摘要
非线性科学是一门研究非线性现象共性的基础科学,其中混沌理论是非线性科学的一个重要分支。本文利用理论分析和数值模拟相结合的方法研究了混沌的控制、同步以及混沌在保密通信中的应用,作者所做的主要研究工作如下。
混沌控制方面：(1)基于离散线性系统的稳定性理论,使用状态反馈法实现了一维Logistic映射和二维Logistic映射的不稳定低周期态的镇定,并且成功控制了这两种系统的分岔点。(2)分别采用直接反馈法、自适应控制法和参数扰动法对Liu系统进行混沌控制,使Liu系统收敛于平衡点,呈现周期态,甚至进入超混沌状态。(3)针对Lorenz系统,添加一个非线性控制器将之构造为超混沌系统,并对新构造的超混沌系统给出相应的电路实现。(4)为一类混沌系统设计普适追踪控制器,使之追踪任意确定的外部参考信号。
完全同步方面：(1)针对异结构混沌系统间的自适应同步,设计了具有一定普适意义的自适应同步控制器和参数更新规则,为一类参数不确定的混沌系统间的自适应同步提供了一种通用的解决方案。(2)通过几个具体例子,研究了如何使用尽量少的信道实现混沌系统的单向耦合同步。(3)为超混沌Lu系统设计简单易行的脉冲同步方案,通过单信道传输实现脉冲同步。(4)基于分数阶系统稳定性理论,研究了分数阶Liu系统在不同分数阶下的动力学行为,提出一种理论计算方法来得到分数阶系统维持混沌态的最低阶数,并给出分数阶系统实现线性耦合同步的简单判据。
广义同步方面：(1)分别用主动控制、全局控制、变量替换三种方法实现了超混沌Chen系统的反同步,并对三种方法加以对比。(2)用变量替换法实现了超混沌Chen系统的单信道投影同步。(3)提出一种分数阶混沌系统基于主动控制实现广义同步的方法,不仅适用于维数相同的混沌系统,也适用维数不同的混沌系统,作为分数阶特例的常微分系统也同样适用该广义同步方法。
混沌保密通信方面：(1)在超混沌系统单信道耦合同步的基础上,设计基于混沌掩盖的保密通信方案,并进行相应的仿真实验。(2)基于参数调制原理,设计了一种保密通信方案,该方案可用于传递模拟信号；若以不同频率的连续信号代替“0”和“1”,辅以滤波方法,还可以用该通信方案传递数字信号。仿真结果验证了该数字保密通信系统的有效性。
本文得到国家自然科学基金(60573172,60973152)、高等学校博士学科点专项科研基金(20070141014)和辽宁省自然科学基金(20082165)资助。
Nonlinear science is a foundational discipline which concerns the common properties of nonlinear phenomena. Chaos theory is one important subdiscipline of nonlinear science. In this paper, some problems about chaotic control, chaotic synchronization and its application in secure communication were studied by theoretical analysis and numerical simulations. The main work in this paper can be summarized as follows.
Chaotic control:(1) Based on the stability theory of discrete system, state feedback method is used to stabilize unstable low-periodic orbits of the Logistic map and the coupled Logistic map, and the bifurcation points of these two system are controlled successfully. (2) Direct feedback, adaptive control and parametric perturbation are used to control Liu system, and Liu system can be guided to equilibrium, periodic motion, even hyperchaos. (3) A new hyperchaotic system is designed by adding a nonlinear controller to Lorenz chaotic system, and the relevant circuit realization is given too. (4) An universal tracking controller is designed for a class of chaotic systems, and the controller can make the output signal track all kinds of reference signals.
