几类混沌系统的界、分岔、控制与同步若干问题研究
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摘要
混沌是非线性系统中存在的一种运动形式,它是在确定性中出现的一种貌似不规则、内在随机的运动、既普遍存在又极其复杂的现象。混沌系统的最大特点就是系统演化对初始条件敏感性,因此混沌系统的未来行为是短期可预测长期不可预测的。混沌运动模糊了确定性运动和随机运动的界限,它具有不可重复性,局部不稳定而整体稳定。在混沌的研究中,混沌系统的有界性是前沿的研究课题之一。混沌系统的最终有界在混沌的定性行为的研究中有着重要的作用,然而估计一个动力系统的全局指数吸引集和最终有界是一项比较困难的任务。一个混沌系统是有界的,意味着混沌吸引子在相空间是有界的,并且所估计的界在混沌控制、同步及其应用中有非常重要的作用。
本学位论文研究混沌系统的界、分岔、控制与同步若干问题,主要内容分为以下几个方面：
1.研究了修正超混沌Lu系统的Hopf分岔。首先,得到Hopf分岔存在的条件。然后,用规范形理论分析了Hopf分岔的类型、分岔的方向、分岔的近似周期解。最后,用数值仿真也验证了理论分析的正确性。
2.统一混沌系统在参数已知和参数未知情况下的同步问题。基于LaSalle不变集原理和线性矩阵不等式,当参数已知时,用一个带有自适应增益的线性反馈控制器实现了两个相同的统一混系统的同步,其次,用一个自适应控制器实现了参数未知时统一混沌系的同步,最后用仿真结果验证了文中的理论分析。
3.利用Shilnikov定理构造一个含有平方项的三维混沌系统,且系统有两个平衡点。构造的过程表明该混沌具有Smale马蹄(同宿轨混沌),满足同宿轨道Shilnikov定理条件,数值仿真验证了该方法的有效性。最后,用待定系数法证明系统中存在Smale马蹄,因而是Shilnikov意义下的混沌。
4.动态输出反馈的方法来控制一类混沌系统。利用李亚普诺夫稳定理论,结合线性矩阵不等式优化方法,分析了系统的稳定性。数值仿真验证了设计方法的可行性。
5.针对一类参数随时间变化,具有外部噪声干扰,不确定混沌系统,用鲁棒自适应方法实现了异结构混沌系统的同步和参数辨识。利用李亚普诺夫稳定理论,设计了鲁棒自适应控制器和参数自适应律。最后以Lorenz系统和Chen系统为例子进行仿真验证,数值仿真验证了所设计方法具有鲁棒性和有效性。
6.引入了一个典型自治混沌系统,该系统包含2个系统变量乘积的非线性函数项。当系统参数变化时,系统属于广义Lorenz系统,又能够不属于广义Lorenz系统.用分岔图,Poincare截面图和Lyapunov指数对典型系统进行了分析。最后,对典型系统的界进行了估计,给出当参数a,b,c, d全为正数时界的表达式。
7.运用参数法给出了一个三维自治系统在满足一定条件下两变量上界的估计并证明,在此基础上给出了该系统三个变量的一个上界估计定理并给予了证明,给出了能产生不稳定的鞍焦平衡点参数的条件,有一个正的李氏指数,从而构造出一个混沌系统。
8.构造了一个具有光滑四次函数的自治混沌系统,系统中包含两个系统变量乘积的非线性函数和三个实平衡点。讨论了平衡点的稳定性、用Lyapunov维数、Poincare截面图、Lyapunov指数谱和分岔图分析了系统的动力学特性。最后,基于拓扑马蹄映射理论,证明自治混沌系统存在拓扑马蹄。
9.提出了一个新的混沌系统,其基本的动力学性质,用平衡点的稳定性、Lyapunov维数、Poincare截面图、Lyapunov指数谱和分岔图进行详尽的分析。然后用脉冲控制将新的系统稳定到原点。
本学位的论文工作得到了禹思敏教授主持的主持的国家自然科学基金(批准号：61172023)和广东省自然科学基金(批准号：S2011010001028)资助。
Chaos is an existing phenomenon in nonlinear systems, which arises from deterministic system in a random and non-reguar fashion. It is ubiquitous and complex. The most reamarkable chaotic character is that chaotic systems exhibit high sensitivity to initial conditions. In other words, a nonlinear system called chaotic system is unpredictable. Chao breaks the rigorous division between deterministic systems and random systems, which is un-repeatable, local un-stable and whole stable. In the study on chaos, The bound of chaotic systems have become front research topics, extremely challenging. The ultimate bound of a chaotic system is important for the investigation of the qualitative behavior of a chaotic system, however, estimating the globally exponentially attractive and ultimate bound set for a dynamic system is a quite challenging task in general. A chaotic system is bounded, in the sense that its chaotic attractor is bounded in the phase space, and estimate of its bound is important in chaos control, chaos synchronization and applications. On the one hand, Hopf bifurcation of a hyperchaotic system is studied in this dissertation, on the other hand, bound, control, synchronization and some issues concerned of several novel chaotic systems are investigated in this paper.
The main content is divided into the following aspects:
1. The Hopf bifurcation of a new modified hyperchaotic Lii system is investigated. Firstly, A detailed set of conditions are derived, which guarantee the existence of the Hopf bifurcation. Furthermore, by applying the normal form theory, the direction and type of the Hopf bifurcation, and the approximate expressions of bifurcating periodic solutions and their periods are determined. In addition, numerical simulation results supporting the theoretical analysis are given.
2. Two novel synchronization criterions of the unified chaotic systems with known or unknown parameters is investigated. Based on LaSalle's invariance principle and linear matrix inequality (LMI) formulation, firstly, a simple linear feedback controller with the updated gain is obtain to make the state of two identical unified chaotic systems asymptotically synchronized, secondly, a simple adaptive controller is proposed for synchronization of the unified chaotic systems with unknown parameters. Finally, simulation results are given to verify the theoretical analysis.
3. Based on the Shilnikov theorem, a new three-dimensional square chaotic system, which has tow equilibrium poits. There is hyperbolic saddle focus. The formation mechanism shows that this chaotic system has Smale horseshoes (homoclinic chaos). Numerical simulations are given to show the effectiveness of the theoretical results. Finally, Smale horseshoses has been found in the system using undetermined-coefficient method, and it is chaotic in the sense of Shilnikov.
4. In this work, a dynamic outputs feedback controller for a class for a class of chaotic systems is developed for the first time. For stability, a well-known Lyapunov stability theorem combining with linear matrix inequality optimizaion is utilized. A numerical simulation is presented to show the feasibility of the proposed scheme.
5. A robust adaptive control is proposed to realize chaos synchronization and parameters identification between two different chaotic systems with uncertainties, bounded time-varying unknown parameters, and noise perturbation. Based on Lyapunov stability theory, a robust control law and a parameters identification scheme are presented. Finally, numerical esults are presented for the Lorenz, Chen systems. Numerical simulations are given to demonstrate the robustness and efficiency of the proposed method.
6. A typical autonomous chaotic system is firstly presented in this paper which contain two smooth quadratic terms of systemic variables. When the parameters of system are changed, the typical system belongs to or does not belong to the generalized Lorenz system. Basic dynamical properties of the typical system are studied in terms of numerical simulation, Lyapunov exponent spectrum, bifurcation diagrams and Poincare section diagrams. At last, the ultimate bound of typical system is estimated and the expression of the bound is derived for all the positive values of its parameters a,b,c,d.
7. At first, under the certain condition, the theorem about an upper bound estimate of two variables was given by parameterization and proved. Secondly, the theorem about and upper bound estimate of three variables was proposed. Then, the condition of generating unstable saddle focus equilibrium points is presented. Therefore, a chaotic system was constructed.
8. A novel chaotic system which contains two smooth biquadratic terms of systemic variables and three real equilibriums is presented. The characters of three equilibriums are discussed. The dynamic properties of the novel system are investigated via theoretical analysis, numerical simulation, Lyapunov dimension, Poincare diagrams, Lyapunov exponent spectrum and bifurcation diagrams. Finally, in the light of topological horseshoe map theory, the paper analyses the existence of topological horseshoe in the presented autonomous chaotic system.
9. A chaotic system is presented and the dynamic properties of the novel system are investigated in term of Lyapunov dimension, Poincare diagrams, Lyapunov exponent spectrum and bifurcation diagrams. At last, the impulsive control of the system is global asymptotical stable at origin.
