混沌系统的广义同步及动态网络同步研究
|
摘要
近年来,混沌及混沌同步吸引了越来越多的科学家和工程技术人员的广泛重视和研究。本文首先简单介绍了混沌、混沌同步的控制和复杂网络动力学同步的发展背景。然后就两个混沌系统间的广义同步和动态网络模型的同步方面进行了深入的研究与探讨。具体的研究工作概括如下:
(1)研究了单向耦合的2个不同的混沌系统两类光滑广义同步流形的存在性。即在响应系统的修正方程在具有渐近稳定平衡点或轨道渐近稳定的周期解的情况下,一定的条件下,可将广义同步化流形存在性问题转化为Lipschitz函数族的压缩不动点问题,理论上严格证明了该广义同步化流形的不变性和指数吸引性。
进一步的,研究了对于单向耦合的2个不同的非线性混沌系统两类H?lder连续广义同步流形的存在性。特定条件下,将广义同步化流形存在性问题转化为H?lder连续函数空间上的Schauder不动点问题,证明广义同步流形的不光滑性和多值性。
(2)研究了更复杂的相互线性耦合的非线性混沌动力系统X和Y广义同步流形的存在性。根据相互耦合的修正系统的性质,将可能的七种情况转化为其中两类的特殊情况分析,即对于Y系统修正方程具有混沌态而X系统修正方程具有渐近稳定平衡点或轨道渐近稳定周期解情况下,理论证明两类双向耦合的光滑广义同步化流形的不变性和指数吸引性。类似的,利用H?lder连续函数空间上的Schauder不动点问题,数学上严格证明了广义H?lder连续同步流形的不光滑性和多值性。
(3)建立了网络间脉冲耦合形式控制实现驱动-响应动态网络的投影同步和投影聚类同步模型,应用脉冲微分方程理论的比较系统方法,给出了驱动-响应网络系统可以实现投影同步和聚类投影同步的数学的充分条件,证明了脉冲间隔可以取任意有限正值。并且由牵制控制可以调节响应网络相对于驱动节点的比例因子数值和各聚类的比例因子数值。
(4)提出并研究了线性控制驱动-响应网络的多因子投影同步模型,基于Lyapunov函数理论方法和矩阵左特征值理论,得到响应网络在对称或不对称耦合下的实现多因子投影同步的充分条件。
(5)建立了投影同步的一个新的一般的网络模型(耦合矩阵非对称且无驱动阵子),利用左特征值理论,设计了耦合强度的一个自适应规则(adaptive scheme)来实现网络多因子投影同步。
本文对所有研究的问题给出了数值仿真例子,仿真结果均很好地验证了相应的理论分析结果。
In the rencent years, the study on chaos and chaos synchronization has attracted increased attention from scientists and engineers. Firstly in this dissertation, a brief introduction to the historical backgrounds of chaos and complex networks, research progress and typical methods for chaotic synchronization are given. Then the dissertation mainly focuses on the generalized synchronization of two chaotic different systems and projective synchronization of complex dynamical networks. The author's main research work and contributions are as follows:
(1) The existence of two types generalized synchronization(GS) of chaotic unidirectional coupled systems is studied. Based on the modified system approach, GS is classified into three types: equilibrium GS, periodic GS, and C-GS, when the modified system has an asymptotically stable equilibrium, asymptotically stable limit cycles, and chaotic attractors, respectively. The existence of the first two type GS can be converted to the problem of compression fixed point under certain conditions. Strict theoretical proofs are given to exponential attractive property of generalized synchronization manifold. By using the Schauder Fixed Point Theorem in H?lder continuous function space, the existence of two types H?lder continuous inertial manifolds, equilibrium GS and periodic GS, with certain conditions will be theoretically proved.
(2) Furthermore, we studies the existence of two types generalized synchronization of bi-directionally coupled chaotic oscillators. More complex situations appear. When the modified system of Y oscillator is chaotic, while the modified system of X oscillator collapses to a stable equilibrium point or periodic orbit, the existence of generalized synchronization can be converted to the problem of Lipschitz function family’s compression fixed point under certain conditions. Proofs are also given to the existence of bi-directionally coupled smooth and H?lder continuous generalized synchronization manifold.
(3) Impulsive projective (cluster) synchronization in 1 + N coupled chaotic systems are investigated with the drive-response dynamical network (DRDN) model. Based on impulsive stability theory, some simple but less conservative criteria are achieved for projective synchronization in DRDNs. It is worth noting that any width of impulsive intervalΔ≥0 can be arbitrarily selected to implement projective synchronization(PS) with desired scaling factor under certain conditions. Furthermore, impulsive pinning scheme is also adopted to direct the (cluster) scaling factor onto the desired value.
(4) By using simple linear control, some sufficient criteria for projective synchronization with different scaling factors of the response networks of symmetrical and asymmetrical coupling are given based on Lyapunov function method.
(5) A new model of N coupled chaotic systems networks without the drive oscillator is established. By designing an adaptive scheme, sufficient criteria for the projective synchronization with different scaling factors PSDF in symmetrical and asymmetrical coupled networks are obtained based on the Lyapunov function method and the left eigenvalue theory.
For all the above theoretical results, we proposed numerical simulations which verity the corresponding theoretical results.
(1)研究了单向耦合的2个不同的混沌系统两类光滑广义同步流形的存在性。即在响应系统的修正方程在具有渐近稳定平衡点或轨道渐近稳定的周期解的情况下,一定的条件下,可将广义同步化流形存在性问题转化为Lipschitz函数族的压缩不动点问题,理论上严格证明了该广义同步化流形的不变性和指数吸引性。
进一步的,研究了对于单向耦合的2个不同的非线性混沌系统两类H?lder连续广义同步流形的存在性。特定条件下,将广义同步化流形存在性问题转化为H?lder连续函数空间上的Schauder不动点问题,证明广义同步流形的不光滑性和多值性。
(2)研究了更复杂的相互线性耦合的非线性混沌动力系统X和Y广义同步流形的存在性。根据相互耦合的修正系统的性质,将可能的七种情况转化为其中两类的特殊情况分析,即对于Y系统修正方程具有混沌态而X系统修正方程具有渐近稳定平衡点或轨道渐近稳定周期解情况下,理论证明两类双向耦合的光滑广义同步化流形的不变性和指数吸引性。类似的,利用H?lder连续函数空间上的Schauder不动点问题,数学上严格证明了广义H?lder连续同步流形的不光滑性和多值性。
(3)建立了网络间脉冲耦合形式控制实现驱动-响应动态网络的投影同步和投影聚类同步模型,应用脉冲微分方程理论的比较系统方法,给出了驱动-响应网络系统可以实现投影同步和聚类投影同步的数学的充分条件,证明了脉冲间隔可以取任意有限正值。并且由牵制控制可以调节响应网络相对于驱动节点的比例因子数值和各聚类的比例因子数值。
(4)提出并研究了线性控制驱动-响应网络的多因子投影同步模型,基于Lyapunov函数理论方法和矩阵左特征值理论,得到响应网络在对称或不对称耦合下的实现多因子投影同步的充分条件。
(5)建立了投影同步的一个新的一般的网络模型(耦合矩阵非对称且无驱动阵子),利用左特征值理论,设计了耦合强度的一个自适应规则(adaptive scheme)来实现网络多因子投影同步。
本文对所有研究的问题给出了数值仿真例子,仿真结果均很好地验证了相应的理论分析结果。
In the rencent years, the study on chaos and chaos synchronization has attracted increased attention from scientists and engineers. Firstly in this dissertation, a brief introduction to the historical backgrounds of chaos and complex networks, research progress and typical methods for chaotic synchronization are given. Then the dissertation mainly focuses on the generalized synchronization of two chaotic different systems and projective synchronization of complex dynamical networks. The author's main research work and contributions are as follows:
(1) The existence of two types generalized synchronization(GS) of chaotic unidirectional coupled systems is studied. Based on the modified system approach, GS is classified into three types: equilibrium GS, periodic GS, and C-GS, when the modified system has an asymptotically stable equilibrium, asymptotically stable limit cycles, and chaotic attractors, respectively. The existence of the first two type GS can be converted to the problem of compression fixed point under certain conditions. Strict theoretical proofs are given to exponential attractive property of generalized synchronization manifold. By using the Schauder Fixed Point Theorem in H?lder continuous function space, the existence of two types H?lder continuous inertial manifolds, equilibrium GS and periodic GS, with certain conditions will be theoretically proved.
(2) Furthermore, we studies the existence of two types generalized synchronization of bi-directionally coupled chaotic oscillators. More complex situations appear. When the modified system of Y oscillator is chaotic, while the modified system of X oscillator collapses to a stable equilibrium point or periodic orbit, the existence of generalized synchronization can be converted to the problem of Lipschitz function family’s compression fixed point under certain conditions. Proofs are also given to the existence of bi-directionally coupled smooth and H?lder continuous generalized synchronization manifold.
(3) Impulsive projective (cluster) synchronization in 1 + N coupled chaotic systems are investigated with the drive-response dynamical network (DRDN) model. Based on impulsive stability theory, some simple but less conservative criteria are achieved for projective synchronization in DRDNs. It is worth noting that any width of impulsive intervalΔ≥0 can be arbitrarily selected to implement projective synchronization(PS) with desired scaling factor under certain conditions. Furthermore, impulsive pinning scheme is also adopted to direct the (cluster) scaling factor onto the desired value.
(4) By using simple linear control, some sufficient criteria for projective synchronization with different scaling factors of the response networks of symmetrical and asymmetrical coupling are given based on Lyapunov function method.
(5) A new model of N coupled chaotic systems networks without the drive oscillator is established. By designing an adaptive scheme, sufficient criteria for the projective synchronization with different scaling factors PSDF in symmetrical and asymmetrical coupled networks are obtained based on the Lyapunov function method and the left eigenvalue theory.
For all the above theoretical results, we proposed numerical simulations which verity the corresponding theoretical results.
引文
[1] Poincare J H.,科学与方法[M].郑太朴译.上海:商务印书馆,1933
[2] H.M.Morse, Recurrent geodesics on a surface of negative curvature[M]. Transactions of the American Mathematical Society, 1921
[3]陈关荣,汪小帆.动力系统的混沌化——理论、方法与应用[M].上海:上海交通大学出版社,2006.1-9
[4]刘曾荣.混沌的微扰判据[M].上海:上海科技教育出版社, 1995
[5]邹恩,李祥飞,陈建国.混沌控制及其优化应用[M].长沙:国防科技大学出版社,2002
[6]黄润生,黄浩.混沌及其应用[M].第二版.武汉:武汉大学出版社,2005
[7]普利高津,斯唐热,从混沌到有序[M].上海:上海译文出版社,1987
[8]柯尔莫哥洛夫,在具有小改变量的哈密顿函数中条件周期运动的保持性[C].国际数学会议:阿姆斯特丹,1954
[9] Lorenz E.N. Deterministic non-periodic flows[J]. J.Atmos.Sci, 1963,20:130-141
[10] Hénon M. A two-dimensional mapping with a strange attractor [J]. Commun Math Phys, 1976, 50: 69-77
[11] Li T Y, Yorke J A. Period three implies chaos [J]. American Mathematical Monthly, 1975, 82: 985-992
[12] Ott E, Grebogi C, Yorke J A. Controlling chaos [J]. Physical Review Letters, 1990, 64: 1196-1199
[13] Pecora L M, Carroll T L. Synchronization in chaotic systems [J]. Physical Review Letters, 1990, 64: 821-824
[14] Komiss G. Synchronization in weighted uncorrelated complex networks in a noisy environment: Optimization and connections with transport efficiency[J]. Phys. Rev. E. 2007,75: 051121
[15] Chavez M, Hwang D.-U, Amann A et al. Synchronizing weighted complex networks[J]. Chaos. 2006,16: 015106
[16] Barahona M, Pecora L M. Synchronization in small-world systems[J]. Phys. Rev. Lett. 2002,89:054101
[17] Wang X F, Chen G R. Synchronization in small-world dynamical networks[J]. International Journal of Bifurcation and Chaos.2002, 12: 187-192
[18] Li C G, Chen G R. Phase synchronization in small-world networks of chaotic oscillators[J]. Physica A., 2004, 341:73-79
[19]刘延柱,陈立群.非线性动力学[M].上海:上海交通大学出版社, 2000
[20] Day R H. The emergence of chaos from classical economic growth[J]. Quarterly Journal of Econimics, 1983, 97(2): 201-213
[21] Devaney R. An introduction to chaotic dynamical systems[M]. New York:Addison-Wesley Publishing Company, 1989
[22] John G, Philip H., Nonlinear oscillations, Dynamical systems and bifurcations of vector fields[M]. New York: Springer, 1983, 166-211
[23] Li C, Chen G. Chaos and hyperchaos in the fractional-order R?ssler equations [J]. Physica A, 2004, 341: 55-61
[24] Yu X. Controlling Lorenz chaos [J]. Int J Syst Sci, 1996, 27(4): 355-359
[25] Madan R N. Chua’s Circuit: A paradigm for chaos [M]. Singapore: World Scientific, 1993
[26] Chen G, Ueta T. Yet another chaotic attractor [J]. Int J Bifurcation chaos, 1990, 9(7): 1465-1466
[27] Hu Manfeng, Xu Zhenyuan. Adaptive projective synchronization of unified chaotic systems and its application to secure communication[J]. Chinese Physics.2007, 16(11):3231-3237
[28] Huygenii G. Horoloquim oscilatorium[J]. France: Parisiis,1673.