Complete synchronization:(1) Universal adaptive synchronization controller and parameter update rule are presented for different-structure chaotic systems. With this method, a universal scheme for adaptive chaotic synchronization between uncertain systems is presented. (2) Through several examples, how to realize unidirectional coupling chaotic synchronization by less channels is discussed. (3) A feasible scheme for the impulsive synchronization of hyperchaotic Lu systems is proposed, and single channel transmitting is realized too. (4) Based on the stability theory of fractional order systems, the dynamic behavior of fractional order Liu system is studied, and a theoretical calculation method is proposed to obtain the lowest chaotic order of a fractional order system. Besides, a simple criterion for the synchronization of two identical fractional order chaotic systems is presented.
Generalized synchronization:(1) The anti-synchronization of hyperchaotic Chen system is studied respectively by active control method, global control method and variable replacement method, and the comparison of these three methods is given too. (2) With variable replacement method, the projective synchronization of hyperchaotic Chen systems is realized by single channel. (3) Based on active control, a scheme is proposed to realize generalized synchronization not only of fractional order systems with same dimension, but also of systems with different dimensions. ODE systems can be seemed as particular fractional order systems, so this scheme is available to generalized synchronization of ODE systems too.
Chaotic secure communication:(1) Based on the unidirectional coupling synchronization of hyperchaotic systems, a chaotic masking secure communication scheme is designed, and the relevant numerical simulation is given. (2) According to parameter modulation theory, a new secure communication scheme is proposed. The useful continuous signal can be transmitted successfully. Choose different-frequency signals as "0" and "1", then this scheme can be used to transmit digital signals via a filter. Numerical simulations show the effectiveness of the digital secure communication.
This research is supported by the National Natural Science Foundation of China (No: 60573172,60973152), the Superior University doctor subject special scientific research foundation of China (No:20070141014) and the National Natural Science Foundation of Liaoning province (No:20082165).
引文
[1]郝柏林.从抛物线谈起——混沌动力学引论[M].上海：上海科技教育出版社,1993.
[2]洛仑兹著.刘式达,刘式适,严中伟译.混沌的本质[M].北京：气象出版社,1997.
[3]吴祥兴,陈忠.混沌学导论[M].上海：上海科学技术文献出版社,1996.
[4]Kuntanapreeda S. Chaos synchronization of unified chaotic systems via LMI[J]. Phys. Lett. A,2009,373(32):2837-2840.
[5]Dadras S, Momeni H R. Control of a fractional-order economical system via sliding mode[J]. Physica A,2010,389(12):2434-2442.
[6]Lin S L, Tung P C. A new method for chaos control in communication systems[J], Chaos, Solitons Fract.,2009,42(5):3234-3241.
[7]Ott E, Grebogi C, Yorke J A. Controlling chaos[J]. Phys. Rev. Lett.,1990,64(11): 1196-1199.
[8]Pecora L M, Carroll T L. Synchronization in chaotic systems[J]. Phys. Rev. Lett. 1990,64(8):821-827.
[9]Pyragas K. Continuous control of chaos by self-controlling feedback [J]. Phys. Rev. A,1992,170(6):421-428.
[10]Namenko A V, Loiko N A. Chaos control in external cavity laser diodes using electronic impulsive delayed feedback[J]. Int. J. Bifur. Chaos,1998,8(9): 1791-1799.
[11]Takashi H, Masato T. Experimental stabilization of unstable periodic orbit in magneto-elastic chaos by delayed feedback control [J]. Int. J. Bifur. Chaos,1997, 7(12):2837-2846.
[12]Gao T G, Chen Z Q, Yuan Z Z, et al. A hyperchaos generated from Chen's system [J]. Int. J. Mod. Phys. C,2006,17(4):471-478.
[13]Ren H P, Liu D. Bifurcation behaviours of peak current controlled PFC boost converter[J]. Chin. Phys.,2005,14(7):1352-1358.
[14]Zheng Y A. Controlling chaos using Takagi-Sugeno fuzzy model and adaptive adjustment[J]. Chin. Phys.,2006,15(11):2549-2552.
[15]Yu Y G. Adaptive synchronization of a unified chaotic system [J]. Chaos, Solitons Fract.,2008,36(2):329-333.
