulsive control and synchronization of spatiotemporalchaos . Chaos, Solitons & Fractals, 2005, 26:615–636
[215] Haeri M, Dehghani M. Impulsive synchronization of Chen’s hyperchaotic system .Physics Letters A, 2006, 356(3):226–230
[216] Farmer J. Chaotic attractors of an infinite-dimensional dynamical system . PhysicaD, 1982, 4(2):366–393
[217] Lu H, He Z. Chaotic behavior in first-order autonomous continuous-time systemswith delay . IEEE transactions on circuits and systems. I, Fundamental theory andapplications, 1996, 43(8):700–702
[218] Liao X F L, Wong K W, Leung C S, et al. Hopf bifurcation and chaos in a singledelayed neuron equation with non-monotonic activation function . Chaos, Solitonsand Fractals, 2001, 12(8):1535–1547
[219] Zhou S, Liao X, Yu J, et al. Chaos and its synchronization in two-neuron systemswith discrete delays . Chaos, Solitons and Fractals, 2004, 21(1):133–142
[220] Levins R. Some demographic and genetic consequences of environmental hetero-geneity for biological control . Bulletin of the Entomology Society of America, 1969,15:237–240
[221] Hosting A. Population Biology . New York: Springer-Verlag, 1997
[222] Li C D, Liao X F, Yu J B. Hopf bifurcation in a prototype delayed system . Solitionand Fractals, 2004, 19:779–787
[223] Sadkowski A. On the application of the logistic differential equation in electrochem-ical dynamics . Journal of Electroanalytical Chemistry, 2000, 486(1):92–94
[224] Kowalczyk R, Forys U. Qualitative analysis on the initial value problem to the logisticequation with delay . Mathematical and computer Modelling, 2002, 35(1):1–13
[225] Hale J, Verduyn S, Lunel I. Introduction to Functional Differential Equations . NewYork: Springer-Verlag, 1993
[226] Schechter M. Principles of functional analysis . New York: Academic Press, 1971
[227] Skarda C, Freeman W. How brains make chaos in order to make sense of the world .Behavioral and Brain Sciences, 1987, 10(1):161–195
[228] Chen L N, Kazuyuki A. Chaotic simulated annealing by a neural network model withtransient chaos . Neural Network, 1995, 8(6):915–930
[229]张学义,胡仕诚,谢荣生.一种混沌神经网络及其在优化计算中的应用.系统工程与电子技术, 2000, 22(7):69–71
[230]毛亚林,张国忠,周明,等.一种改进混沌神经网络及其在组合优化问题中的应用.山东大学学报, 2005, 35(2):72–76
[231] Lewis J E, Glass L. Steady states, limit cycles, and chaos in models of complexbiological networks . International Journal of Bifurcation and Chaos, 1991, 1:477–483
[232] Renals S, Rohwer R. A study of network dynamics . Journal of Statistical Physics,1990, 58:825–849
[233] Das P K, Schieve W C, Zeng Z. Chaos in an effective four-neuron neural network .Physics Letters A, 1991, 161(1):60–66
[234] Das P K, Schieve W C. A bifurcation analysis of the four dimensional generalizedHopfield neural network . Physica D: Nonlinear Phenomena, 1995, 88(1):14–28
[235] Das A, Pritha D, Roy A. Chaos in three dimensional neural network. Applications inMathematical Modelling, 2000, 24:511–522
[236] Das A, Das P, Roy A B. Chaos in a threedimensional general model of neural network.International Journal of Bifurcation and Chaos, 2002, 12(10):2271–2281
[237] Yang X S, Yuan Q. Chaos and transient chaos in simple Hopfield neural networks .Neurocomputing, 2005, 69(1-3):232–241
[238] Yang X S, Huang Y. Complex dynamics in simple Hopfield neural networks . Chaos,2006, 16:033114
[239] Li Q D, Yang X S, Yang H Y. Hyperchaos in Hopfield-type neural networks . Neuro-computing, 2005, 67:275–280
[240] Park J H, Won S. Stability analysis for neutral delay-differential systems . Journal ofThe Franklin Institute, 2000, 337(1):1–9
[241] Hu G D, Mitsui T. Stability analysis of numerical methods for systems of neutraldelay-differential equations . BIT Numerical Mathematics, 1995, 35(4):504–515
[242] Niculescu S I. On delay-dependent stability under model transformations of someneutral linear systems . International Journal of Control, 2001, 74(6):609–617
[243] Marcus C M, Westervelt R M. Stability of analog neutral networks with delay . Phys-ical Review A-General Physics, 3 rd Series, 1989, 39:347–359
[244] Han Q. Robust stability of uncertain delay-differential systems of neutral type . Au-tomatica, 2002, 38(4):719–723
[245] Thompson S, Shampine L F. A friendly Fortran DDE solver. Applied NumericalMathematics, 2006, 56(3-4):503–516
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |