摘要
混沌系统的混沌同步与反同步是近年来非线性科学领域研究的热点问题,也是混沌保密通信的关键。本文用理论推导和数值模拟相结合的方法研究了混沌系统的反同步以及混沌在保密通信中的应用。
对于阶数相同的混沌系统,先设计了一类主动控制器,把误差系统的项全部抵消,实现了参数已知的Lorenz系统和Chen系统的反同步；然后采用参数自适应辨识与反馈控制的方法,设计出一种自适应控制器,实现了参数未知的Lorenz混沌系统和Lu系统的反同步。对不同阶数的两个混沌系统,先采用降阶的思想,把多余的阶数当成干扰项看待,从而把不同阶数混沌系统的反同步问题转化为相同阶数混沌系统的反同步,结合自适应控制方法,实现了参数未知的超混沌Lorenz系统与新系统的完全反同步,并估计出系统的部分未知参数；然后在不改变系统混沌特性的基础上,用升阶的方法把系统变为相同的阶数,也实现了超混沌Lorenz系统与新系统的完全反同步,并使系统所有的“未知参数”都得到估计。每种方法都用Matlab软件进行数值模拟,并画出反同步的误差图。最后,基于混沌掩盖和混沌参数调制技术,构造了混沌保密通信系统,数值仿真表明,在接收端,有用信号能被有效地恢复出来。
The synchronization and anti-synchronization of chaotic systems are hotspots in nonlinear science fields in recent years and also be the key point in secure communication. Using the methods of theoretical derivation and numerical simulation, this dissertation studies the anti-synchronization of chaotic systems as well as the application of chaos in secure communication.
For two chaotic systems with the same order, firstly, an active controller is designed to realize the anti-synchronization of Lorenze chaotic system and Chen chaotic system with the known parameter by eliminating the items of erroneous system. Then, using the strategies of adaptive control for parameters and feedback control method, an adaptive controller is designed, which realizes the anti-synchronization of Lorenze chaotic system and Lu chaotic system with the unknown parameters.
For two chaotic systems with different order, firstly, using the depressed-order method, we view the redundant order as distractors to translate the problem of anti-synchronization with different orders into the same order. At the same time, combining adaptive control method, the anti-synchronization of hyper-chaotic Lorenze system and the New chaotic system is realized and parts of parameters are estimated. Then, without changing the chaos characteristics of the system, using add-order method, the anti-synchronization of hyper-chaotic Lorenze system and the New chaotic system is also realized, and all of the unknown parameters are estimated. Numerical simulations with Matlab software is carried out for each method and the anti-synchronized charts of the erroneous system are drawn. Finally, the secure communication systems are set up by chaotic masking and chaotic parameter modulation, and the simulation results show that, useful signals can be recovered from the receiver system effectively.
引文
[1]Poincare H.,李醒民,科学的价值,北京：光明日报出版社,1998
[2]Lorenz E.N., Deterministic non-periodic flow, Journal of Atmospheric Sciences,1963, 20(2):130-141
[3]卢侃,孙建华编译.混沌学传奇[M].上海：上海翻译出版公司,1991
[4]Ruelle D., Takens F., On the nature of turbulence, Common Mathematics Physics,1971, 20(1):167-192,23(2):343-344
[5]Li T.Y, Yorke J.A., Period three implies chaos, American Mathematical Monthly,1975, 82:985-992
[6]May IL, Simple mathematical models with very complicated dynamics, Nature,1976, 261(5560):459-467
[7]Feigenbaum M.J., Quantitative universality for a class of nonlmear transformations, Journal of Statistical Physics,1978,19(1):25—52
[8]Grassberger P and Pracaccia., Characterization of strange attractors. Phys. Rev. Lett.50 346-349(1983)
[9]Devaney R. L. An introduction to chaotic dynamical systems. Asseion-Wesley.1987