[16]Sun H J, Cao H J. Chaos control and synchronization of a modified chaotic system [J]. Chaos, Solitons Fract.,2008,37(5):1442-1455.
[17]Salarieh H, Alasty A. Adaptive control of chaotic systems with stochastic time varying unknown parameters[J]. Chaos, Solitons Fract.,2008,38(1):168-177.
[18]Chen B, Liu X P, Tong S C. Robust fuzzy control of nonlinear systems with input delay[J]. Chaos, Solitons Fract.,2008,37(3):894-901.
[19]Lu J F, Ma Z X, Li L. Double delayed feedback control for the stabilization of unstable steady states in chaotic systems[J]. Commun. Nonlinear Sci. Numer. Simul.,2009, 14(7):3037-3045.
[20]Li R H. Exponential generalized synchronization of uncertain coupled chaotic systems by adaptive control[J]. Commun. Nonlinear Sci. Numer. Simul.,2009,14(6): 2757-2764.
[21]Tang F. An adaptive synchronization strategy based on active control for demodulating message hidden in chaotic signals[J]. Chaos, Solitons Fract.,2008,37(4): 1090-1096.
[22]Haeri M, Dehghani M. Impulsive synchronization of different hyperchaotic (chaotic) systems[J]. Chaos, Solitons Fract.,2008,38(1):120-131.
[23]Behzad M, Salarieh H, Alasty A. Chaos synchronization in noisy environment using nonlinear filtering and sliding mode control[J]. Chaos, Solitons Fract.,2008, 36(5):1295-1304.
[24]Kocarev L, Halle K S, Eckert K, et al. Experimental demonstration of secure communications via chaotic synchronization[J]. Int. J. Bifur. Chaos,1993,2(3): 709-713.
[25]Cuomo K M, Oppenheim A V, Strogatz S H. Synchronization of lorenzed-based chaotic circuits with applications to communications [J]. IEEE Trans. Circuits Systems-II, 1993,40(10):626-633.
[26]Dedieu H, Kennedy M P, Hasler M. Chaos shift keying:modulation and demodulation of a chaotic carrier using self-synchronizing chua's circuit[J]. IEEE Trans. Circuits Systems-II,1993,40(10):634-642.
[27]Sushchik M, Tsimring L S, Volkovskii A R. Performance analysis of correlation-based communication schemes utilizing chaos[J]. IEEE Trans. Circuits Systems-I,2001, 48(12):1684-1691.
[28]Dedieu H, Kennedy M P, Hasler M. Chaos shift keying:modulation and demodulation of a chaotic carrier using self-synchronizing Chua's circuits[J]. IEEE Trans. Circuits Systems-II,1993,40:634-642.
[29]Abel A, Schwatz W, Gotz M. Noise performance of chaotic communication systems [J]. IEEE Trans. Circuits Systems-I,2000,47(12):1726-1732.
[30]Kennedy M P, Kolumban G, Kis G, et al. Performance evaluation of FM-DCSK modulation in multipath environments [J]. IEEE Trans. Circuits Systems-I,2001,48(12): 1702-1717.
[31]Halle K S, Wu C W, Itoh M, et al. Spread spectrum communications through modulation of chaos[J]. Int. J. Bifur. Chaos,1993,3(1):469-477.
[32]Bai E W, Lonngren K E, Ucar A. Secure communication via multiple parameter modulation in a delayed chaotic system[J]. Chaos, Solitons Fract.,2005,23(3):1071-1076.
[33]Cannas B, Cincotti S, Usai E. A chaotic modulation scheme based on algebraic observability and sliding mode differentiators [J]. Chaos, Solitons Fract.,2005, 26(2):363-377.
[34]Zheng G, Boutat D, Floquet T, et al. Secure communication based on multi-input multi-output chaotic system with large message amplitude[J]. Chaos, Solitons Fract.,2009,41(3):1510-1517.