本学位论文研究混沌系统的界、分岔、控制与同步若干问题,主要内容分为以下几个方面：
1.研究了修正超混沌Lu系统的Hopf分岔。首先,得到Hopf分岔存在的条件。然后,用规范形理论分析了Hopf分岔的类型、分岔的方向、分岔的近似周期解。最后,用数值仿真也验证了理论分析的正确性。
2.统一混沌系统在参数已知和参数未知情况下的同步问题。基于LaSalle不变集原理和线性矩阵不等式,当参数已知时,用一个带有自适应增益的线性反馈控制器实现了两个相同的统一混系统的同步,其次,用一个自适应控制器实现了参数未知时统一混沌系的同步,最后用仿真结果验证了文中的理论分析。
3.利用Shilnikov定理构造一个含有平方项的三维混沌系统,且系统有两个平衡点。构造的过程表明该混沌具有Smale马蹄(同宿轨混沌),满足同宿轨道Shilnikov定理条件,数值仿真验证了该方法的有效性。最后,用待定系数法证明系统中存在Smale马蹄,因而是Shilnikov意义下的混沌。
4.动态输出反馈的方法来控制一类混沌系统。利用李亚普诺夫稳定理论,结合线性矩阵不等式优化方法,分析了系统的稳定性。数值仿真验证了设计方法的可行性。
5.针对一类参数随时间变化,具有外部噪声干扰,不确定混沌系统,用鲁棒自适应方法实现了异结构混沌系统的同步和参数辨识。利用李亚普诺夫稳定理论,设计了鲁棒自适应控制器和参数自适应律。最后以Lorenz系统和Chen系统为例子进行仿真验证,数值仿真验证了所设计方法具有鲁棒性和有效性。
6.引入了一个典型自治混沌系统,该系统包含2个系统变量乘积的非线性函数项。当系统参数变化时,系统属于广义Lorenz系统,又能够不属于广义Lorenz系统.用分岔图,Poincare截面图和Lyapunov指数对典型系统进行了分析。最后,对典型系统的界进行了估计,给出当参数a,b,c, d全为正数时界的表达式。
7.运用参数法给出了一个三维自治系统在满足一定条件下两变量上界的估计并证明,在此基础上给出了该系统三个变量的一个上界估计定理并给予了证明,给出了能产生不稳定的鞍焦平衡点参数的条件,有一个正的李氏指数,从而构造出一个混沌系统。
8.构造了一个具有光滑四次函数的自治混沌系统,系统中包含两个系统变量乘积的非线性函数和三个实平衡点。讨论了平衡点的稳定性、用Lyapunov维数、Poincare截面图、Lyapunov指数谱和分岔图分析了系统的动力学特性。最后,基于拓扑马蹄映射理论,证明自治混沌系统存在拓扑马蹄。
9.提出了一个新的混沌系统,其基本的动力学性质,用平衡点的稳定性、Lyapunov维数、Poincare截面图、Lyapunov指数谱和分岔图进行详尽的分析。然后用脉冲控制将新的系统稳定到原点。
本学位的论文工作得到了禹思敏教授主持的主持的国家自然科学基金(批准号：61172023)和广东省自然科学基金(批准号：S2011010001028)资助。
Chaos is an existing phenomenon in nonlinear systems, which arises from deterministic system in a random and non-reguar fashion. It is ubiquitous and complex. The most reamarkable chaotic character is that chaotic systems exhibit high sensitivity to initial conditions. In other words, a nonlinear system called chaotic system is unpredictable. Chao breaks the rigorous division between deterministic systems and random systems, which is un-repeatable, local un-stable and whole stable. In the study on chaos, The bound of chaotic systems have become front research topics, extremely challenging. The ultimate bound of a chaotic system is important for the investigation of the qualitative behavior of a chaotic system, however, estimating the globally exponentially attractive and ultimate bound set for a dynamic system is a quite challenging task in general. A chaotic system is bounded, in the sense that its chaotic attractor is bounded in the phase space, and estimate of its bound is important in chaos control, chaos synchronization and applications. On the one hand, Hopf bifurcation of a hyperchaotic system is studied in this dissertation, on the other hand, bound, control, synchronization and some issues concerned of several novel chaotic systems are investigated in this paper.
The main content is divided into the following aspects:
1. The Hopf bifurcation of a new modified hyperchaotic Lii system is investigated. Firstly, A detailed set of conditions are derived, which guarantee the existence of the Hopf bifurcation. Furthermore, by applying the normal form theory, the direction and type of the Hopf bifurcation, and the approximate expressions of bifurcating periodic solutions and their periods are determined. In addition, numerical simulation results supporting the theoretical analysis are given.
2. Two novel synchronization criterions of the unified chaotic systems with known or unknown parameters is investigated. Based on LaSalle's invariance principle and linear matrix inequality (LMI) formulation, firstly, a simple linear feedback controller with the updated gain is obtain to make the state of two identical unified chaotic systems asymptotically synchronized, secondly, a simple adaptive controller is proposed for synchronization of the unified chaotic systems with unknown parameters. Finally, simulation results are given to verify the theoretical analysis.
3. Based on the Shilnikov theorem, a new three-dimensional square chaotic system, which has tow equilibrium poits. There is hyperbolic saddle focus. The formation mechanism shows that this chaotic system has Smale horseshoes (homoclinic chaos). Numerical simulations are given to show the effectiveness of the theoretical results. Finally, Smale horseshoses has been found in the system using undetermined-coefficient method, and it is chaotic in the sense of Shilnikov.
4. In this work, a dynamic outputs feedback controller for a class for a class of chaotic systems is developed for the first time. For stability, a well-known Lyapunov stability theorem combining with linear matrix inequality optimizaion is utilized. A numerical simulation is presented to show the feasibility of the proposed scheme.
5. A robust adaptive control is proposed to realize chaos synchronization and parameters identification between two different chaotic systems with uncertainties, bounded time-varying unknown parameters, and noise perturbation. Based on Lyapunov stability theory, a robust control law and a parameters identification scheme are presented. Finally, numerical esults are presented for the Lorenz, Chen systems. Numerical simulations are given to demonstrate the robustness and efficiency of the proposed method.
6. A typical autonomous chaotic system is firstly presented in this paper which contain two smooth quadratic terms of systemic variables. When the parameters of system are changed, the typical system belongs to or does not belong to the generalized Lorenz system. Basic dynamical properties of the typical system are studied in terms of numerical simulation, Lyapunov exponent spectrum, bifurcation diagrams and Poincare section diagrams. At last, the ultimate bound of typical system is estimated and the expression of the bound is derived for all the positive values of its parameters a,b,c,d.
7. At first, under the certain condition, the theorem about an upper bound estimate of two variables was given by parameterization and proved. Secondly, the theorem about and upper bound estimate of three variables was proposed. Then, the condition of generating unstable saddle focus equilibrium points is presented. Therefore, a chaotic system was constructed.
8. A novel chaotic system which contains two smooth biquadratic terms of systemic variables and three real equilibriums is presented. The characters of three equilibriums are discussed. The dynamic properties of the novel system are investigated via theoretical analysis, numerical simulation, Lyapunov dimension, Poincare diagrams, Lyapunov exponent spectrum and bifurcation diagrams. Finally, in the light of topological horseshoe map theory, the paper analyses the existence of topological horseshoe in the presented autonomous chaotic system.
9. A chaotic system is presented and the dynamic properties of the novel system are investigated in term of Lyapunov dimension, Poincare diagrams, Lyapunov exponent spectrum and bifurcation diagrams. At last, the impulsive control of the system is global asymptotical stable at origin.
引文
[1]B. L. Hao. Chao[M].Singapore: World Sience,1984.
[2]B. L. Hao. Chao Ⅱ [M]. Singapore: World Sience,1990.
[3]吕金虎,陆君安,陈士华.混沌时间序列分析及应用[M].武汉：武汉出版社,2002.
[4]陈关荣,吕金虎.Lorenz系统族的动力学分析、控制与同步[M].北京：科学出版社,2003.
[5]E. N. Lorenz.Determinitic nonperiodic flow[J].J.Atmospheric Sci.,1963,20:130-141.
[6]R.Barboza.Dynamics of a hyperchaotic Lorenz system[J].Int.J.Bifurcat.Chaos, 2007,17(12):4285-4294.
[7]Y. X. Li, W. K. Tang, G. R. Chen. Generating hyperchaos via state feedback control[J]. IntJ.Bifurcat. Chaos,2005,15(10):3367-3375.
[8]B.C.Bao,Z.Liu.A hyperchaotic attractor coin from chaotic Lu system[J].Chin.Phy.Lett. 2008,25 (7):2396-2399.
[9]G. Y. Wang,X. Zhang,Y. Zheng,Y. X. Li. A new modified hyperchaotic Lu attrator via state feedback control[J]. Physica A,371(2):260-272.
[10]Q. G. Yang, Y. J. Liu. A hyperchaotic system from a chaotic system with one saddle and two stable node-foci[J]. J.Math.Anal.Appl.,2009,360(1):293-306.
[11]Q. G. Yang, K. M. Zhang, G. Chen. Hyperchaotic attrators from a linearly controlled Lorenz system[J]. Nonlinear Analysis:Real Word Applications,2009,10(3): 1601-1617
[12]王兴元,王明军.超混沌Lorenz系统[J].物理学报,2007,56(9)：5236-5141.
[13]刘扬正.超混沌Lu系统的电路实现[J].物理学报,2008,57(3)：1439-1443.
[14]张晓丹,崔丽娟.一类四维超混沌系统的界及同步的研究[J].物理学报,2011,60(11)：110511-110218.
[15]刘明华,冯久超.一个新的超混沌系统[J].物理学报,2009,58(7)：4457-4462.
[16]禹思敏,禹之鼎.一个新的五阶超混沌电路及其研究[J].物理学,2008,57(11)：6589-6867.
[17]Z.H.Wang, S.J. Cang, E. O. Ochola, Y. X. Sun, A hyperchaotic system without equilibrium[J].Nonlinear Dyn.,2012,69(2):531-537.
[18]S.Zheng,X.J.Han,Q.S.Bi,Bifurcation and fast-slow behaviors in a hyperchaotic dynamical system[J].Commun.Nonlinear Sci. Numer.Simulat.,2011,16(4):1998-2005.
[19]K.M.Zhang,Q.G.Yang, Hopf bifurcation analysis in a 4D-hyperchaotic system[J]. Journal of Systems Science and Complexity.2010,23(4) 748-758.
[20]E. Ott, C. Grebogi, J. A. Yorke. Controlling chaos [J]. Phys. Rev. Lett.,1990,60(11): 1196-1199.
[21]L. M. Pecora, T. L. Carroll. Synchronization in chaotic systems [J]. Phys. Rev. Lett., 1990,64(8):821-824.
[22]H. Abarbanel, N. F. Rulkov, M. Sushchik. Generalized synchronization of chaos:The auxiliary system approach[J]. Phys. Rev. E,1996,53(5):4528-4535.
[23]J. P. Yan, C. P. Li. Generalized projective synchronization of a unified chaotic system[J]. Chaos, Solit. Fract.,2005,26(40):1119-1124.
[24]J. P. Yan, C. P. Li. Generalized projective synchronization for the chaotic Lorenz system and Chen system[J]. J. Shanghai University.2006,10(4):299-304.
[25]G H. Li. Modified projedtive synchronization of chaotic system[J]. Chao, Solit. Fract.,2007,32(5):1786-1790.
[26]G.A. Leonov, S. M. Abramovich, A. I. Bunin. Problems of nonlinear and turbulent process in physics. In:Proceeding of second international working group, Kiev (in Russian), Part Ⅱ,1985,75-77.
[27]G Leonov, V. Reitmann. Attraktoreingrenzuny fur Nichtlineare System. Leipzing:Teuber-Verlag; 1987.
[28]G Leonov, A. Bunin, N. Koksch. Attractor localization of Lorenz system. ZAMM 1987,67:649-656.
[29]S. D. Peter. Bounds for trajectories of the Lorenz equations: an illustration of how to choose Liapunov fuctions [J]. Phys. Lett. A.,2001,281:161-167.
[30]X. X. Liao. On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization [J]. Sci China Ser E,2004,34 (12):404-419.