[29]郑志刚.耦合非线性系统的时空动力学与合作行为[M].北京:高等教育出版,2004
[30] Rayleigh J. The theory of sound[M]. New York: Dover Publishers,1945
[31] Winfree A T. Geometry of biological time[M]. New York: Springer-Verlag, 1990
[32] Glass L, Mackay M C.From clocks to chaos: the rhythms of life[M]. Princeton NJ: Princeton University Press, 1988
[33] Lu J Q, Cao J D. Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters[J]. Chaos, 2005, 15:043901
[34] Zhan M, Wang X G, Gong X F et al. Complete synchronization and generalized synchronization of one-way coupled time-delay systems[J]. Phys. Rev. E, 2003, 68: 036208
[35] Rulkov N F, Sushchik M M, Tsimring L S et al. Generalized synchronization of chaos in directionally coupled chaotic systems[J]. Phys. Rev. E., 1995, 51:980-994
[36] Kocarev L, Parlitz U. Generalized Synchronization, Predictability, and Equivalence of Unidirectionally Coupled Dynamical Systems[J]. Phys. Rev. Lett. 1996,76:1816 - 1819
[37] Abarbanel H D I, Rulkov N F, Sushchik M M. Generalized synchronization of chaos: The auxiliary system approach[J]. Phys. Rev. E.1996, 53:4528-4535
[38] Rosenblum M G, Pikovsky A S, Kurths J., Phase Synchronization of Chaotic Oscillators[J]. Phys. Rev. Lett.1996, 76:1804-1807
[39] Blasius B, Huppert A, Stone L., Complex dynamics and phase synchronization in spatially extended ecological systems[J]. Nature. 1999, 399(6734): 354-359
[40]Parlitz U, Junge L, Lauterborn W., Experimental observation of phase synchronization[J]. Phys. Rev. E. 1996,54:2115-2117
[41] Mainieri R, Rehacek J., Projective synchronization in three-dimensional chaotic systems[J]. Phys. Rev. Lett, 1999, 82:3042-3045
[42] Guo Liuxiao and Xu Zhenyuan, Projective Synchronization in Drive--Response Networks via Impulsive Control[J]. Chinese Physics Letters, 2008, 25:2816-2819
[43] Guo Liuxiao and Xu Zhenyuan,Adaptive projective synchronization with differentscaling factors in networks[J], Chinese Physics B, 2008, 17:4067-4072
[44] Wen G L, Xu D. Nonlinear observer control for full-state projective synchronization in chaotic continuous-time systems[J], Chaos, Solitons and Fractals, 2005, 26:71-77
[45] Hu M F, Xu Z Y, Zhang R. Full state hybrid projective synchronization in continuous-time chaotic (hyperchaotic) systems[J]. Communications in Nonlinear Science and Numerical Simulation, 2008,13(2):456-464
[46] Manfeng Hu, Zhenyuan Xu, Rong Zhang, Full state hybrid projective synchronization in continuous-time chaotic (hyperchaotic) systems[J], Communications in Nonlinear Science and Numerical Simulation, 2008, 13(2): 456-464.
[47] Hu M F, Xu Z Y, Zhang R., Parameters identification and adaptive full state hybrid projective synchronization of chaotic (hyper-chaotic) systems[J]. Phys Lett.A., 2007, 361: 231-237
[48] Hu M F, Xu Z Y, Zhang R., Adaptive full state hybrid projective synchronization of chaotic systems with the same and different order[J]. Phys Lett.A., 2007, 365: 315-327
[49] Yan Z Y. Q-S (lag or anticipated) synchronization backstepping scheme in a class of continuous-time hyperchaotic systems—A symbolic-numeric computation approach[J]. Chaos, 2005,15: 023902
[50]Yan Z Y. Chaos Q-S synchronization between R?ssler system and a new unified chaotic system[J]. Phys. Lett. A, 2005, 334:406-412
[51] Hu M F, Xu Z Y, Yang Y Q, Projective cluster synchronization in drive-response dynamical networks[J]. Physica A: Statistical Mechanics and its Applications, 2008, 387: 3759-3768.
[52] Brown R., Kocarev L. A unifying definition of synchronization for dynamical systems[J]. Chaos, 2000,10(2):344-349
[53] Murali K,Lakshmanan M. Drive-response scenario of chaos synchronization in identical nonlinear systems[J]. Phys. Rev. E. 1994,49:4882 - 4885
[54] Yassen M T. Controlling chaos and synchronization for new chaotic system using linear feedback control[J]. Chaos, Solitons and Fractals.2005, 26(3): 913-920
[55] Wang F Q, Liu C X. A new criterion for chaos and hyperchaos synchronization using linear feedback control[J]. Phys.Lett.A.2006, 360:274-278
[56]王发强,刘崇新. Liu混沌系统的线性反馈同步控制及电路实验的研究[J].物理学报,2006,55(10): 5055-5060
[57] Wu X F, Chen G, Cai J P. Chaos synchronization of the master–slave generalized Lorenz systems via linear state error feedback control[J]. Physica D.2007, 229:52-80
[58] Lu J A, Wu X Q, Han X P et al. Adaptive feedback synchronization of a unified chaotic system[J], Phys.Lett.A.2004, 329:327-333
[59] Lei Y M, Xu W, Shen J W. Robust synchronization of chaotic non-autonomous systems using adaptive-feedback control[J]. Chaos, Solitons and Fractal, 2007, 31:371-379
[60] Yassen M T. Feedback and adaptive synchronization of chaotic Lüsystem[J], Chaos, Solitons and Fractal, 2005, 25:379-386
[61] Tao C H, Liu X F. Feedback and adaptive control and synchronization of a set of chaotic and hyperchaotic systems[J]. Chaos, Solitons and Fractal, 2007,32:1572-1581
[62] Yan Z Y, Yu P. Linear feedback control, adaptive feedback control and theircombination for chaos (lag) synchronization of LC chaotic systems[J]. Chaos, Solitons andFractal, 2007, 33: 419-435
[63] Ali M K, Fang J Q. Synchronization of chaos and hyperchaos using linear andnonlinear feedback functions[J]. Phys. Rev. E.1997, 55:5285-5290
[64]陈志盛,孙克辉,张泰山. Liu混沌系统的非线性反馈同步控制[J].物理学报，2005,54(6): 2580-2583
[65]刘扬正,费树岷. Genesio-Tesi和Coullet混沌系统之间的非线性反馈同步[J].物理学报，2005,54(8): 3486-3490
[66] Bainov D D, Simeonov P S. Systems with impulsive effect: stability, theory andapplications[M]. New York: Halsted Press, 1989.