[10]张化光,王智良,黄伟.混沌系统的控制理论[M].沈阳：东北大学出版社,2003
[11]高铁杠,混沌系统同步及其在信息安全中的应用：[博士学位论文],天津：南开大学,2005
[12]Wolf A, Swift J B, Swinney H L, and Vastano J A Determining Lyapunov exponents from a time series. Physica 1985,16 D:285-317
[13]吕金虎,陆君安,陈士化,混沌时间序列分析及其应用,武汉大学出版社,武汉,2002
[14]刘国刚,赵怡.参数不确定混沌系统的自适应同步控制[J].电路与系统学,2005,10(1)：24-27
[15]齐冬莲,魏金岭,赵光宇.基于系统辨别的自适应混沌同步控制研究[J].控制与决策,2001,16(1)：120-122
[16]Anil Maybhate, R. E. Amtitkar. Use of synchronization and adaptive control in paramete estimation form a time series[J]. Phys. Rev. Lett.,1999,59(1):284-292
[17]Moez Feki. An adaptive chaos synchronization scheme applied to secure communication[J]. Chaos, Solitions and Fractals,2003,18:141-148
[18]Zhou S B, Liao X F, Yu J B, et al. Chaos and its synchronization in two-neuron systems with discrete delays[J]. Chaos,Solitions and Fractals,2004,21(1):133-142
[19]周尚波,何松柏,虞厥邦等.具有时延的神经元模型耦合系统的混沌同步[J].电子与信息学报,2002,24(10)：1428-1432
[20]Ju H. Park. Adaptive synchronization of Rossler system with uncertain parameters[J]. Chaos Solitions and Fractals,2005,25(2):325-331
[21]姚明海.混沌控制的研究进展和展望[J].浙江工业大学报,2001,29(4)：332-336
[22]Pikovsky A, Rosemblum M, and Kurths J. Synchronization, a universal concept in nonlinear sciences. Cambridge University Press:Cambridge.2001
[23]Boccaletti S, Kurths J, Osipov G, Valladares D L, and Zhou C S. The synchronization of chaotic systems. Phys. Reports,2002,366:1-101
[24]李辉.混沌数字通信[M].北京：清华大学出版社,2006.11
[25]赵耿,方锦清.混沌通信分类及其保密通信研究[J].自然杂志,2003,25(1)：21-30
[26]赵耿,方锦清.现代信息安全与混沌保密通信应用研究的进展[J].物理学进展,2003,23(2)：212-255
[27]瞿少成,王永骥.基于主动滑模控制的一类不确定混沌系统的同步[J].计算机工程与应用,2006,01：180-182
[28]KIM C M, RIM S H, KEY W. Anti-synchronization of chaotic oscillatrs[J].Phys. Lett; A,2003,320(1):39-46
[29]Hu J, CHEN S H, CHEN L. Adaptive control for anti-synchronization of Chua's chaotic system[J]. Phys. Lett. A,2005,339(4):455-460
[30]刘秉正,彭建华,非线性动力学,北京：高等教育出版社,2004
[31]Pecora L M and Carroll T L. Svnchronization in chaotic systems. Phys. Rev. Lett. 1990,64:821-824
[32]贾飞蕾.非线性动力系统混沌同步的研究：[硕士学位论文].陕西：西北工业大学,2007
[33]Kocarev L and Parl i tz U. General Approach for Chaotic Synchronization with Applications to Communication. Phys. Rev. Lett.1995,74:5028-5031
[34]Ho M.C., Hung Y.C., Synchronization of two different systems by using generalized active control, Physics Leners A,2002,301(5-6):424-428
[35]YassenM,T., Chaos synchronization between two different chaotic systems using active control, Chaos, Solitons and Fractals,2005,23(1):131-140
[36]Huberman B A Lumer E., Dynamics of adaptive systems, IEEE Transaction Circuits and Systems I,1990,37(4):547-550
[37]SinhaS., Ramaswany R., Rao J., Adaptive Control in nonlinear dynamics, Physical D, 1990,43(1):115-128
[38]Li Z.G Wen C.Y, Soh Y.c., et al, The stabilization and synchronization of Chua's socillators via impulsive control, IEEE Transactions on Circuits and systems I,2001,48(11): 1351-1355
[39]Xie W. X., Wen C.Y., Li Z. G, Impulsive control for the stabilization and synchronization of Lorenz systems, Physics Letters A.2000,275(1-2):67-72
[40]陈菊芳,张入元,彭建华,脉冲驱动离散混沌系统同步的实验与理论研究,物理学报,2003,52(7)：1589-594