[31]D.M. Li, J.A. Lu, X.Q. Wu, G.R. Chen. Estimating the bounds for the Lorenz family of chaotic systems[J]. Chaos, Solit. Fract.,2005,23(2):529-534.
[32]D.M.Li,J.A.Lu,X.Q.Wu,GR. Chen. Estimating the bound and positively invariant set for the Lorenz system and a unified chaotic system [J].Math Anal Appl.2006,333(2):844-853.
[33]P.Yu, X.X.Liao. New estimations for globally attractive and positive invariant set of the family of the Lorenz system[J]. Int.J.Bifurcat. Chaos,2006,16(11):3383-3390.
[34]Y. J. Shun. Solution bounds of generalized Lorenz chaotic systems [J]. Chaos, Solit. Fract.,2009,40(2):691-696.
[35]P. Wang, D.M. Li, Q. L. Hu, Bounds of the hyper-chaotic Lorenz-Stenflo system[J]. Commun. Nonlinear Sci.Numer. Simulat.,2010,15(9):2514-2520.
[36]P.Yu, X.X.Liao. On the study of globally exponentially attractive set of a general chaotic system[J]. Commun. Nonlinear Sci. Numer. Simulat.,2008,13(8):1498-1507.
[37]X. X. Liao, P.Yu. Study on the global property of the smooth Chua's system[J]. Int. J. Bifurcat. Chaos,2006,16(10):2815-2841.
[38]D. M. Li, X. Q. Xu, J. A. Lu, Estimating the ultimate bound and poitively invariant set for the hyperchaotic Lorenz-Haken system[J]. Chaos, Solit. Fract.,2009,39(3): 1290-1296.
[39]X.W.Qin, G.R. Chen. On the boundedness of solution of the Chen system[J].Math Anal Appl.2007,329(1):445-451.
[40]O. E. Rossler. An equation for hyperchaos[J]. Phys. Lett. A.,1979,71 (2-3):155-157.
[41]T. Matsumoto, L. O. Chua, K. Kobayashi. Hyperchaos:Laboratory experiment and numerical confirmation [J]. IEEE Trans. Circ. and Syst.,1986,33 (11):1143-1147.
[42]C.Z. Ning, H.Haken, Detuned lasers and the complex Lorenz equations:subcitical and super-critical Hopf bifurcations[J]. Phys.Rev. A.,1990,41 (7):3826-3837.
[43]T. Kapitaniak, L. O. Chua. Hyper-chaotic attractor of unidirectionally-coupled Chua's circuit[J].Int. J. Bifurcat. Chaos,1994,4(2):477-482.
[44]K. Thamilmaran, M. Lakshmanan, A. Venkatesan. Hyperchaos in a Modified Canonical Chua's Circuit, Int.J.Bifurcat. Chaos,2004,14 (221):221-243.
[45]A. Cenys, A. Tamasvisus, A. Baziliauskas. Hyperchaos in coupled Colpitts oscillators[J]. Chaos, Solit. Fract.,2003,17 (2):349-353.
[46]V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J.B. Cuenot, P. Levy, J.B. Cuenot, P. Levy, W.T. Rhodes. Communicating with hyperchaos:The dynamics of a DNLF emitter and recovery of transmitted information[J]. Optics and Spectroscopy,2003,95 (1):114-118.
[47]R. Vicente, J. Dauden, P. Colet, R. Toral Analysis and Characterization of the Hyperchaos Generated by a Semiconductor Laser Subject to delayed feedback Loop[J]. IEEE J. Quantum Electronics,2005,41(4):541-548.
[48]P. Arena, S. Baglio, L. Fortuna, G. Manganaro. Hyperchaos from cellular networks[J] Electronics Letters,1995,31(4):250-251.
[49]J.Y. Hsieh, C.C. Hwang, A.P.Li, W.J. Li, Controlling hyper-chaos of the Rossler system, Int. J. Control,1999,72 (10):882-886.
[50]P.Q. Jiang, B.H.Wang, Bu S.L,Q.H. Xia, X.S. Luo. Hyperchaotic Sysnchronization in Deterministic Small-World Dynamical Networks[J]. Internat. J. Modern Phys.B., 2004,18 (30):2674-2679.
[51]C.P. Li, GR. Chen, A note on bifurcation in Chen's system[J]. Int. J. Bifurcat. Chaos.,2003,13(6):1609-1615.
[52]T.Ueta, GR.Chen. Bifurcation analysis of Chen's equation[J]. Int.J. Bifurcat. Chaos., 2000,10 (8):1917-1931.
[53]X.B. Zhou, Y. Wu,Y. Li, Z.X. Wei. Hopf bifurcation analysis of the Liu system[J]. Chaos, Solit. Fract.,2008,36 (5):1385-1391.
[54]B.Hassard, N. Kazarinoff, Y. Wan. Theory and Application of Hopf Bifurcation[M]. first edition, Cambridge, New York,1982.
[55]G.R. Chen, T. Ueta. Yet another chaotic attractor[J]. Int. J. Bifurcat. Chaos., 1999,9 (7):1465-1466.
[56]J.H. Lii, G. R. Chen. A new chaotic attractor coined [J]. Int. J. Bifurcat. Chaos., 2002,12 (3):659-651.
[57]J.H. Lu, G. R. Chen, D. Z. Cheng. Bridge the gap between the Lorenz system and the Chen system[J]. Int. J. Bifurcat. Chaos.,2002,12 (12):2918-2926.
[58]C. Huygens. Horologium Oscillatorium[M]. Parisiis:Apud F. Muguet 1673.
[59]Z. M. Ge, J. K. Lee. Chaos synchronization and parameter identification for gyroscope system [J].Appl. Math. Comput.,2005,163(2):667-682.
[60]Y. W. Wang, Z. H. Guan. Generalized synchronization of continuous chaotic system[J]. Chaos Solit. Fract.,2006,27(l):97-101.
[61]L. Junge, U. Parlitz, Phase synchronization of coupled Ginzburg-landau Equation [J]. Phys. Rev. E.,2000,62(1):438-441.
[62]E. M. Shahverdiev, S. Sivaprakasam, K. A. Shore. Lag sysnchronization in time-delayed systems [J]. Phys. Lett. A.,2002,292(6):320-324.
[63]B. Wang, G. J. Wen. On the synchronization of a class o chaotic systems based on backstepping method[J]. Phys. Lett. A.,2007,370(1):35-39.
[64]J. M. Nazzal, A. N. Natsheh. Chaos control using sliding-mode theory [J]. Chaos, Solit. Fract.,2007,33(2):695-702.
[65]S. ozoguz, A. S. Elwakil, K.N. Salama. N-scroll chaos generator using nonlinear transconductor[J].Electron Lett.,2002,38(14):685-686.
[66]C. C. Penga, C. L. Chen. Robust chaotic control of Lorenz system by backstepping design[J]. Chaos, Solit. Fract.,2008,37(2):598-608.
[67]M. Haeri, M. S. Tavazoei, M. R. Naseh. Synchronization of uncertain chaotic systems using active sliding mode control[J]. Chaos, Solit. Fract.,2007,33(4):1230-1239.
[68]J.H. Park. On synchronization of unified chaotic systems via nonlinear control[J]. Chaos, Soliton. Fract.,2005,25(3):699-794.
[69]L.L. Zhang,.L. H. Huang, Z. Z. Zhang, Z. Y. Wang. Fuzzy adaptive synchronization of uncertain chaotic system via delayed feedback control[J].Phys. Lett. A.2008, 372(29):6082-6086.
[70]X. Liu, J. Z. Wang, L. Huan. Global synchronization for a class of dynamical complex networks[J]. Physica A.,2007,386(1):543-556.
[71]L. Wang, X. J. Kong, H. Shi, H.P.Dai, Y. X, Sun.LMI-based criteria for Synchronization of complex dynamical networks[J]. J. Phys. A:Math. Thror.,2008, 41(28):28102.
[72]J. H. Park, S. M. Lee, H. Y. Jung. LMI optimization approach to synchronization of stochastic delayed discrete-time complex networks[J]. J. Optim. Theory Appl.,2009, 143(2):357-367.
[73]S. Kuntanapreeda. Chaos synchronization of unified chaos via LMI[J]. Phys. Lett. A. 2009,372(2):2837-2840.
[74]C. X. Liu, L. L. Liu, T. Liu. A novel three-dimensional autonomous chaos system[J]. Chaos, Solit. Fract.,2009,39(4):1950-1958.
[75]张建雄,唐万生,徐勇.一个新的三维混沌系统[J].物理学报,2008,57(11)：6799-6809.
[76]冯朝文,蔡理,唐强,张立森.一种新的三维自治混沌系统[J].物理学报,2011,60(3)：1-7.
[77]C. Silva. Si'lnikov theorem-a tutorial[J]. IEEE Trans. Circ. Syst. I.1993,40(10):68-82
[78]T. S. Zhou, G. R. Chen, Q. G. Yang. Constructing a new chaotic system based on S'ilnicov criterion[J]. Chaos, Solit. Fract.,2004,19(4):985-993.
[79]T. S. Zhou, Y. Tang, G. R. Chen. Chen's attractor exists[J]. Int. J. Bifurcat. Chaos., 2004,14(9):3167-3177.
[80]Y. X. Yang, J. H. Sun. Si'lnikov homoclinic orbit in a new chaotic system[J]. Chaos, Solit. Fract.,2007,39(1):150-159.
[81]B. Y. Chen, T. S. Zhou. S'lnikov homoclinic orbit in two class of 3D autonomous nonlinear[J]. Int J Bifurcat modern Phys B,2011,25(20):2697-2712.
[82]B. Y. Chen, T. S. Zhou, G. R. Chen.An extended Si'lnikov homoclinic theorem arid its applications[J]. Int. J. Bifurcat. Chaos.,2009,19(5):1679-1693.
[83]L. Q. Zhou, F. Q. Chen. Hopf bifurcation and Si'lnikov chaos of Genesio systems[J]. Chaos, Solit. Fract.,2009,40(3):1413-1422.
[84]X. Wang. S'lnikov chaos and Hopf bifurcation analysis of Rucklidge system[J]. Chaos, Solit. Fract.,2009,42(4):2208-2217.
[85]L. Q. Zhou, Q. F. Chen. S'lnikov chaos of Liu system[J]. Chaos,2008,18(1): 013133.
[86]S. H. Chen, J. H. L Lu. Synchronization of of an uncertain unified chaotic system via adaptive control[J]. Chaos, Solit. Fract.,2002,14(2):643-647.