[67] Lakshmikantham V, Bainov D D, Simeonov P S. Theory of impulsive differentialequations[M]. Singapore: World scientific, 1989.
[68] E. M. Shahverdiev, S. Sivaprakasam, and K. A. Shore, Parameter mismatches andperfect anticipating synchronization in bidirectionally coupled external cavity laserdiodes[J]. Phys. Rev. E, 2002,66: 017206
[69] Kres imir Josi?, Invariant Manifolds and Synchronization of Coupled DynamicalSystems[J]. Phys. Rev. Lett. 1998, 80: 3053 - 3056
[70] Moore P K, Horsthemke W. Localized patterns in homogeneous networks ofdiffusively coupled reactors [J]. Physica D, 2005, 206: 121-144
[71] Chen G, Zhou J, Liu Z. Global synchronization of coupled delayed neural networksand applications to chaotic models [J]. Int J Bifurc Chaos, 2004, 14: 2229-2240
[72] Zeng Z, Wang J. Completely stability of cellular neural networks with time-varyingdelays [J]. IEEE Trans. Circuits Syst.-I, 2006, 53(4): 944-955
[73] Zhou J, Chen T. Synchronization in general complex delayed dynamical networks [J].IEEE Trans Circuits Syst-I, 2006, 53(3): 733-744
[74] Cao J D, Li P, Wang W W. Global synchronization in arrays of delayed neuralnetworks with constant and delayed coupling [J]. Phys Lett A, 2006, 353: 318-325
[75] Lu W, Chen T. New approach to synchronization analysis of linearly coupled ordinarydifferential systems [J]. Physica D, 2006, 213: 214-230
[76] Lu W, Chen T. Synchronization of coupled connected neural networks with delays [J].IEEE Trans Circuits Syst-I, 2004, 51(12): 2491-2503
[77] Wang X F, Chen G R. Synchronization in small-world dynamical networks [J]. Int JBifurc Chaos, 2002, 12: 187-192
[78] Watts D J. Small Worlds: The Dynamics of Networks between Order and Randomness[M]. Princeton: Princeton University Press, 1999
[79] Gade P M, Hu C K. Synchronous chaos in coupled map with small-world interactions[J]. Phys Rev E, 2000, 62(5): 6409-6413
[80] Wu C W, Chua L O. Synchronization in an array of linearly coupled dynamicalsystems [J]. IEEE Trans Circuits Syst-I, 1995, 42(8): 430-447
[81] Wang X F, Chen G. Synchronization in scale-free dynamical networks: Robustness andFragility [J]. IEEE Trans Circuits Syst-I, 2002, 49: 54-62
[82] Hu G, Qu Z L. Controlling spatiotemporal chaos in coupled map lattice systems[J],Phys. Rev. Lett. 1994,72:68-71
[83]汪小帆，李翔，陈关荣．复杂网络理论及其应用[M].北京：清华大学出版社，2006
[84] Manfeng Hu, Yongqing Yang, Zhenyuan Xu, Impulsive control of projectivesynchronization in chaotic systems[J]. Physics Letters A, 2008, 372(18), 3228-3233.
[85] Nikolai F Rulkov, M M Sushchik, L S Tsimring, H DI Abarbanel, Generalizedsynchronization of chaos in directionally coupled chaotic systems[J]. Phys.Rev.E, 1995, 51980-994.
[86] Henry D.I Abarbanel, Nikolai F Rulkov, M.M Sushchik, Generalized Synchronizationof Chaos: The Auxiliary System Approach[J]. Phys.Rev.E, 1996, 53 4528-4535.
[87] Brain R Hunt, E Ott, J A Yorke，Differentiable generalized synchronization of chaos[J].Phys.Rev.E,1997, 55 4029-4034.
[88] Z Zheng, G Hu, Generalized synchronization versus phase synchronization[J].Phys.Rev.E 2000, 62: 7882-7885.
[89] Nikolai F.Rulkov, Valentin S.Afraimovich, Clifford Tureman Lewis, Jean-ReneChazottes and Albert Cordonet , Multi-valued mappings in the generalized synchronizationof chaos[J]. Phys.Rev.E, 2001, 64: 016217.
[90] Paul So, Ernest Barreto, Kresimir Josic, Evelyn Sander and Steven J.Schiff, Limits tothe experimental detection of nonlinear synchrony[J]. Phys.Rev.E, 2002, 65: 046225.
[91] Zheng Z G, Wang X G and M.C.Cross, Transitions from partial to completegeneralized synchronizations in bi-directionally coupled chaotic oscillators[J]. Phys.Rev.E,2002, 65: 056211.
[92] A.E Hramov, A.A Koronovskii , Generalized synchronization: A modified systemapproach[J]. Phys.Rev.E, 2005, 71: 067201.
[93] J Lu，Y Xi, Linear generalized synchronization of continuous-time chaotic systems[J].chaos solitons and fractols, 2003,17: 825-843.
[94] D He, Z Zheng, L Stone, Detecting generalized synchrony: An improved approach[J].Phys.Rev.E, 2003, 67: 026223.
[95] S Boccaletti, J Kurths, G Osipov, DL Valladares,, The synchronization of chaoticsystem[J]. Physics Reports, 2002, 366: 1-101.