[41]Yan J.J., Hung M.L, Liao T.L., Adaptive sliding mode control for synchronization of chaotic gyros with fully unknown parameters,Journal of Sound and Vibration,2006, 298(1-2):298-06
[42]Yah J.J., Lin J, S Liao T, L, Synchronization of a modified Chua s circuit system via adaptive sliding mode control, Chaos, Solitons and Fractals,2006
[43]Etemadi S., Alasty A., Salarieh H., Synchronization of chaotic systems with parameter uncertainties via variable structure control, PhysicsLetters A,2006,357(1):17-21
[44]Yu Y.G, Zhang S.C., Controlling unto'rain Lu system using baekstepping design. Chaos, Solitons and Fractals,2003,15(5):897-902
[45]Ln J.H., Zhang S.C, Controlling Chen's chaotic attractor using backstepping design based on parameters identification, Physical Letter A,2001,286(2.3):48-152
[46]关新平,陈彩莲,范正平,基于backstepping方法的混沌系统自适应模糊控制,物理学报,2002,51(4)：753-758
[47]杨涛,邵惠鹤,基于遗传基因算法混沌系统同步的研究,控制理论与应用,2002,19(5)：789-792
[48]Oliveira A. GD., Jones A J., Synchronization of chaotic maps by feedback control and application to secure communications using chaotic neural networks, International Journal of Bifurcation of Chaos,1998,8(12):2225-2237
[49]Lai Y.C., Grebogi C., Synchronization of chaotic trajectories using control, Physical Review E,1993,47(4):2357-2360
[50]Chen G, Ueta T. Yet another chaotic attractor[J]. Int. J. Bifurcation and Chaos,1999, 9(7):1465-1466
[51]关新平,范正平等.混沌控制及其在保密通信中的应用[M].北京：国防工业出版社,2002
[52]Lu J H, Chen G, Zhang S. Dynamical analysis of a new chaotic attractor[J]. Int. J. Bifurcat Chaos,2002,12:1001-1015
[53]Lu J H, Chen G, Cheng D Z, Celikovsky S. Bridge the gap between the Lorenz system and the Chen system[J]. Int. J. Bifur Chaos,2002,12:2917-26
[54]Chen H.K., Lee C.I., Anti-control of chaos in rigid body motion,Chaos,Solitons and Fractals,2004,21(4):957-965
[55]Rossler O E. An equation for continuous chaos[J]. Phys. Lett A, 1976,57(5):397-398.
[56]Chen G, Dong X. From Chaos to Order:Methdologies. Perspectives and Applications (Singapore:World Scientific) chapt,1998.
[57]王光瑞,于熙龄,陈式刚.混沌的控制,同步与利用(北京：国防工业出版社)2001
[58]王兴元.复杂非线性系统中的混沌(北京：电子工业出版社)第二章.2003
[59]Li Y X, Tang W K S, Chen G R 2005 Int. J Bifur. Chaos 153-367
[60]Chen A M, Lu J A, Lu J H, Yu S M 2006 Physian A.64-103
[61]Nikolov S, Clodong S. Chaos, Solitions Fract.2004,22-40
[62]王兴元,王明军.超混沌Lorenz系统[J].物理学报.2007(09)
[63]Bowong S. Stability analysis for the synchronization of chaotic systems with different order:application to secure communications. Physics Letters A,2004,326(1-2):102-113
[64]Bowong S. and McCl intock P V E. Adaptive synchronization between chaotic dynamical systems of different order. Physics Letters A,2006,358(2):134-141.
[65]Oppenheim A V, Womell G W, Isabelle S H. Signal processing in the context of chaotic signals[J]. IEEE ICASSP-92,1992 4(23-26):117-120
[66]Kocariv L, Halle K S, Eckert K.Experimental demonstration of secure communication via chaotic synchronization[J]. Int.J. of Bifurcation and Chaos,1992,2(3):120-125
[67]李辉.混沌数字通信[M].北京：清华大学出版社,2006.11.2(3)：709-713
[68]杨秀丽,王陆唐,黄肇明.混沌通信技术概述[J].微计算机信息,2004,20(12)：120-121
[69]张涛,刘宗才,刘佩田等.利用相移键控实现混沌通信[J].量子电子学报,2002,19(4)：334-336
[70]Halle K S,Wh. C.W Itoh M, et al. Spread spectrum communication modulation of chaos. Tnt.J. Bifurcation and Chaos,1993,3(2):469-477