[35]Hyun C H, Park C W, Kim J H. Synchronization and secure communication of chaotic systems via robust adaptive high-gain fuzzy observer[J]. Chaos, Solitons Fract. 2009,40(5):2200-2209.
[36]Zhu F L. Observer-based synchronization of uncertain chaotic system and its application to secure communications[J]. Chaos, Solitons Fract.,2009,40(5): 2384-2391.
[37]Abarbanel H D, Linsay P S. Secure communication and unstable periodic orbit of strange attractors[J]. IEEE Trans. Circuits Systems-II,1993,40(1):576-587.
[38]Rulkov N F, Sushchik M M, Tsimring L S, et al. Digital communication using chaotic pulse-position modulation [J]. IEEE Trans. Circuits Systems-I,2001,48(12): 1436-1444.
[39]王玫,焦李成.一种基于混沌序列相关同步的DS-CDMA通信系统[J].通信学报,2002,23(8)：121-127.
[40]格莱克著.张淑誉译.混沌：开创新科学[M].上海：上海译文出版社,1990.
[41]Lorenz E N. Deterministic nonperodic flow[J]. J. Atmos. Sci.,1963,20:130-141.
[42]Li T Y, Yorke J A. Period three implies chaos[J]. Amer Math Monthly,1975,82: 985-992.
[43]May R M. Simple mathematical models with very complicated dynamics[J]. Nature, 1976,261:459-467.
[44]Feigenbaum M J. Quantitative universality for a class of nonlinear transformations[J]. J. Stat. Phys.,1978,19(1):25-52.
[45]Takens F. Detecting strange attractors in turbulence [J]. Lect Notes in Math.,1981, 898:366-381.
[46]Eckmann J P. Roads to turbulence in dissipative dynamics system [J]. Rev. Mod. Phys. 1981,53:643-649.
[47]Packard N H, Crutchfield J P, Farmer J D, et al. Geometry from a time series[J]. Phys. Rev. Lett.,1980,45:712-716.
[48]Welch P D. The use of fast fourier transform for the estimation for the estimation of power spectra:a method based on time averaging over short, modified periodograms[J]. IEEE Trans. Audio and Electroacoust,1967,15(2):70-73.
[49]Benettin G, Galgani L, Giorgilli A, et al. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems:a method for computing all of them[J]. Meccanica,1980,15:9-20.
[50]Wolf A, Swift J B, Swinney H L, et al. Determining Lyapunov exponents from a time series[J]. Physica D,1985,16:285-298.
[51]Wu X Y, Zhang H M. Synchronization of two hyperchaotic systems via adaptive control[J]. Chaos, Solitons Fract.,2009,39(5):2268-2273.
[52]Zhou X B, Wu Y, Li Y, et al. Adaptive control and synchronization of a new modified hyperchaotic Lu system with uncertain parameters [J]. Chaos, Solitons Fract.,2009, 39(5):2477-2483.
[53]Zhao J C, Lu J A. Using sampled-data feedback control and linear feedback synchronization in a new hyperchaotic system[J]. Chaos, Solitons Fract.,2008, 35(2):376-382.
[54]Yue D, Lam J, Wang Z D. Persistent disturbance rejection via state feedback for networked control systems[J]. Chaos, Solitons Fract.,2009,40(1):382-391.
[55]Cuomo K M, Oppenheim A V. Circuit implementation of synchronized chaos with application to communications[J]. Phys. Rev. Lett.,1993,71:65-68.
[56]Milanovic V, Zaghloul M E. Improved masking algorithm chaotic communications systems[J]. Elec. Lett.,1996,1:11-12.
[57]Yu P, Lookman T. Extract recovery from masked chaotic signal. Mini SymPosium Cryptograghy[M]. Toronto:Canadian Applied Mathematics Society,1997.
[58]刘峰,陈小利,穆肇骊,等.混沌系统的反馈同步及其在保密通信中的应用[J].电子学报,2000,28(8)：46-48.