[96] Gálvarez, S Li, F Montoya, G Pastor, M Romera，Breaking projective chaossynchronization secure communication using filtering and generalized synchronization[J].Chaos, Solitons and Fractals 2005, 24: 775-783.
[97] Z X Dan, Z L Li, M L Quan, Construction of Generalized Synchronization for a Kindof Array Differential Equations and Applications[J]. Chin.Phys.Lett, 2003, 12: 2114-2117.
[98] K Josic, Synchronization of chaotic systems and invariant manifolds, Nonlinearity[J],2000, 13: 1321-1336.
[99]李芳,胡爱花,徐振源.两个不相同系统的广义同步化[J].物理学报, 2006, 55(2):590-597
[100]王柔怀,伍卓群.常微分方程讲义[M].北京：人民教育出版社, 1979
[101]林振声，概周期微分方程与积分流形[M].上海科学技术出版社, 1986
[102] Hu M F, Yang Y Q, Xu Z Y et al. Projective synchronization in drive-response dynamical networks[J], Physica A. 2007,381:457-466
[103] Yang T. Impulsive control[J]. IEEE Trans.Automat.Contr.1999, 44:1081-1083
[104] Chen D L, Sun J T, Wu Q D. Impulsive control and its application to Lü's chaotic system[J], Chaos, Solitons and Fractals.2004, 21:1135-1142
[105] Yang T. Impulsive control theory[M]. Berlin: Springer.2001
[106] Yang T, Yang L B, Yang C M. Impulsive synchronization of Lorenz systems[J]. Phys. Lett.A.1997, 226:349-354
[107] Itoh M, Yang T, Chua L O. Conditions for impulsive synchronization of chaotic and hyperchaotic systems[J]. Int. J. Bifurcation and Chaos. 2001,11:551-560
[108] Li C D, Liao X F, Zhang X Y. Impulsive synchronization of chaotic systems[J]. Chaos. 2005, 15:023104
[109] Bainov D D, Simeonov P S. Systems with impulsive effect: stability, theory and applications[M]. New York: Halsted Press, 1989.
[110] Lakshmikantham V, Bainov D D, Simeonov P S. Theory of impulsive differential equations[M]. Singapore: World scientific, 1989.
[111]郭大均.非线性分析中的半序方法[M].济南:山东科技出版社.1997.
[112]傅希林,闫宝强,刘衍胜.脉冲微分系统引论[M].北京:科学出版社.2005.
[113] Sun J T. Impulsive control of a new chaotic system[J]. Mathematics and Computers in Simulation.2004, 64:669-677
[114] Sun J T, Zhang Y P. Impulsive control of a nuclear spin generator[J]. Journal of Computational and Applied Mathematics.,2003, 157:235-242
[115] Akhmetov M U, Zafer A., Stability of the Zero Solution of Impulsive Differential Equations by the Lyapunov Second Method[J]. Journal of Mathematical Analysis and Applications., 2000,248:69-82
[116] Zhao H Q, Feng E M., Stability of impulsive system by perturbing Lyapunov functions[J]. Applied Mathematics Letters, 2007, 20:194-198
[117] Lakshmikantham V, Liu X Z., Stability criteria for impulsive differential equations in terms of two measures[J]. J. Math. Anal. Appl. 1989,137:591–604
[118] Wang P G, Liu X., Practical stability of impulsive hybrid differential systems in terms of two measures on time scales[J]. Nonlinear Analysis.2006, 65:2035-2042
[119] Yang T., A survey of chaotic secure communication systems[J]. International Journal of Computational Cognition., 2004, 2:81-130
[120] Bainov D D, Simeonov P S., Systems with impulsive effect: stability, theory and applications[M]. New York: Halsted Press, 1989
[121] Li Z G, Wen C Y, Soh Y C., Analysis and design of impulsive control systems[J]. IEEE Trans.Automat.Contr.2001, 46:894-897
[121] Chen T P Liu X W Lu W L, Pinning Complex Networks by a Single Controller[J]. IEEE Trans. Circuits Syst.-I, 2007, 54: 1317-1326.
[122] Lu W L Chen T P, New approach to synchronization analysis of linearly coupled ordinary differential systems[J]. Physica D, 2006, 213: 214-230.
[123] Grigoriev R O, Cross M C, Schuster H G. Pinning Control of SpatiotemporalChaos[J], Phys. Rev. Lett. 1997,79:2795-2798
[124] Wang X F, Chen G R. Pinning control of scale-free dynamical networks[J]. Physica A.2002, 310:521-531
[125] Xiang L Y, Liu Z X, Chen Z Q et al. Pinning control of complex dynamical networks with general topology[J], Physica A.2007, 379:298-306
[126] Ma Z J, Liu Z R, Zhang G. A new method to realize cluster synchronization in connected chaotic networks[J], Chaos.2006, 16:023103
[127]马忠军.混沌动力学网络上的几类同步现象研究[D][博士学位论文].上海:上海大学,2006
[128]匡继昌.常用不等式[M].(第三版)济南:山东科学技术出版社.2004,447-458
[2] H.M.Morse, Recurrent geodesics on a surface of negative curvature[M]. Transactions of the American Mathematical Society, 1921
[3]陈关荣,汪小帆.动力系统的混沌化——理论、方法与应用[M].上海:上海交通大学出版社,2006.1-9
[4]刘曾荣.混沌的微扰判据[M].上海:上海科技教育出版社, 1995
[5]邹恩,李祥飞,陈建国.混沌控制及其优化应用[M].长沙:国防科技大学出版社,2002
[6]黄润生,黄浩.混沌及其应用[M].第二版.武汉:武汉大学出版社,2005
[7]普利高津,斯唐热,从混沌到有序[M].上海:上海译文出版社,1987
[8]柯尔莫哥洛夫,在具有小改变量的哈密顿函数中条件周期运动的保持性[C].国际数学会议:阿姆斯特丹,1954
[9] Lorenz E.N. Deterministic non-periodic flows[J]. J.Atmos.Sci, 1963,20:130-141
[10] Hénon M. A two-dimensional mapping with a strange attractor [J]. Commun Math Phys, 1976, 50: 69-77
[11] Li T Y, Yorke J A. Period three implies chaos [J]. American Mathematical Monthly, 1975, 82: 985-992
[12] Ott E, Grebogi C, Yorke J A. Controlling chaos [J]. Physical Review Letters, 1990, 64: 1196-1199
[13] Pecora L M, Carroll T L. Synchronization in chaotic systems [J]. Physical Review Letters, 1990, 64: 821-824
[14] Komiss G. Synchronization in weighted uncorrelated complex networks in a noisy environment: Optimization and connections with transport efficiency[J]. Phys. Rev. E. 2007,75: 051121