[87]张兴华,丁守刚.非均匀气隙永磁同步电机的自适应混沌同步[J].控制理论与应用,2009,26(6)：661-664.
[88]C. C. Wang, S. Pain, T. Yauh. Chaos control in AFM system using sliding mode control by backstepping[J]. Commun. Nonlinear Sci. Numer. Simulat.,2010, 15(3):741-751.
[89]H. L. Guo. Generalized projective synchronization between two different chaotic systems using active backstepping control[J]. Phys. Lett. A,2006,355(4):326-330.
[90]L.L. Wang, M. C. Kuo. Robust synchronization of drive-response chaotic system via adaptive sliding mode control[J]. Chaos, Solit. Fract.,2009,39(5):2086-2092.
[91]J.J. Yan, Y. S. Yang, Y. Y. Chang, C. Y. Chen. Roboust synchronization of uncertain chaotic system via sliding mode[J]. Chaos, Solit. Fract.,2007,34(3):947-954.
[92]N. Cai, Y. W. Jing, S. Y. Zhang. Modified projective synchronization of chaotic systems with disturbances via active sliding mode control[J]. Commun. Nonlinear Sci. Numer. Simulat.,2010,15(6):1613-1620.
[93]X. Z. Liu. Impulsive synchronization of chaotic systems subject to time delay[J]. Nonliear Analysis,2009,71(12):1320-1327.
[94]M. F. Hu, Y. Q. Yang. Impulsive control of projective synchronization in chaotic Systems[J]. Phys. Lett. A,2008,372 (18):3228-323.
[95]J. H. Park, O.M. Kwon, S. M. Lee. LMI optimization approach to stabilization of Genesio-Tesi chaotic system via dynamic controller[J]. Appl. Math. Comput.,2008, 196(1):200-206.
[96]C.D. Li, X.F. Liao, R. Zhang. Impulsive synchronization of nonlinear coupled chaotic systems[J]. Phys. Lett. A,2004,328(1):47-50.
[97]H. G. Yu, S. M. Zhong, R.P, Agarwal. Mathematics analysis and chaos in an ecological model with an impulsive control strategy[J]. Commun. Nonlinear Sci. Numer. Simulat.,2011,16(2):776-786.
[98]G.X. Chen, A simple adaptive feedback control method for chaos and hyper-chaos control[J]. Appl. Math. Comput.,2011,217 (17):7258-7264.
[99]R.W. Guo, A simple adaptive controller for chaos and hyperchaos synchronization[J]., Phys. Lett. A,2008,372 (34):5593-5597.
[100]H.T. Yu, J. Wang, B. Deng, X.L, Wei, Adaptive synchronization control of coupled chaotic neurons in the external electrical stimulation[J] Chin. Phys. B.,2013, 22(5):058701.
[101]F. X. Chen, L. Chen, W.D. Zhang, Stabilization of parameters perturbation chaotic system via adaptive backstepping technique [J]. Appl. Math. Comput.,2008,200(1): 101-109.
[102]李雨珊,吕翎,刘烨,刘硕,闫兵兵,常欢,周佳楠.复杂网络时空混沌同步的Backstepping设计[J].物理学报,2013,62(2)：0205513.
[103]G.P. Jiang, W.X. Zheng. A LMI criterion fo linear-state-feedback based chaos synchronization of a class of chaotic systems [J].Chaos, Solit. Fract,2005,26 (2): 437-443.
[104]J.H. Kim, C.W. Park, E. Kim, M. Park. Fuzzy adaptive synchronization of uncertain chaotic systems[J]. Phys. Lett. A,2005,334 (4):295-305.
[105]M.G. Mahmoud, E.E. Mahmoud, A.A. Arafa. Controlling hyperchaotic complex systems with unkown parameters based on adaptive passive method[J]. Chin. Phys. B.,2013,22(6):060508.
[106]T.C. Lin, C.H. Kuo. H8 synchronization of uncertain fractional order chaotic systems[J].Adaptive fuzzy approach, IAS Transactions,2011 50 (4):548-556.
[107]C.K. Ahn, S.T. Jung, S.K. Kang, S.C. Joo. Adaptive H8 synchronization of uncertain chaotic systems with external disturbance[J]. Commun. Nonlinear Sci. Numer. Simulat.,2010,15 (8):2168-2177.
[108]C. K. Ahn.An H8 approach to anti-synchronization for chaotic systems [J]. Phys. Lett. A,2009,373 (20):1729-1733.
[109]R.M. Guerra, W. YU.Chaotic synchronization and secure communication via sliding-mode observer[J]. Int. J. Bifurcat. Chaos.,2008,18 (1):235-243.
[110]M.R. Naseh, M. Haeri. Robust synchronization of chaotic systems using active sliding mode control with minimum control effort[J].Int.J. Modern Physic B,2011, 25(17):2271-2288.
[111]M. Pourmahmood, S. Khanmohammadi, G. Alizadeh. Synchronization of two different uncertain chaotic systems with unknown paramiters using robust adaptive sliding mode controller[J]. Commun. Nonlinear Sci. Numer. Simulat.,2011,16 (7):2583-2868.
[112]刘金琨.滑模变结构MATLAB仿真[M].北京：清华大学出版社,2005.
[113]G. Y. Qi, G. R. Chen, M. Wyk, B. Wyk, Y. H. Zhang. A four-wing chaotic attractor generated from a new 3D quadratic autonomous system[J].Chaos, Solit. Fract.,2008, 38 (3):705-721.
[114]C. X. Liu. A novel chaotic attractor [J]. Chaos, Solit. Fract.,2009,39(3):1037-1045.
[115]X.H.Xiong, J. W. Wang. Conjugate Lorenz-type chaotic attractors [J].Chaos, Solit. Fract.,2009,40(2):923-929.
[116]Y.J.Liu, Q.G.Yang. Dynamics of a new Lorenz-like chaotic system[J]. Nonliear Anlysis:Real world Appl.,2010,11(4):2563-2572.
[117]B. Munmuangsaen, B. Srisuchinwong. A new five-time simple chaotic attractor[J]. Phys. lett.A,2009,373:4038-4043.
[118]J. Kennedy, S. Kocak, J. A. York. A chaos lemma [J]. Amer Math Mon,2001,108 (5):411-423.
[119]J. Kennedy, J. A. York.Topological horseshoes [J]. Trans Amer Math Soc,2001, 353(6)2513-2530.
[120]X. S. Yang, Y. Tang. Horseshoes in piecewise continuous maps [J]. Chaos, Solit. Fract,2004,19(14)841-845.
[121]B. H. Li, Y. Huang. Horseshoes in a biochemical attractor[J]. Chaos, Solit Fract.,40 (2009)1207-1212.
[122]M.C. Li, M. Malkin. Topological horseshoes for perturbations of singular difference equations[J]. Nonlinearity,2006,19 (4):795-811.
[123]W. Z. Huang, Y. Huang. Chaos, bifurcation and robustness of a class of Hopfield neural networks[J]. Int. J. Bifurcat. Chaos.,2011,21 (3):885-895.
[124]Q. Yuan, Q.D. Li, X.S. Yang. Horseshoe chaos in a class of simple Hopfield neural networks [J]. Chaos Solit Fract.,2009,39 (4):1552-1529.
[125]C. Ma, X.Y. Wang. Hopf bifurcation and topological horseshoe of a novel finance chaotic system[J] Commun. Nonlinear Sci. Numer. Simulat.,2012,17(2):721-730.
[126]Q.D. Li, X.S. Yang. A simple method for finding topological horseshoes[J]. Int. J. Bifurcat. Chaos.,2010,20 (2):467-478.
[127]X.S. Yang. Topological horseshoes in continuous maps[J]. Chaos Solit Fract.,2007, 33(1):225-233.
[128]G. G. Dong, S. Zhang, L. X. Tian, R. J. Du. Spectrum analysis and circuit Implementation of a new 3D chaotic system with novel chatic attractors[J]. Chin. Phys. Lett.,2010,27(2):02057.
[129]S. Celikovsky, G. R. Chen. On the generalized Lorenz canonical form [J]. Chaos Solit Fract.,2005,26(5):1271-1276.
[130]S. J. Schiff, K. Jerger, D. H. Duong, T. Chang, M. L. Spano, W. L. Ditto. Controllingchaos brains in the brain[J]. Nature,1994,370 (6491):615-620.
[131]G. R. Cheng, D.J. Lai. Feedback control of Lyapunov exponents for discrete-time dynamical[J].Int. J. Bifurcat. Chaos.1996,6 (7):1341-1349.
[132]G. R. Cheng, D. J. Lai. Feedback anticontrol of chaos[J]. Int. J. Bifurcat. Chaos.,1998, 8(7):1585-1590.
[133]X. F. Wang, G. R. Cheng. On feedback anticontrol of discrete chaos[J]. Int. J. Bifurcat. Chaos.,1998,9(7):1435-1441.
[134]X. F. Wang, G R. Cheng. Yet another algorithm for chaotifying control of discrete chaos[J].Control Theory Appl,2000,17(3):336-341.
[135]X. F. Wang, G. R. Cheng. Chaotification via arbitrarily small feedback control[J]. Int. J. Bifurcat. Chaos.,2000,10(3):549-575.
[136]Q.G Yang, Z.C. Zhou, C.R.Chen. An unusual 3D autonomous quadratic chaotic system with two stable node-foci[J]. Int. J. Bifurcat. Chaos.,2010,20(4):1061-1083.
[137]Q.J.Zhang, J. A. Lu. Chaos synchronization of a new chaotic via nonlinear control[J]. Chaos, Solit. Fract.,2008,37(1):175-179.
[138]王春华,胡燕,余飞,徐洁.一类混沌系统时间可控的自适应投影同步[J].物理学报,2013,62(11)：110509.
[139]J.Zhou, J.A.Lu, J.H. Lii. Pinning adaptive synchronization of a general complex dynamical network[J]. Automatica,2008,44(4):996-1003.
[140]J.Zhou, J.A.Lu, J.H.Lu. Adaptive synchronization of an uncertain complex dynamatic network[J]. IEEE Trans, on Automatic control,2006,51(4):652-656.
[141]于海涛,王江.FitzHugh-Nagumo基于反演自适应动态滑模的神经元混沌同步控制[J].物理学报,2013,62(17)：170511.