[59]Liao T, Huang N S. An obsevered-based approach for chaotic synchronization with applications to secure communications[J]. IEEE Trans. Circuits Systems-I,1999, 46(9):1144-1150.
[60]李建芬,李农.一种新的蔡氏混沌掩盖通信方法[J].系统工程与电子技术,2002,24(4)：41-43.
[61]李农,李建芬,张智军.一种改进的混沌掩盖通信方法[J].系统工程与电子技术,2004,26(5)：583-586.
[62]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)：212-256.
[63]Yang T, Chua L 0. Secure communication via chaotic parameter modulation[J]. IEEE Trans. Circuits Systems-I,1996,43(9):817-819.
[64]Tang F. An adaptive synchronization strategy based on active control for demodulating message hidden in chaotic signals[J]. Chaos, Solitons Fract.,2008, 37(4):1090-1096.
[65]Banerjee S. Synchronization of time-delayed systems with chaotic modulation and cryptography[J]. Chaos, Solitons Fract.,2009,42(2):745-750.
[66]Rulkov N F, Sushchik M M, Tsimring L S, et al. Digital communication using chaotic pulse-position modulation[J]. IEEE Trans. Circuits Systems-I,2001,48(12): 1436-1444.
[67]Maggio G M, Rulkov N, Reggiani L. Pseudo-Chaotic time hopping for UWB impulse radio[J]. IEEE Trans. Circuits Systems-I,2001,48(12):1424-1435.
[68]Chen G, Lai D. Feedback anticontroller of discrete chaos [J]. Int. J. Bifur. Chaos, 1998,8(7):1585-1590.
[69]Fuh C C, Tsai H H. Control of discrete-time chaotic system via feedback linearization[J]. Chaos, Solitons Fract.,2002,13(2):285-294.
[70]Verduzco F. Control of oscillations from the k-zero bifurcation[J]. Chaos, Solitons Fract.,2007,33(2):492-504.
[71]Yu P, Leung A Y T. The simplest normal form and its application to bifurcation control[J]. Chaos, Solitons Fract.,2007,33(3):845-863.
[72]Pyragas K, Tamasiavicius A. Experimental control of chaos by delayed self-controlling feedback[J]. Phys. Lett. A,1993,180(2):99-102.
[73]Layeghi H, Arjmand M T, Salarieh H, et al. Stabilizing periodic orbits of chaotic systems using fuzzy adaptive sliding mode control [J]. Chaos, Solitons Fract.,2008, 37(4):1125-1135.
[74]Sun Y J. Robust tracking control of uncertain Duffing-Holmes control systems[J]. Chaos, Solitons Fract.,2009,40(3):1282-1287.
[75]Li Y X, Tang W K S, Chen G R. Generating hyperchaos via state feedback control [J]. Int. J. Bifur. Chaos,2005,15(10):3367-3375.
[76]Chen A M, Lu J A, Lu J H, et al. Generating hyperchaotic Lu attractor via state feedback control[J]. Physica A,2006,364:103-110.
[77]李国辉,周世平,徐得名,等.间隙线性反馈控制混沌[J].物理学报,2000,49(11)：2123-2128.
[78]王兴元,骆超.二维Logistic映射的分岔与分形[J].力学学报,2005,37(3)：346-355.
[79]Liu C X, Liu T, Liu L, et al. A new chaotic attractor[J]. Chaos, Solitons Fract. 2004,22(5):1031-1038.
[80]Ramasubramanian K, Sriram M S. A comparative study of computation of Lyapunov spectra with different algorithms[J]. Physica D,2000,139:72-86.
[81]Wu Z M, Xie J Y, Fang Y Y, et al. Controlling chaos with periodic parametric perturbations in Lorenz system[J]. Chaos, Solitons Fract.,2007,32(1):104-112.
[82]Rossler O E. An equation for hyperchaos [J]. Phys. Lett. A,1979,71(2,3):155-156.
[83]Chen A M, Lu J A, Lii J H, et al. Generating hyperchaotic Lu attractor via state feedback control[J]. Physica A,2006,364:103-110.