[15] Chavez M, Hwang D.-U, Amann A et al. Synchronizing weighted complex networks[J]. Chaos. 2006,16: 015106
[16] Barahona M, Pecora L M. Synchronization in small-world systems[J]. Phys. Rev. Lett. 2002,89:054101
[17] Wang X F, Chen G R. Synchronization in small-world dynamical networks[J]. International Journal of Bifurcation and Chaos.2002, 12: 187-192
[18] Li C G, Chen G R. Phase synchronization in small-world networks of chaotic oscillators[J]. Physica A., 2004, 341:73-79
[19]刘延柱,陈立群.非线性动力学[M].上海:上海交通大学出版社, 2000
[20] Day R H. The emergence of chaos from classical economic growth[J]. Quarterly Journal of Econimics, 1983, 97(2): 201-213
[21] Devaney R. An introduction to chaotic dynamical systems[M]. New York:Addison-Wesley Publishing Company, 1989
[22] John G, Philip H., Nonlinear oscillations, Dynamical systems and bifurcations of vector fields[M]. New York: Springer, 1983, 166-211
[23] Li C, Chen G. Chaos and hyperchaos in the fractional-order R?ssler equations [J]. Physica A, 2004, 341: 55-61
[24] Yu X. Controlling Lorenz chaos [J]. Int J Syst Sci, 1996, 27(4): 355-359
[25] Madan R N. Chua’s Circuit: A paradigm for chaos [M]. Singapore: World Scientific, 1993
[26] Chen G, Ueta T. Yet another chaotic attractor [J]. Int J Bifurcation chaos, 1990, 9(7): 1465-1466
[27] Hu Manfeng, Xu Zhenyuan. Adaptive projective synchronization of unified chaotic systems and its application to secure communication[J]. Chinese Physics.2007, 16(11):3231-3237
[28] Huygenii G. Horoloquim oscilatorium[J]. France: Parisiis,1673.
[29]郑志刚.耦合非线性系统的时空动力学与合作行为[M].北京:高等教育出版,2004
[30] Rayleigh J. The theory of sound[M]. New York: Dover Publishers,1945
[31] Winfree A T. Geometry of biological time[M]. New York: Springer-Verlag, 1990
[32] Glass L, Mackay M C.From clocks to chaos: the rhythms of life[M]. Princeton NJ: Princeton University Press, 1988
[33] Lu J Q, Cao J D. Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters[J]. Chaos, 2005, 15:043901
[34] Zhan M, Wang X G, Gong X F et al. Complete synchronization and generalized synchronization of one-way coupled time-delay systems[J]. Phys. Rev. E, 2003, 68: 036208
[35] Rulkov N F, Sushchik M M, Tsimring L S et al. Generalized synchronization of chaos in directionally coupled chaotic systems[J]. Phys. Rev. E., 1995, 51:980-994
[36] Kocarev L, Parlitz U. Generalized Synchronization, Predictability, and Equivalence of Unidirectionally Coupled Dynamical Systems[J]. Phys. Rev. Lett. 1996,76:1816 - 1819
[37] Abarbanel H D I, Rulkov N F, Sushchik M M. Generalized synchronization of chaos: The auxiliary system approach[J]. Phys. Rev. E.1996, 53:4528-4535
[38] Rosenblum M G, Pikovsky A S, Kurths J., Phase Synchronization of Chaotic Oscillators[J]. Phys. Rev. Lett.1996, 76:1804-1807
[39] Blasius B, Huppert A, Stone L., Complex dynamics and phase synchronization in spatially extended ecological systems[J]. Nature. 1999, 399(6734): 354-359
[40]Parlitz U, Junge L, Lauterborn W., Experimental observation of phase synchronization[J]. Phys. Rev. E. 1996,54:2115-2117
[41] Mainieri R, Rehacek J., Projective synchronization in three-dimensional chaotic systems[J]. Phys. Rev. Lett, 1999, 82:3042-3045
[42] Guo Liuxiao and Xu Zhenyuan, Projective Synchronization in Drive--Response Networks via Impulsive Control[J]. Chinese Physics Letters, 2008, 25:2816-2819
[43] Guo Liuxiao and Xu Zhenyuan,Adaptive projective synchronization with differentscaling factors in networks[J], Chinese Physics B, 2008, 17:4067-4072
[44] Wen G L, Xu D. Nonlinear observer control for full-state projective synchronization in chaotic continuous-time systems[J], Chaos, Solitons and Fractals, 2005, 26:71-77
[45] Hu M F, Xu Z Y, Zhang R. Full state hybrid projective synchronization in continuous-time chaotic (hyperchaotic) systems[J]. Communications in Nonlinear Science and Numerical Simulation, 2008,13(2):456-464
[46] Manfeng Hu, Zhenyuan Xu, Rong Zhang, Full state hybrid projective synchronization in continuous-time chaotic (hyperchaotic) systems[J], Communications in Nonlinear Science and Numerical Simulation, 2008, 13(2): 456-464.
[47] Hu M F, Xu Z Y, Zhang R., Parameters identification and adaptive full state hybrid projective synchronization of chaotic (hyper-chaotic) systems[J]. Phys Lett.A., 2007, 361: 231-237
[48] Hu M F, Xu Z Y, Zhang R., Adaptive full state hybrid projective synchronization of chaotic systems with the same and different order[J]. Phys Lett.A., 2007, 365: 315-327
[49] Yan Z Y. Q-S (lag or anticipated) synchronization backstepping scheme in a class of continuous-time hyperchaotic systems—A symbolic-numeric computation approach[J]. Chaos, 2005,15: 023902
[50]Yan Z Y. Chaos Q-S synchronization between R?ssler system and a new unified chaotic system[J]. Phys. Lett. A, 2005, 334:406-412
[51] Hu M F, Xu Z Y, Yang Y Q, Projective cluster synchronization in drive-response dynamical networks[J]. Physica A: Statistical Mechanics and its Applications, 2008, 387: 3759-3768.