[142]Z.Y.Shun, W.Z, Zhu, G. Q. Si, Y.Ge, Y.B. Zhang. Adaptive synchronization design for uncertain chaotic systems in the presence of unknown system parameters:a revisist [J]. Nonlinear Dyn.,2013,72(4):729-749.
[143]C.C.Yang, C.J.Ou. Adaptive teminal sliding mode control subject to input nonlinearity for synchronization of chaotic gyros[J] Commun.Nonlinear Sci. Numer. Simulat.,2013,18(3):682-691.
[144]马铁东,江伟波,浮洁,柴毅,陈立平,薛方正.一类分数阶混沌系统的自适应同步[J].物理学报,2012,61(16)：160506.
[145]祝大伟,涂俐兰.随机扰动下Lorenz混沌系统的自适应同步与参数识别[J].物理学报,2013,62(5):050508.
[146]邓玮,方洁,吴振军,吴艳军,含有不确定项的混沌系统自适应修正函数投影同步[J].物理学报，2012,61(14):140503.
[147]X.Y. Wang, Y.Q. Wang. Adaptive control for synchronization of a four-dimensional chaotic system via a single variable[J]. Nonlinear Dyn,2011,65(3):311-316.
[148]L.Q.Shen, M. Wang. Adaptive control of chaotic system based on a single layer neural network[J]. Phys. Lett. A.,2007,368(5):379-382.
[149]S. Kuntanpreeda. An observer-based neural network controller for chaotic Lorenz system[J]. Lect. Notes Comput. Sci.,2008,5370:608-617.
[150]H.T. Yau, C.S. Shieh. Chaos synchronization using fuzzy logic controller [J]. Nonlinear Anal.:Real World Appl.,2008,9(4):1808-1810.
[151]J.Wen, C.S. Jiang. Adaptive fuzzy control for a class of chaotic systems with nonaffine inputs [J]. Commun.Nonlinear Sci. Numer. Simulat.,2011,16(1):475-492.
[152]J.G Lu, D.J. Hill. Global asymptotical synchronization of chaotic Lur'e systems using sampled data: A linear matrix inequality approach[J]. IEEE Trans, irc. Syst. II:Exp Brief,2008,55(6):586-590.
[153]Y. He, G. L. Wen, Q. G. Wang. Delay-dependent synchronization criterion for Lur'e systems with delay feedback control [J].Int. J. Bifur. Chaos,2006,16(10):3087-3091.
[154]Y.Z, Sun, W.Li, J. Ruan. Generalized outer synchronization between complex dynamical nerworks with time delay and noise perturbation [J]. Commun. Nonlinear Sci. Numer. Simulat.,2013,18(4):989-998.
[155]M.M. Asheghan, M. T. H. Beheshti. An LMI approach to robust synchronization of a class of chaotic systems with gain variations[J]. Chaos Solit. Fract.,2009, 42(2):1106-1111.
[156]R. C. Wu, W. W. Zhang. Synchronization of delayed chaotic neural networks by LMI[C].Proceedings of the 2008 Second International Conference on Future Generation Communication and Networking Symposia(FGCNS), Hinan, China,2008, 55-59.
[157]F.X. Chen, W. D. Zhang. LMI criteria for robust chaos synchronization of a class of chaotic systems[J]. Nonlinear Ananlysis:Theroy, Methods and Applications,2007, 67(12):3384-3393.
[158]B. Li, D. Yang, X. H. Zhang, L. T. Ma. Chaotic lag synchronization of coupled time-delayed neural network with two neurons using LMI approach[J].Acta Automatica Sinica,2007,33(11):1196-1200.
[159]任涛,朱志良,于海,.王猛.基于Min-Max方法的混沌系统采样同步控制研究[J]物理学报,2013,62(17)：170501.
[160]K. Tanaka, T. IKeda, H. O. Wang. A unified approach to controlling chaos via an LMI-based fuzzy control system design[J]. IEEE Trans. Circ. And Syst,-I,1998, 45(10):1021-1040.
[161]S.Kuntanapreeda. Simple LMI-based synchronization of fractional-order chaotic systems[J]. Int. J. Bifur. Chaos,2013,23(1):1350011.
[162]吴忠强,杨阳,徐纯华,混沌状态下永磁同步发电机的故障诊断—LMI法研究[J].物理学报,2013,62(15)：150507.
[163]李水平,许志军.利用小增益定理实现Liu系统的同步[J].吉林师范大学学报,2010,1(1)：45-48.
[164]姜玉秋.Lorenz混沌系统的线性同步控制[J].吉林大学学,2011,29(6)：571-575.
[165]J.Xiao. Anti-synchronization of a new hyperchaotic system via small-gain theorem[J]. Chin. Phys. B.,2010,19(10):100505.
[166]J.T. Li, W.L.Li, Q.P.Li, Sliding mode control for uncertain chaotic system with input Nonlinearty[J]. Commun.Nonlinear Sci. Numer. Simulat.,2012,17(1):341-348.
[167]F. Y. Liu, T.S.Li. Z.F.Li, R.H. Li. Adaptive terminal sliding mode control for anti-synchronization of uncertain chaotic system[J]. Nonlinear.Dyn.,2013,74(4): 991-1002.
[168]X.Y. Wang, D.Lin, Z.J.Wang. Controlling the uncertain multi-scroll critical chaotic system with input nonlinear using sliding mode control[J]. Mordern Physics letters B, 2009,23(16):2021-2034.
[169]M.P.Aghababa, H.P. Aghababa. A novel finite-time sliding mode controller for synchronization with input nonlinearity[J]. Arab. J. Sci. Eng,2013,38(11): 3221-3232.
[170]倪骏康,刘崇新,庞霞.电力系统混沌振荡的等效快速终端模糊滑模控制[J].物理学报,2013,62(19)：190507.
[171]吕翎,李雨珊,韦琳玲,于淼,张檬.基于滑模控制法实现规则网络混沌同步[J].物理学报,2012,61(12)：120504.
[172]吴忠强,谭拂晓,王绍仙.基于无源化的细胞神经网络超混沌系统同步[J].物理学报,2006,60(11)：1651-1658.
[173]韦笃取,张波.基于无源理论自适应镇定具有输入的永磁同步电动机的混沌运动[J].物理学报,2012,61(3)：030505.
[174]H. Hamiche, K. Kemih, M. Ghanes, G. Zhang, S. Djennoune. Passive and impulsive synchronization of a new four-dimensional chaotic system[J]. Nonlinear Analysis, 2011,74(4):1146-1154.
[175]关治洪,廖俊锋,廖锐全.不确定脉冲系统的鲁棒H∞控制[J].控制理论与应用2002,19(4)：623-626.
[176]陈贵词,张芙蓉.一类不确定时滞系统的脉冲控制[J].武汉大学学报(自然科学版),2006,29(5)：517-519.
[177]成新明,关治洪,刘新芝.一类不确定的脉冲控制[J].电机与控制学报,.2002,6(4)：310-313.
[178]C. D. Li, X. F. Liao, R. Zhang. Impulsive synchronization of nonlinear coupled chaotic systems[J]. Phys. Lett. A.,2004,328(1):47-50.
[179]G.J. Liu. A novel study for impulsive synchronization of fractional-order chaotic systems[J]. Chin. Phys. B.,2013,22(6):060510.
[180]C.D. Li, X.F. Liao, X.Y. Zhang. Impulsive synchronization of chaotic systems[J]. Chaos,2005,15:023104.
[181]Y.S. Chen, C.C. Chang. Adaptive impulsive synchronization of nonlinear chaotic systems[J]. Nonlinear Dvn,2012,70(3):1795-1803.
[182]胡汉平,于志良,刘凌锋.光电反馈混沌系统脉冲同步特性研究[J].物理学报,2012,61(19)：10504.
[183]罗永健,于茜,张东.参数不确定时延超混沌系统的脉冲同步方法研究[J].物理学报,2011,60 (11):0504.C. Ma, X.Y. Wang, Impulsive control and synchronization of a new unified hyperchaotic system with varying control gains and impulsive intervals[J]. Nonlinear Dyn,2012,70(1):551-558.
[2]B. L. Hao. Chao Ⅱ [M]. Singapore: World Sience,1990.
[3]吕金虎,陆君安,陈士华.混沌时间序列分析及应用[M].武汉：武汉出版社,2002.
[4]陈关荣,吕金虎.Lorenz系统族的动力学分析、控制与同步[M].北京：科学出版社,2003.
[5]E. N. Lorenz.Determinitic nonperiodic flow[J].J.Atmospheric Sci.,1963,20:130-141.
[6]R.Barboza.Dynamics of a hyperchaotic Lorenz system[J].Int.J.Bifurcat.Chaos, 2007,17(12):4285-4294.
[7]Y. X. Li, W. K. Tang, G. R. Chen. Generating hyperchaos via state feedback control[J]. IntJ.Bifurcat. Chaos,2005,15(10):3367-3375.
[8]B.C.Bao,Z.Liu.A hyperchaotic attractor coin from chaotic Lu system[J].Chin.Phy.Lett. 2008,25 (7):2396-2399.
[9]G. Y. Wang,X. Zhang,Y. Zheng,Y. X. Li. A new modified hyperchaotic Lu attrator via state feedback control[J]. Physica A,371(2):260-272.
[10]Q. G. Yang, Y. J. Liu. A hyperchaotic system from a chaotic system with one saddle and two stable node-foci[J]. J.Math.Anal.Appl.,2009,360(1):293-306.
[11]Q. G. Yang, K. M. Zhang, G. Chen. Hyperchaotic attrators from a linearly controlled Lorenz system[J]. Nonlinear Analysis:Real Word Applications,2009,10(3): 1601-1617
[12]王兴元,王明军.超混沌Lorenz系统[J].物理学报,2007,56(9)：5236-5141.
[13]刘扬正.超混沌Lu系统的电路实现[J].物理学报,2008,57(3)：1439-1443.
[14]张晓丹,崔丽娟.一类四维超混沌系统的界及同步的研究[J].物理学报,2011,60(11)：110511-110218.
[15]刘明华,冯久超.一个新的超混沌系统[J].物理学报,2009,58(7)：4457-4462.
[16]禹思敏,禹之鼎.一个新的五阶超混沌电路及其研究[J].物理学,2008,57(11)：6589-6867.
[17]Z.H.Wang, S.J. Cang, E. O. Ochola, Y. X. Sun, A hyperchaotic system without equilibrium[J].Nonlinear Dyn.,2012,69(2):531-537.