[84]Zhang T J, Feng G. Output tracking and synchronization of chaotic Chua's circuit with disturbances via model predictive regulator [J]. Chaos, Solitons Fract.,2009, 39(2):810-820.
[85]Sun Y J. Composite tracking control for a class of uncertain nonlinear control systems with uncertain deadzone nonlinearities[J]. Chaos, Solitons Fract.,2008, 35(2):383-389.
[86]Rossler 0 E. A equation for continuous chaos[J]. Phys. Lett. A,1976,57(5): 397-398.
[87]Chen G, Ueta T. Yet another chaotic attractor[J]. Int. J. Bifur. Chaos.,1999, 9(7):1465-1466.
[88]Jiang G P, Tang K S, Chen G. A simple global synchronization criterion for coupled chaotic systems [J]. Chaos, Solitons Fract.,2003,15(5):925-935
[89]Zhou X B, Wu Y, Li Y, et al. Adaptive control and synchronization of a new modified hyperchaotic Lu system with uncertain parameters [J]. Chaos, Solitons Fract.,2009, 39(5):2477-2483.
[90]Gao T G, Chen Z Q, Yuan Z Z, et al. Adaptive synchronization of a new hyperchaotic system with uncertain parameters[J].Chaos, Solitons Fract.,2007,33 (3):922-928.
[91]蔡国梁,黄娟娟.超混沌Chen系统和超混沌Rossler系统的异结构同步[J].物理学报,2006,55(8)：3997-4004.
[92]王兴元,武相军.变形耦合发电机系统中的混沌控制[J].物理学报,2006,55(10)5083-5093.
[93]Wang F Q, Liu C X. A new criterion for chaos and hyperchaos synchronization using linear feedback control[J]. Phys. Lett. A,2006,360(2):274-278.
[94]Peng J H, Ding E J, Ding M, et al. Sychronizing hyperchaos with a scalar transmitted signal [J]. Phys. Rev. Lett.,1996,76(6):904-907.
[95]Chen D L, Sun J T, Huang C S. Impulsive control and synchronization of general chaotic system[J]. Chaos, Solitons Fract.,2006,28(1):213-218.
[96]Chen Y S, Chang C C. Impulsive synchronization.of Lipschitz chaotic systems [J]. Chaos, Solitons Fract.,2009,40(3):1221-1228.
[97]Zhu W, Xu D Y, Huang Y M. Global impulsive exponential synchronization of time-delayed coupled chaotic systems[J]. Chaos, Solitons Fract.,2008,35(5): 904-912.
[98]Podlubny I. Fractional differential equations[M]. New York:Academic Press,1999.
[99]Hilfer R. Applications of fractional calculus in physics[M]. New Jersey:World Scientific,2001.
[100]Bagley R L, Calico R A. Fractional order state equations for the control of viscoelastically damped structures[J]. J. Guid. Control Dynam.,1991,14(2): 304-311.
[101]Koeller R C. Application of fractional calculus to the theory of viscoelasticity [J]. J. Appl. Mech.,1984,51(2):294-298.
[102]Koeller R C. Polynomial operators, Stieltjes convolution, and fractional calculus in hereditary mechanics[J]. Acta Mechanica,1986,58(3-4):251-264.
[103]Heaviside O. Electromagnetic theory[M]. New York:Chelsea,1971.
[104]Grigorenko I, Grigorenko E. Chaotic dynamics of the fractional Lorenz system [J]. Phys. Rev. Lett.,2003,91(3):034101.
[105]Gao X, Yu J B. Chaos in the fractional order periodically forced complex Duffing's oscillators[J]. Chaos, Solitons Fract.,2005,24(4):1097-1104
[106]Hartley T T, Lorenzo C F, Qammer H K. Chaos in a fractional order Chua's system [J]. IEEE Trans. CAS-I,1995,42(8):485-490.