[52] Brown R., Kocarev L. A unifying definition of synchronization for dynamical systems[J]. Chaos, 2000,10(2):344-349
[53] Murali K,Lakshmanan M. Drive-response scenario of chaos synchronization in identical nonlinear systems[J]. Phys. Rev. E. 1994,49:4882 - 4885
[54] Yassen M T. Controlling chaos and synchronization for new chaotic system using linear feedback control[J]. Chaos, Solitons and Fractals.2005, 26(3): 913-920
[55] Wang F Q, Liu C X. A new criterion for chaos and hyperchaos synchronization using linear feedback control[J]. Phys.Lett.A.2006, 360:274-278
[56]王发强,刘崇新. Liu混沌系统的线性反馈同步控制及电路实验的研究[J].物理学报,2006,55(10): 5055-5060
[57] Wu X F, Chen G, Cai J P. Chaos synchronization of the master–slave generalized Lorenz systems via linear state error feedback control[J]. Physica D.2007, 229:52-80
[58] Lu J A, Wu X Q, Han X P et al. Adaptive feedback synchronization of a unified chaotic system[J], Phys.Lett.A.2004, 329:327-333
[59] Lei Y M, Xu W, Shen J W. Robust synchronization of chaotic non-autonomous systems using adaptive-feedback control[J]. Chaos, Solitons and Fractal, 2007, 31:371-379
[60] Yassen M T. Feedback and adaptive synchronization of chaotic Lüsystem[J], Chaos, Solitons and Fractal, 2005, 25:379-386
[61] Tao C H, Liu X F. Feedback and adaptive control and synchronization of a set of chaotic and hyperchaotic systems[J]. Chaos, Solitons and Fractal, 2007,32:1572-1581
[62] Yan Z Y, Yu P. Linear feedback control, adaptive feedback control and theircombination for chaos (lag) synchronization of LC chaotic systems[J]. Chaos, Solitons andFractal, 2007, 33: 419-435
[63] Ali M K, Fang J Q. Synchronization of chaos and hyperchaos using linear andnonlinear feedback functions[J]. Phys. Rev. E.1997, 55:5285-5290
[64]陈志盛,孙克辉,张泰山. Liu混沌系统的非线性反馈同步控制[J].物理学报，2005,54(6): 2580-2583
[65]刘扬正,费树岷. Genesio-Tesi和Coullet混沌系统之间的非线性反馈同步[J].物理学报，2005,54(8): 3486-3490
[66] Bainov D D, Simeonov P S. Systems with impulsive effect: stability, theory andapplications[M]. New York: Halsted Press, 1989.
[67] Lakshmikantham V, Bainov D D, Simeonov P S. Theory of impulsive differentialequations[M]. Singapore: World scientific, 1989.
[68] E. M. Shahverdiev, S. Sivaprakasam, and K. A. Shore, Parameter mismatches andperfect anticipating synchronization in bidirectionally coupled external cavity laserdiodes[J]. Phys. Rev. E, 2002,66: 017206
[69] Kres imir Josi?, Invariant Manifolds and Synchronization of Coupled DynamicalSystems[J]. Phys. Rev. Lett. 1998, 80: 3053 - 3056
[70] Moore P K, Horsthemke W. Localized patterns in homogeneous networks ofdiffusively coupled reactors [J]. Physica D, 2005, 206: 121-144
[71] Chen G, Zhou J, Liu Z. Global synchronization of coupled delayed neural networksand applications to chaotic models [J]. Int J Bifurc Chaos, 2004, 14: 2229-2240
[72] Zeng Z, Wang J. Completely stability of cellular neural networks with time-varyingdelays [J]. IEEE Trans. Circuits Syst.-I, 2006, 53(4): 944-955
[73] Zhou J, Chen T. Synchronization in general complex delayed dynamical networks [J].IEEE Trans Circuits Syst-I, 2006, 53(3): 733-744
[74] Cao J D, Li P, Wang W W. Global synchronization in arrays of delayed neuralnetworks with constant and delayed coupling [J]. Phys Lett A, 2006, 353: 318-325
[75] Lu W, Chen T. New approach to synchronization analysis of linearly coupled ordinarydifferential systems [J]. Physica D, 2006, 213: 214-230
[76] Lu W, Chen T. Synchronization of coupled connected neural networks with delays [J].IEEE Trans Circuits Syst-I, 2004, 51(12): 2491-2503
[77] Wang X F, Chen G R. Synchronization in small-world dynamical networks [J]. Int JBifurc Chaos, 2002, 12: 187-192
[78] Watts D J. Small Worlds: The Dynamics of Networks between Order and Randomness[M]. Princeton: Princeton University Press, 1999
[79] Gade P M, Hu C K. Synchronous chaos in coupled map with small-world interactions[J]. Phys Rev E, 2000, 62(5): 6409-6413
[80] Wu C W, Chua L O. Synchronization in an array of linearly coupled dynamicalsystems [J]. IEEE Trans Circuits Syst-I, 1995, 42(8): 430-447
[81] Wang X F, Chen G. Synchronization in scale-free dynamical networks: Robustness andFragility [J]. IEEE Trans Circuits Syst-I, 2002, 49: 54-62
[82] Hu G, Qu Z L. Controlling spatiotemporal chaos in coupled map lattice systems[J],Phys. Rev. Lett. 1994,72:68-71
[83]汪小帆，李翔，陈关荣．复杂网络理论及其应用[M].北京：清华大学出版社，2006
[84] Manfeng Hu, Yongqing Yang, Zhenyuan Xu, Impulsive control of projectivesynchronization in chaotic systems[J]. Physics Letters A, 2008, 372(18), 3228-3233.
[85] Nikolai F Rulkov, M M Sushchik, L S Tsimring, H DI Abarbanel, Generalizedsynchronization of chaos in directionally coupled chaotic systems[J]. Phys.Rev.E, 1995, 51980-994.
[86] Henry D.I Abarbanel, Nikolai F Rulkov, M.M Sushchik, Generalized Synchronizationof Chaos: The Auxiliary System Approach[J]. Phys.Rev.E, 1996, 53 4528-4535.
[87] Brain R Hunt, E Ott, J A Yorke，Differentiable generalized synchronization of chaos[J].Phys.Rev.E,1997, 55 4029-4034.
[88] Z Zheng, G Hu, Generalized synchronization versus phase synchronization[J].Phys.Rev.E 2000, 62: 7882-7885.