[18]S.Zheng,X.J.Han,Q.S.Bi,Bifurcation and fast-slow behaviors in a hyperchaotic dynamical system[J].Commun.Nonlinear Sci. Numer.Simulat.,2011,16(4):1998-2005.
[19]K.M.Zhang,Q.G.Yang, Hopf bifurcation analysis in a 4D-hyperchaotic system[J]. Journal of Systems Science and Complexity.2010,23(4) 748-758.
[20]E. Ott, C. Grebogi, J. A. Yorke. Controlling chaos [J]. Phys. Rev. Lett.,1990,60(11): 1196-1199.
[21]L. M. Pecora, T. L. Carroll. Synchronization in chaotic systems [J]. Phys. Rev. Lett., 1990,64(8):821-824.
[22]H. Abarbanel, N. F. Rulkov, M. Sushchik. Generalized synchronization of chaos:The auxiliary system approach[J]. Phys. Rev. E,1996,53(5):4528-4535.
[23]J. P. Yan, C. P. Li. Generalized projective synchronization of a unified chaotic system[J]. Chaos, Solit. Fract.,2005,26(40):1119-1124.
[24]J. P. Yan, C. P. Li. Generalized projective synchronization for the chaotic Lorenz system and Chen system[J]. J. Shanghai University.2006,10(4):299-304.
[25]G H. Li. Modified projedtive synchronization of chaotic system[J]. Chao, Solit. Fract.,2007,32(5):1786-1790.
[26]G.A. Leonov, S. M. Abramovich, A. I. Bunin. Problems of nonlinear and turbulent process in physics. In:Proceeding of second international working group, Kiev (in Russian), Part Ⅱ,1985,75-77.
[27]G Leonov, V. Reitmann. Attraktoreingrenzuny fur Nichtlineare System. Leipzing:Teuber-Verlag; 1987.
[28]G Leonov, A. Bunin, N. Koksch. Attractor localization of Lorenz system. ZAMM 1987,67:649-656.
[29]S. D. Peter. Bounds for trajectories of the Lorenz equations: an illustration of how to choose Liapunov fuctions [J]. Phys. Lett. A.,2001,281:161-167.
[30]X. X. Liao. On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization [J]. Sci China Ser E,2004,34 (12):404-419.
[31]D.M. Li, J.A. Lu, X.Q. Wu, G.R. Chen. Estimating the bounds for the Lorenz family of chaotic systems[J]. Chaos, Solit. Fract.,2005,23(2):529-534.
[32]D.M.Li,J.A.Lu,X.Q.Wu,GR. Chen. Estimating the bound and positively invariant set for the Lorenz system and a unified chaotic system [J].Math Anal Appl.2006,333(2):844-853.
[33]P.Yu, X.X.Liao. New estimations for globally attractive and positive invariant set of the family of the Lorenz system[J]. Int.J.Bifurcat. Chaos,2006,16(11):3383-3390.
[34]Y. J. Shun. Solution bounds of generalized Lorenz chaotic systems [J]. Chaos, Solit. Fract.,2009,40(2):691-696.
[35]P. Wang, D.M. Li, Q. L. Hu, Bounds of the hyper-chaotic Lorenz-Stenflo system[J]. Commun. Nonlinear Sci.Numer. Simulat.,2010,15(9):2514-2520.
[36]P.Yu, X.X.Liao. On the study of globally exponentially attractive set of a general chaotic system[J]. Commun. Nonlinear Sci. Numer. Simulat.,2008,13(8):1498-1507.
[37]X. X. Liao, P.Yu. Study on the global property of the smooth Chua's system[J]. Int. J. Bifurcat. Chaos,2006,16(10):2815-2841.
[38]D. M. Li, X. Q. Xu, J. A. Lu, Estimating the ultimate bound and poitively invariant set for the hyperchaotic Lorenz-Haken system[J]. Chaos, Solit. Fract.,2009,39(3): 1290-1296.
[39]X.W.Qin, G.R. Chen. On the boundedness of solution of the Chen system[J].Math Anal Appl.2007,329(1):445-451.
[40]O. E. Rossler. An equation for hyperchaos[J]. Phys. Lett. A.,1979,71 (2-3):155-157.
[41]T. Matsumoto, L. O. Chua, K. Kobayashi. Hyperchaos:Laboratory experiment and numerical confirmation [J]. IEEE Trans. Circ. and Syst.,1986,33 (11):1143-1147.
[42]C.Z. Ning, H.Haken, Detuned lasers and the complex Lorenz equations:subcitical and super-critical Hopf bifurcations[J]. Phys.Rev. A.,1990,41 (7):3826-3837.
[43]T. Kapitaniak, L. O. Chua. Hyper-chaotic attractor of unidirectionally-coupled Chua's circuit[J].Int. J. Bifurcat. Chaos,1994,4(2):477-482.
[44]K. Thamilmaran, M. Lakshmanan, A. Venkatesan. Hyperchaos in a Modified Canonical Chua's Circuit, Int.J.Bifurcat. Chaos,2004,14 (221):221-243.
[45]A. Cenys, A. Tamasvisus, A. Baziliauskas. Hyperchaos in coupled Colpitts oscillators[J]. Chaos, Solit. Fract.,2003,17 (2):349-353.
[46]V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J.B. Cuenot, P. Levy, J.B. Cuenot, P. Levy, W.T. Rhodes. Communicating with hyperchaos:The dynamics of a DNLF emitter and recovery of transmitted information[J]. Optics and Spectroscopy,2003,95 (1):114-118.
[47]R. Vicente, J. Dauden, P. Colet, R. Toral Analysis and Characterization of the Hyperchaos Generated by a Semiconductor Laser Subject to delayed feedback Loop[J]. IEEE J. Quantum Electronics,2005,41(4):541-548.
[48]P. Arena, S. Baglio, L. Fortuna, G. Manganaro. Hyperchaos from cellular networks[J] Electronics Letters,1995,31(4):250-251.
[49]J.Y. Hsieh, C.C. Hwang, A.P.Li, W.J. Li, Controlling hyper-chaos of the Rossler system, Int. J. Control,1999,72 (10):882-886.
[50]P.Q. Jiang, B.H.Wang, Bu S.L,Q.H. Xia, X.S. Luo. Hyperchaotic Sysnchronization in Deterministic Small-World Dynamical Networks[J]. Internat. J. Modern Phys.B., 2004,18 (30):2674-2679.
[51]C.P. Li, GR. Chen, A note on bifurcation in Chen's system[J]. Int. J. Bifurcat. Chaos.,2003,13(6):1609-1615.
[52]T.Ueta, GR.Chen. Bifurcation analysis of Chen's equation[J]. Int.J. Bifurcat. Chaos., 2000,10 (8):1917-1931.
[53]X.B. Zhou, Y. Wu,Y. Li, Z.X. Wei. Hopf bifurcation analysis of the Liu system[J]. Chaos, Solit. Fract.,2008,36 (5):1385-1391.
[54]B.Hassard, N. Kazarinoff, Y. Wan. Theory and Application of Hopf Bifurcation[M]. first edition, Cambridge, New York,1982.
[55]G.R. Chen, T. Ueta. Yet another chaotic attractor[J]. Int. J. Bifurcat. Chaos., 1999,9 (7):1465-1466.
[56]J.H. Lii, G. R. Chen. A new chaotic attractor coined [J]. Int. J. Bifurcat. Chaos., 2002,12 (3):659-651.
[57]J.H. Lu, G. R. Chen, D. Z. Cheng. Bridge the gap between the Lorenz system and the Chen system[J]. Int. J. Bifurcat. Chaos.,2002,12 (12):2918-2926.
[58]C. Huygens. Horologium Oscillatorium[M]. Parisiis:Apud F. Muguet 1673.
[59]Z. M. Ge, J. K. Lee. Chaos synchronization and parameter identification for gyroscope system [J].Appl. Math. Comput.,2005,163(2):667-682.
[60]Y. W. Wang, Z. H. Guan. Generalized synchronization of continuous chaotic system[J]. Chaos Solit. Fract.,2006,27(l):97-101.
[61]L. Junge, U. Parlitz, Phase synchronization of coupled Ginzburg-landau Equation [J]. Phys. Rev. E.,2000,62(1):438-441.
[62]E. M. Shahverdiev, S. Sivaprakasam, K. A. Shore. Lag sysnchronization in time-delayed systems [J]. Phys. Lett. A.,2002,292(6):320-324.
[63]B. Wang, G. J. Wen. On the synchronization of a class o chaotic systems based on backstepping method[J]. Phys. Lett. A.,2007,370(1):35-39.
[64]J. M. Nazzal, A. N. Natsheh. Chaos control using sliding-mode theory [J]. Chaos, Solit. Fract.,2007,33(2):695-702.
[65]S. ozoguz, A. S. Elwakil, K.N. Salama. N-scroll chaos generator using nonlinear transconductor[J].Electron Lett.,2002,38(14):685-686.
[66]C. C. Penga, C. L. Chen. Robust chaotic control of Lorenz system by backstepping design[J]. Chaos, Solit. Fract.,2008,37(2):598-608.
[67]M. Haeri, M. S. Tavazoei, M. R. Naseh. Synchronization of uncertain chaotic systems using active sliding mode control[J]. Chaos, Solit. Fract.,2007,33(4):1230-1239.
[68]J.H. Park. On synchronization of unified chaotic systems via nonlinear control[J]. Chaos, Soliton. Fract.,2005,25(3):699-794.
[69]L.L. Zhang,.L. H. Huang, Z. Z. Zhang, Z. Y. Wang. Fuzzy adaptive synchronization of uncertain chaotic system via delayed feedback control[J].Phys. Lett. A.2008, 372(29):6082-6086.
[70]X. Liu, J. Z. Wang, L. Huan. Global synchronization for a class of dynamical complex networks[J]. Physica A.,2007,386(1):543-556.
[71]L. Wang, X. J. Kong, H. Shi, H.P.Dai, Y. X, Sun.LMI-based criteria for Synchronization of complex dynamical networks[J]. J. Phys. A:Math. Thror.,2008, 41(28):28102.
[72]J. H. Park, S. M. Lee, H. Y. Jung. LMI optimization approach to synchronization of stochastic delayed discrete-time complex networks[J]. J. Optim. Theory Appl.,2009, 143(2):357-367.