[107]Li C G, Chen G. Chaos in the fractional order Chen system and its control[J]. Chaos, Solitons Fract.,2004,22(3):549-554.
[108]Li C P, Peng G J. Chaos in Chen's system with a fractional order[J]. Chaos, Solitons Fract.,2004,22(2):443-450.
[109]Deng W H, Li C P. Chaos synchronization of the fractional Lu system[J]. Physica A,2005,353:61-72.
[110]Caputo M. Linear models of dissipation whose Q is almost frequency independent [J]. Goephys. J. R. Atr. Soc.,1967,13:529-539.
[111]Matignon D. Stability results for fractional differential equations with application to control processing[J]. Computational Eng. in Sys. Appl.,1996, 2:963-968
[112]薛定宇,陈阳泉.高等应用数学问题的MATLAB求解[M].北京：清华大学出版社,2004.
[113]Zhou J, Chen Z. Further results on complete synchronization for noise-perturbed chaotic systems[J]. Phys. Lett. A,2008,372(33):5394-5401.
[114]Li G H, Zhou S P. Anti-synchronization in different chaotic systems[J]. Chaos, Solitons Fract.,2007,32(2):516-520.
[115]Ge Z M, Chen C C. Phase synchronization of coupled chaotic multiple time scales systems[J]. Chaos, Solitons Fract.,2004,20(3).:639-647.
[116]Erjaee G H. On analytical justification.of phase synchronization in different chaotic systems[J]. Chaos, Solitons Fract.,2009,39(3):1195-1202.
[117]Huang Y H, Wang Y W, Xiao J W. Generalized lag-synchronization of continuous chaotic system[J]. Chaos, Solitons Fract.,2009,40(2):766-770.
[118]Li G H, Xiong C A, Sun X N. Projective synchronization based on suitable separation[J].Chaos, Solitons Fract.,2007,32(2):561-565.
[119]Khadra A, Liu X Z, Shen X. Impulsively synchronizing chaotic systems with delay and applications to secure communication[J]. Automatica,2005,41(9):1491-1502.
[120]Zhou J, Huang H B, Qi G X, et al. Communication with spatial periodic chaos synchronization[J]. Phys. Lett. A,2005,335(2,3): 191-196.
[121]Li X J, Xu Z G, Zhou D H. Chaotic secure communication based on strong tracking filtering[J]. Phys. Lett. A,2008,372(44):6627-6632.
[122]Sun Y H, Cao J D, Feng G. An adaptive chaotic secure communication scheme with channel noises[J]. Phys. Lett. A,2008,372(33):5442-5447.
[123]Hoang T M, Nakagawa M. A secure communication system using projective-lag and/or projective-anticipating synchronizations of coupled multidelay feedback systems[J]. Chaos, Solitons Fract.,2008,38(5):1423-1438.
[124]Cruz-Hernandez C, Romero-Haros N. Communicating via synchronized time-delay Chua's circuits[J]. Commun. Nonlinear Sci. Numer. Simul.,2008,13(3):645-659.
[125]Fallahi K, Raouf i R, Khoshbin H. An application of Chen system for secure chaotic communication based on extended Kalman filter and multi-shift cipher algorithm. Commun[J]. Nonlinear Sci. Numer. Simul.,2008,13(4):763-781.
[126]Tian Y C, Gao F R. Adaptive control of chaotic continuous-time systems with delay [J]. Physica D,1998,117(1-4):1-12.
[127]Yang T, Yang L B, Yang C M. Breaking chaotic secure communications using a spectogram[J]. Phys. Lett. A,1998,247(1,2):105-111.
[128]Alvarez G, Montoya F, Romera M, et al. Breaking parameter modulated chaotic secure communication system[J]. Chaos, Solitons Fract.,2004,21(4):783-787.
[129]Mackey M C, Glass L. Oscillation and chaos in physiological control system[J]. Science,1977,197(4300):287-289.
[130]关新平,范正平,陈彩莲,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,2004.