[89] Nikolai F.Rulkov, Valentin S.Afraimovich, Clifford Tureman Lewis, Jean-ReneChazottes and Albert Cordonet , Multi-valued mappings in the generalized synchronizationof chaos[J]. Phys.Rev.E, 2001, 64: 016217.
[90] Paul So, Ernest Barreto, Kresimir Josic, Evelyn Sander and Steven J.Schiff, Limits tothe experimental detection of nonlinear synchrony[J]. Phys.Rev.E, 2002, 65: 046225.
[91] Zheng Z G, Wang X G and M.C.Cross, Transitions from partial to completegeneralized synchronizations in bi-directionally coupled chaotic oscillators[J]. Phys.Rev.E,2002, 65: 056211.
[92] A.E Hramov, A.A Koronovskii , Generalized synchronization: A modified systemapproach[J]. Phys.Rev.E, 2005, 71: 067201.
[93] J Lu，Y Xi, Linear generalized synchronization of continuous-time chaotic systems[J].chaos solitons and fractols, 2003,17: 825-843.
[94] D He, Z Zheng, L Stone, Detecting generalized synchrony: An improved approach[J].Phys.Rev.E, 2003, 67: 026223.
[95] S Boccaletti, J Kurths, G Osipov, DL Valladares,, The synchronization of chaoticsystem[J]. Physics Reports, 2002, 366: 1-101.
[96] Gálvarez, S Li, F Montoya, G Pastor, M Romera，Breaking projective chaossynchronization secure communication using filtering and generalized synchronization[J].Chaos, Solitons and Fractals 2005, 24: 775-783.
[97] Z X Dan, Z L Li, M L Quan, Construction of Generalized Synchronization for a Kindof Array Differential Equations and Applications[J]. Chin.Phys.Lett, 2003, 12: 2114-2117.
[98] K Josic, Synchronization of chaotic systems and invariant manifolds, Nonlinearity[J],2000, 13: 1321-1336.
[99]李芳,胡爱花,徐振源.两个不相同系统的广义同步化[J].物理学报, 2006, 55(2):590-597
[100]王柔怀,伍卓群.常微分方程讲义[M].北京：人民教育出版社, 1979
[101]林振声，概周期微分方程与积分流形[M].上海科学技术出版社, 1986
[102] Hu M F, Yang Y Q, Xu Z Y et al. Projective synchronization in drive-response dynamical networks[J], Physica A. 2007,381:457-466
[103] Yang T. Impulsive control[J]. IEEE Trans.Automat.Contr.1999, 44:1081-1083
[104] Chen D L, Sun J T, Wu Q D. Impulsive control and its application to Lü's chaotic system[J], Chaos, Solitons and Fractals.2004, 21:1135-1142
[105] Yang T. Impulsive control theory[M]. Berlin: Springer.2001
[106] Yang T, Yang L B, Yang C M. Impulsive synchronization of Lorenz systems[J]. Phys. Lett.A.1997, 226:349-354
[107] Itoh M, Yang T, Chua L O. Conditions for impulsive synchronization of chaotic and hyperchaotic systems[J]. Int. J. Bifurcation and Chaos. 2001,11:551-560
[108] Li C D, Liao X F, Zhang X Y. Impulsive synchronization of chaotic systems[J]. Chaos. 2005, 15:023104
[109] Bainov D D, Simeonov P S. Systems with impulsive effect: stability, theory and applications[M]. New York: Halsted Press, 1989.
[110] Lakshmikantham V, Bainov D D, Simeonov P S. Theory of impulsive differential equations[M]. Singapore: World scientific, 1989.
[111]郭大均.非线性分析中的半序方法[M].济南:山东科技出版社.1997.
[112]傅希林,闫宝强,刘衍胜.脉冲微分系统引论[M].北京:科学出版社.2005.
[113] Sun J T. Impulsive control of a new chaotic system[J]. Mathematics and Computers in Simulation.2004, 64:669-677
[114] Sun J T, Zhang Y P. Impulsive control of a nuclear spin generator[J]. Journal of Computational and Applied Mathematics.,2003, 157:235-242
[115] Akhmetov M U, Zafer A., Stability of the Zero Solution of Impulsive Differential Equations by the Lyapunov Second Method[J]. Journal of Mathematical Analysis and Applications., 2000,248:69-82
[116] Zhao H Q, Feng E M., Stability of impulsive system by perturbing Lyapunov functions[J]. Applied Mathematics Letters, 2007, 20:194-198
[117] Lakshmikantham V, Liu X Z., Stability criteria for impulsive differential equations in terms of two measures[J]. J. Math. Anal. Appl. 1989,137:591–604
[118] Wang P G, Liu X., Practical stability of impulsive hybrid differential systems in terms of two measures on time scales[J]. Nonlinear Analysis.2006, 65:2035-2042
[119] Yang T., A survey of chaotic secure communication systems[J]. International Journal of Computational Cognition., 2004, 2:81-130
[120] Bainov D D, Simeonov P S., Systems with impulsive effect: stability, theory and applications[M]. New York: Halsted Press, 1989
[121] Li Z G, Wen C Y, Soh Y C., Analysis and design of impulsive control systems[J]. IEEE Trans.Automat.Contr.2001, 46:894-897
[121] Chen T P Liu X W Lu W L, Pinning Complex Networks by a Single Controller[J]. IEEE Trans. Circuits Syst.-I, 2007, 54: 1317-1326.
[122] Lu W L Chen T P, New approach to synchronization analysis of linearly coupled ordinary differential systems[J]. Physica D, 2006, 213: 214-230.
[123] Grigoriev R O, Cross M C, Schuster H G. Pinning Control of SpatiotemporalChaos[J], Phys. Rev. Lett. 1997,79:2795-2798
[124] Wang X F, Chen G R. Pinning control of scale-free dynamical networks[J]. Physica A.2002, 310:521-531
[125] Xiang L Y, Liu Z X, Chen Z Q et al. Pinning control of complex dynamical networks with general topology[J], Physica A.2007, 379:298-306
[126] Ma Z J, Liu Z R, Zhang G. A new method to realize cluster synchronization in connected chaotic networks[J], Chaos.2006, 16:023103
[127]马忠军.混沌动力学网络上的几类同步现象研究[D][博士学位论文].上海:上海大学,2006
[128]匡继昌.常用不等式[M].(第三版)济南:山东科学技术出版社.2004,447-458
© 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 |