[73]S. Kuntanapreeda. Chaos synchronization of unified chaos via LMI[J]. Phys. Lett. A. 2009,372(2):2837-2840.
[74]C. X. Liu, L. L. Liu, T. Liu. A novel three-dimensional autonomous chaos system[J]. Chaos, Solit. Fract.,2009,39(4):1950-1958.
[75]张建雄,唐万生,徐勇.一个新的三维混沌系统[J].物理学报,2008,57(11)：6799-6809.
[76]冯朝文,蔡理,唐强,张立森.一种新的三维自治混沌系统[J].物理学报,2011,60(3)：1-7.
[77]C. Silva. Si'lnikov theorem-a tutorial[J]. IEEE Trans. Circ. Syst. I.1993,40(10):68-82
[78]T. S. Zhou, G. R. Chen, Q. G. Yang. Constructing a new chaotic system based on S'ilnicov criterion[J]. Chaos, Solit. Fract.,2004,19(4):985-993.
[79]T. S. Zhou, Y. Tang, G. R. Chen. Chen's attractor exists[J]. Int. J. Bifurcat. Chaos., 2004,14(9):3167-3177.
[80]Y. X. Yang, J. H. Sun. Si'lnikov homoclinic orbit in a new chaotic system[J]. Chaos, Solit. Fract.,2007,39(1):150-159.
[81]B. Y. Chen, T. S. Zhou. S'lnikov homoclinic orbit in two class of 3D autonomous nonlinear[J]. Int J Bifurcat modern Phys B,2011,25(20):2697-2712.
[82]B. Y. Chen, T. S. Zhou, G. R. Chen.An extended Si'lnikov homoclinic theorem arid its applications[J]. Int. J. Bifurcat. Chaos.,2009,19(5):1679-1693.
[83]L. Q. Zhou, F. Q. Chen. Hopf bifurcation and Si'lnikov chaos of Genesio systems[J]. Chaos, Solit. Fract.,2009,40(3):1413-1422.
[84]X. Wang. S'lnikov chaos and Hopf bifurcation analysis of Rucklidge system[J]. Chaos, Solit. Fract.,2009,42(4):2208-2217.
[85]L. Q. Zhou, Q. F. Chen. S'lnikov chaos of Liu system[J]. Chaos,2008,18(1): 013133.
[86]S. H. Chen, J. H. L Lu. Synchronization of of an uncertain unified chaotic system via adaptive control[J]. Chaos, Solit. Fract.,2002,14(2):643-647.
[87]张兴华,丁守刚.非均匀气隙永磁同步电机的自适应混沌同步[J].控制理论与应用,2009,26(6)：661-664.
[88]C. C. Wang, S. Pain, T. Yauh. Chaos control in AFM system using sliding mode control by backstepping[J]. Commun. Nonlinear Sci. Numer. Simulat.,2010, 15(3):741-751.
[89]H. L. Guo. Generalized projective synchronization between two different chaotic systems using active backstepping control[J]. Phys. Lett. A,2006,355(4):326-330.
[90]L.L. Wang, M. C. Kuo. Robust synchronization of drive-response chaotic system via adaptive sliding mode control[J]. Chaos, Solit. Fract.,2009,39(5):2086-2092.
[91]J.J. Yan, Y. S. Yang, Y. Y. Chang, C. Y. Chen. Roboust synchronization of uncertain chaotic system via sliding mode[J]. Chaos, Solit. Fract.,2007,34(3):947-954.
[92]N. Cai, Y. W. Jing, S. Y. Zhang. Modified projective synchronization of chaotic systems with disturbances via active sliding mode control[J]. Commun. Nonlinear Sci. Numer. Simulat.,2010,15(6):1613-1620.
[93]X. Z. Liu. Impulsive synchronization of chaotic systems subject to time delay[J]. Nonliear Analysis,2009,71(12):1320-1327.
[94]M. F. Hu, Y. Q. Yang. Impulsive control of projective synchronization in chaotic Systems[J]. Phys. Lett. A,2008,372 (18):3228-323.
[95]J. H. Park, O.M. Kwon, S. M. Lee. LMI optimization approach to stabilization of Genesio-Tesi chaotic system via dynamic controller[J]. Appl. Math. Comput.,2008, 196(1):200-206.
[96]C.D. Li, X.F. Liao, R. Zhang. Impulsive synchronization of nonlinear coupled chaotic systems[J]. Phys. Lett. A,2004,328(1):47-50.
[97]H. G. Yu, S. M. Zhong, R.P, Agarwal. Mathematics analysis and chaos in an ecological model with an impulsive control strategy[J]. Commun. Nonlinear Sci. Numer. Simulat.,2011,16(2):776-786.
[98]G.X. Chen, A simple adaptive feedback control method for chaos and hyper-chaos control[J]. Appl. Math. Comput.,2011,217 (17):7258-7264.
[99]R.W. Guo, A simple adaptive controller for chaos and hyperchaos synchronization[J]., Phys. Lett. A,2008,372 (34):5593-5597.
[100]H.T. Yu, J. Wang, B. Deng, X.L, Wei, Adaptive synchronization control of coupled chaotic neurons in the external electrical stimulation[J] Chin. Phys. B.,2013, 22(5):058701.
[101]F. X. Chen, L. Chen, W.D. Zhang, Stabilization of parameters perturbation chaotic system via adaptive backstepping technique [J]. Appl. Math. Comput.,2008,200(1): 101-109.
[102]李雨珊,吕翎,刘烨,刘硕,闫兵兵,常欢,周佳楠.复杂网络时空混沌同步的Backstepping设计[J].物理学报,2013,62(2)：0205513.
[103]G.P. Jiang, W.X. Zheng. A LMI criterion fo linear-state-feedback based chaos synchronization of a class of chaotic systems [J].Chaos, Solit. Fract,2005,26 (2): 437-443.
[104]J.H. Kim, C.W. Park, E. Kim, M. Park. Fuzzy adaptive synchronization of uncertain chaotic systems[J]. Phys. Lett. A,2005,334 (4):295-305.
[105]M.G. Mahmoud, E.E. Mahmoud, A.A. Arafa. Controlling hyperchaotic complex systems with unkown parameters based on adaptive passive method[J]. Chin. Phys. B.,2013,22(6):060508.
[106]T.C. Lin, C.H. Kuo. H8 synchronization of uncertain fractional order chaotic systems[J].Adaptive fuzzy approach, IAS Transactions,2011 50 (4):548-556.
[107]C.K. Ahn, S.T. Jung, S.K. Kang, S.C. Joo. Adaptive H8 synchronization of uncertain chaotic systems with external disturbance[J]. Commun. Nonlinear Sci. Numer. Simulat.,2010,15 (8):2168-2177.
[108]C. K. Ahn.An H8 approach to anti-synchronization for chaotic systems [J]. Phys. Lett. A,2009,373 (20):1729-1733.
[109]R.M. Guerra, W. YU.Chaotic synchronization and secure communication via sliding-mode observer[J]. Int. J. Bifurcat. Chaos.,2008,18 (1):235-243.
[110]M.R. Naseh, M. Haeri. Robust synchronization of chaotic systems using active sliding mode control with minimum control effort[J].Int.J. Modern Physic B,2011, 25(17):2271-2288.
[111]M. Pourmahmood, S. Khanmohammadi, G. Alizadeh. Synchronization of two different uncertain chaotic systems with unknown paramiters using robust adaptive sliding mode controller[J]. Commun. Nonlinear Sci. Numer. Simulat.,2011,16 (7):2583-2868.
[112]刘金琨.滑模变结构MATLAB仿真[M].北京：清华大学出版社,2005.
[113]G. Y. Qi, G. R. Chen, M. Wyk, B. Wyk, Y. H. Zhang. A four-wing chaotic attractor generated from a new 3D quadratic autonomous system[J].Chaos, Solit. Fract.,2008, 38 (3):705-721.
[114]C. X. Liu. A novel chaotic attractor [J]. Chaos, Solit. Fract.,2009,39(3):1037-1045.
[115]X.H.Xiong, J. W. Wang. Conjugate Lorenz-type chaotic attractors [J].Chaos, Solit. Fract.,2009,40(2):923-929.
[116]Y.J.Liu, Q.G.Yang. Dynamics of a new Lorenz-like chaotic system[J]. Nonliear Anlysis:Real world Appl.,2010,11(4):2563-2572.
[117]B. Munmuangsaen, B. Srisuchinwong. A new five-time simple chaotic attractor[J]. Phys. lett.A,2009,373:4038-4043.
[118]J. Kennedy, S. Kocak, J. A. York. A chaos lemma [J]. Amer Math Mon,2001,108 (5):411-423.
[119]J. Kennedy, J. A. York.Topological horseshoes [J]. Trans Amer Math Soc,2001, 353(6)2513-2530.
[120]X. S. Yang, Y. Tang. Horseshoes in piecewise continuous maps [J]. Chaos, Solit. Fract,2004,19(14)841-845.
[121]B. H. Li, Y. Huang. Horseshoes in a biochemical attractor[J]. Chaos, Solit Fract.,40 (2009)1207-1212.
[122]M.C. Li, M. Malkin. Topological horseshoes for perturbations of singular difference equations[J]. Nonlinearity,2006,19 (4):795-811.
[123]W. Z. Huang, Y. Huang. Chaos, bifurcation and robustness of a class of Hopfield neural networks[J]. Int. J. Bifurcat. Chaos.,2011,21 (3):885-895.
[124]Q. Yuan, Q.D. Li, X.S. Yang. Horseshoe chaos in a class of simple Hopfield neural networks [J]. Chaos Solit Fract.,2009,39 (4):1552-1529.
[125]C. Ma, X.Y. Wang. Hopf bifurcation and topological horseshoe of a novel finance chaotic system[J] Commun. Nonlinear Sci. Numer. Simulat.,2012,17(2):721-730.
[126]Q.D. Li, X.S. Yang. A simple method for finding topological horseshoes[J]. Int. J. Bifurcat. Chaos.,2010,20 (2):467-478.
[127]X.S. Yang. Topological horseshoes in continuous maps[J]. Chaos Solit Fract.,2007, 33(1):225-233.
[128]G. G. Dong, S. Zhang, L. X. Tian, R. J. Du. Spectrum analysis and circuit Implementation of a new 3D chaotic system with novel chatic attractors[J]. Chin. Phys. Lett.,2010,27(2):02057.
[129]S. Celikovsky, G. R. Chen. On the generalized Lorenz canonical form [J]. Chaos Solit Fract.,2005,26(5):1271-1276.
[130]S. J. Schiff, K. Jerger, D. H. Duong, T. Chang, M. L. Spano, W. L. Ditto. Controllingchaos brains in the brain[J]. Nature,1994,370 (6491):615-620.
[131]G. R. Cheng, D.J. Lai. Feedback control of Lyapunov exponents for discrete-time dynamical[J].Int. J. Bifurcat. Chaos.1996,6 (7):1341-1349.
[132]G. R. Cheng, D. J. Lai. Feedback anticontrol of chaos[J]. Int. J. Bifurcat. Chaos.,1998, 8(7):1585-1590.
[133]X. F. Wang, G. R. Cheng. On feedback anticontrol of discrete chaos[J]. Int. J. Bifurcat. Chaos.,1998,9(7):1435-1441.
[134]X. F. Wang, G R. Cheng. Yet another algorithm for chaotifying control of discrete chaos[J].Control Theory Appl,2000,17(3):336-341.
[135]X. F. Wang, G. R. Cheng. Chaotification via arbitrarily small feedback control[J]. Int. J. Bifurcat. Chaos.,2000,10(3):549-575.
[136]Q.G Yang, Z.C. Zhou, C.R.Chen. An unusual 3D autonomous quadratic chaotic system with two stable node-foci[J]. Int. J. Bifurcat. Chaos.,2010,20(4):1061-1083.
[137]Q.J.Zhang, J. A. Lu. Chaos synchronization of a new chaotic via nonlinear control[J]. Chaos, Solit. Fract.,2008,37(1):175-179.
[138]王春华,胡燕,余飞,徐洁.一类混沌系统时间可控的自适应投影同步[J].物理学报,2013,62(11)：110509.
[139]J.Zhou, J.A.Lu, J.H. Lii. Pinning adaptive synchronization of a general complex dynamical network[J]. Automatica,2008,44(4):996-1003.
[140]J.Zhou, J.A.Lu, J.H.Lu. Adaptive synchronization of an uncertain complex dynamatic network[J]. IEEE Trans, on Automatic control,2006,51(4):652-656.
[141]于海涛,王江.FitzHugh-Nagumo基于反演自适应动态滑模的神经元混沌同步控制[J].物理学报,2013,62(17)：170511.
[142]Z.Y.Shun, W.Z, Zhu, G. Q. Si, Y.Ge, Y.B. Zhang. Adaptive synchronization design for uncertain chaotic systems in the presence of unknown system parameters:a revisist [J]. Nonlinear Dyn.,2013,72(4):729-749.
[143]C.C.Yang, C.J.Ou. Adaptive teminal sliding mode control subject to input nonlinearity for synchronization of chaotic gyros[J] Commun.Nonlinear Sci. Numer. Simulat.,2013,18(3):682-691.
[144]马铁东,江伟波,浮洁,柴毅,陈立平,薛方正.一类分数阶混沌系统的自适应同步[J].物理学报,2012,61(16)：160506.
[145]祝大伟,涂俐兰.随机扰动下Lorenz混沌系统的自适应同步与参数识别[J].物理学报,2013,62(5):050508.
[146]邓玮,方洁,吴振军,吴艳军,含有不确定项的混沌系统自适应修正函数投影同步[J].物理学报，2012,61(14):140503.
[147]X.Y. Wang, Y.Q. Wang. Adaptive control for synchronization of a four-dimensional chaotic system via a single variable[J]. Nonlinear Dyn,2011,65(3):311-316.
[148]L.Q.Shen, M. Wang. Adaptive control of chaotic system based on a single layer neural network[J]. Phys. Lett. A.,2007,368(5):379-382.
[149]S. Kuntanpreeda. An observer-based neural network controller for chaotic Lorenz system[J]. Lect. Notes Comput. Sci.,2008,5370:608-617.
[150]H.T. Yau, C.S. Shieh. Chaos synchronization using fuzzy logic controller [J]. Nonlinear Anal.:Real World Appl.,2008,9(4):1808-1810.
[151]J.Wen, C.S. Jiang. Adaptive fuzzy control for a class of chaotic systems with nonaffine inputs [J]. Commun.Nonlinear Sci. Numer. Simulat.,2011,16(1):475-492.
[152]J.G Lu, D.J. Hill. Global asymptotical synchronization of chaotic Lur'e systems using sampled data: A linear matrix inequality approach[J]. IEEE Trans, irc. Syst. II:Exp Brief,2008,55(6):586-590.
[153]Y. He, G. L. Wen, Q. G. Wang. Delay-dependent synchronization criterion for Lur'e systems with delay feedback control [J].Int. J. Bifur. Chaos,2006,16(10):3087-3091.
[154]Y.Z, Sun, W.Li, J. Ruan. Generalized outer synchronization between complex dynamical nerworks with time delay and noise perturbation [J]. Commun. Nonlinear Sci. Numer. Simulat.,2013,18(4):989-998.
[155]M.M. Asheghan, M. T. H. Beheshti. An LMI approach to robust synchronization of a class of chaotic systems with gain variations[J]. Chaos Solit. Fract.,2009, 42(2):1106-1111.
[156]R. C. Wu, W. W. Zhang. Synchronization of delayed chaotic neural networks by LMI[C].Proceedings of the 2008 Second International Conference on Future Generation Communication and Networking Symposia(FGCNS), Hinan, China,2008, 55-59.
[157]F.X. Chen, W. D. Zhang. LMI criteria for robust chaos synchronization of a class of chaotic systems[J]. Nonlinear Ananlysis:Theroy, Methods and Applications,2007, 67(12):3384-3393.
[158]B. Li, D. Yang, X. H. Zhang, L. T. Ma. Chaotic lag synchronization of coupled time-delayed neural network with two neurons using LMI approach[J].Acta Automatica Sinica,2007,33(11):1196-1200.
[159]任涛,朱志良,于海,.王猛.基于Min-Max方法的混沌系统采样同步控制研究[J]物理学报,2013,62(17)：170501.
[160]K. Tanaka, T. IKeda, H. O. Wang. A unified approach to controlling chaos via an LMI-based fuzzy control system design[J]. IEEE Trans. Circ. And Syst,-I,1998, 45(10):1021-1040.
[161]S.Kuntanapreeda. Simple LMI-based synchronization of fractional-order chaotic systems[J]. Int. J. Bifur. Chaos,2013,23(1):1350011.
[162]吴忠强,杨阳,徐纯华,混沌状态下永磁同步发电机的故障诊断—LMI法研究[J].物理学报,2013,62(15)：150507.
[163]李水平,许志军.利用小增益定理实现Liu系统的同步[J].吉林师范大学学报,2010,1(1)：45-48.
[164]姜玉秋.Lorenz混沌系统的线性同步控制[J].吉林大学学,2011,29(6)：571-575.
[165]J.Xiao. Anti-synchronization of a new hyperchaotic system via small-gain theorem[J]. Chin. Phys. B.,2010,19(10):100505.
[166]J.T. Li, W.L.Li, Q.P.Li, Sliding mode control for uncertain chaotic system with input Nonlinearty[J]. Commun.Nonlinear Sci. Numer. Simulat.,2012,17(1):341-348.
[167]F. Y. Liu, T.S.Li. Z.F.Li, R.H. Li. Adaptive terminal sliding mode control for anti-synchronization of uncertain chaotic system[J]. Nonlinear.Dyn.,2013,74(4): 991-1002.
[168]X.Y. Wang, D.Lin, Z.J.Wang. Controlling the uncertain multi-scroll critical chaotic system with input nonlinear using sliding mode control[J]. Mordern Physics letters B, 2009,23(16):2021-2034.
[169]M.P.Aghababa, H.P. Aghababa. A novel finite-time sliding mode controller for synchronization with input nonlinearity[J]. Arab. J. Sci. Eng,2013,38(11): 3221-3232.
[170]倪骏康,刘崇新,庞霞.电力系统混沌振荡的等效快速终端模糊滑模控制[J].物理学报,2013,62(19)：190507.
[171]吕翎,李雨珊,韦琳玲,于淼,张檬.基于滑模控制法实现规则网络混沌同步[J].物理学报,2012,61(12)：120504.
[172]吴忠强,谭拂晓,王绍仙.基于无源化的细胞神经网络超混沌系统同步[J].物理学报,2006,60(11)：1651-1658.
[173]韦笃取,张波.基于无源理论自适应镇定具有输入的永磁同步电动机的混沌运动[J].物理学报,2012,61(3)：030505.
[174]H. Hamiche, K. Kemih, M. Ghanes, G. Zhang, S. Djennoune. Passive and impulsive synchronization of a new four-dimensional chaotic system[J]. Nonlinear Analysis, 2011,74(4):1146-1154.
[175]关治洪,廖俊锋,廖锐全.不确定脉冲系统的鲁棒H∞控制[J].控制理论与应用2002,19(4)：623-626.
[176]陈贵词,张芙蓉.一类不确定时滞系统的脉冲控制[J].武汉大学学报(自然科学版),2006,29(5)：517-519.
[177]成新明,关治洪,刘新芝.一类不确定的脉冲控制[J].电机与控制学报,.2002,6(4)：310-313.
[178]C. D. Li, X. F. Liao, R. Zhang. Impulsive synchronization of nonlinear coupled chaotic systems[J]. Phys. Lett. A.,2004,328(1):47-50.
[179]G.J. Liu. A novel study for impulsive synchronization of fractional-order chaotic systems[J]. Chin. Phys. B.,2013,22(6):060510.
[180]C.D. Li, X.F. Liao, X.Y. Zhang. Impulsive synchronization of chaotic systems[J]. Chaos,2005,15:023104.
[181]Y.S. Chen, C.C. Chang. Adaptive impulsive synchronization of nonlinear chaotic systems[J]. Nonlinear Dvn,2012,70(3):1795-1803.
[182]胡汉平,于志良,刘凌锋.光电反馈混沌系统脉冲同步特性研究[J].物理学报,2012,61(19)：10504.
[183]罗永健,于茜,张东.参数不确定时延超混沌系统的脉冲同步方法研究[J].物理学报,2011,60 (11):0504.C. Ma, X.Y. Wang, Impulsive control and synchronization of a new unified hyperchaotic system with varying control gains and impulsive intervals[J]. Nonlinear Dyn,2012,70(1):551-558.