混沌噪声背景下谐波参数估计方法研究
|
摘要
混沌信号处理是非线性信号处理的一个崭新的学科分支,不仅在海洋、生物、医学、化学等自然和工程领域有着丰富的研究课题,而且在保密通信、数字水印和电子对抗中有广阔的应用前景。混沌噪声背景下谐波信号参数估计是混沌信号处理的一个重要研究内容,对它的研究有着重大的理论意义和重要的实用价值。例如,海洋杂波中目标信号的特征参数估计、生物医学中母体胎儿心电信号的检测和视觉诱发脑电信号中微弱信号的检测和提取,以及反混沌干扰过程等都可以归结为混沌噪声背景下信号参数估计问题。
本文首先介绍了混沌理论和混沌信号处理的相关知识,在此基础上提出了基于单变量驱动混沌同步的谐波频率估计方法,并利用互四阶累积量参数估计方法提高频率估计精度。具体研究内容包括:基于确定系统混沌同步的谐波频率估计方法;基于系统模型已知但参数摄动混沌同步的谐波频率估计方法;基于系统模型已知但系统参数完全未知混沌同步的谐波频率估计方法;基于结构互异混沌同步的谐波频率估计方法。本文方法能实现强混沌噪声和强随机噪声共存背景下谐波频率估计;随机噪声可以为相关的高斯(白色或有色)噪声,也可为互不相关的(白色或有色、高斯或非高斯)噪声。理论分析和仿真实验表明,本文提出的混沌噪声背景下谐波信号频率估计方法,具有噪声抑制能力强、估计精度高的优点。
Chaotic signal is come from a determinate system which looks like ruleless andseemingly stochastic behavior , it is a ubiquitous complicated dynamics and naturalphenomena. the determinate system mean the chaotic function is known and unchanged.The stochastic character of chaos is that the evolvement of chaos function depend oninitial conditions sensitively and is unpredictable for long time . Chaos theory mainlystudy on nonlinear dynamics system , the purpose of chaotic theory is discovering pos-sible basic rules which hide in the seemingly random phenomena .In fact ,many naturephenomena is nonlinear and traditional linear theory can not explain the phenomena.Chaotic theory study has achieve many fruits since 1970’s,including mechanism researchof chaos ,how to produce chaos ,the judgement of chaos and phase space reconstruc-tion etc. Today ,the new trend of chaotic research is change from abstract mathematicsproblem to engineering field.
Chaotic signal processing is a fire-new branch of non-linear signal processing. It hasabundant research tasks in natural and engineering field,such as ocean, geology ,biologyand chemistry etc, it also can be used in chaotic secure communication , digital water-mark and electronic warfare. At present, chaotic signal processing becomes a popularscience research field. As an important branch of chaotic signal processing, harmonicsignal parameter estimation from chaotic background has important theory and appli-cation value. For example, the target signal character parameter estimation from seaclutter(which has been proved to be chaotic),fetal electrocardiogram extracted from ma-trix in biomedicine ,weak signal extracted from visual evoked potential and resistingchaotic interference all belong to signal extraction character parameter from chaoticbackground. The theory of harmonic parameter estimation in the chaotic noise back-ground is a new research field in recent ten years ,thus it has no systemic theory. Themethod of harmonic parameter estimation in chaotic background must closely dependon the character and attribute of chaotic signal which is different from other signal pro-cess method .The attribute of chaotic signal including predictable for short time, phase space reconstruction, controllable, fraction etc. Now, most of chaotic signal processingtheory are based on predictable for short time and phase space reconstruction, a fewresearcher use chaotic fraction to study signal detection. While the paper put forwarda new method to chaotic signal processing based on chaotic controllable character (infact , chaotic synchronization belong to chaotic control theory ). In this thesis , manynew theoretical methods ,which are used to harmonic parameter estimation from chaoticbackground under different conditions such as signal to noise , frequency ranges ,havebeen put forward by application of chaotic synchronization and modern signals pro-cessing theories ( cross power spectrum dense analysis , cross-higher-order cumulant ).Imitations by computer have proved effectiveness , robustness and usefulness of thesemethods.
The following are the creative works of this thesis:
1. This paper presents a frequency parameter estimation method for weak harmonicsignal embedded in chaotic’noise’based on chaotic synchronization. First of all ,wechanged the chaotic synchronization problem to stabilization of chaotic synchronizationerror system .second , we consider the chaotic synchronization driving by mixed signals(chaotic signal ,harmonic signal and other noise) as the stabilization of interfered system,and give some proof of the theory. Last, when chaotic synchronization occurred , theerror data series between drive and response signal have the harmonic signal , using signalprocessing method ,it’s easy to estimate the harmonic frequency. The main characteristicof the method can be generalized as: (1) the method is mix of controlling theory andsignal processing theory , because chaotic synchronization actually belongs to chaoticcontrol theory. (2)the method only study the chaotic synchronization driving by onevariable.(3) three methods are presented to analyzing the synchronization error whichare cross spectrum ,MUSIC and SV D ff MN. simulation results show that the twoalgorithms can realize higher accurate and higher reliable frequency parameter estimationthan power spectrum used by other chaotic signal processing method ,another meritof these methods is that they can process more background noise such as Gaussiancolor noise and non-gaussian color noise.(4) The method is suit for the situation wherebackground interference signals is mixture of strong chaotic signal and strong other kindof noise, which is advantage of other kind of chaotic signal methods .
2. the thesis present harmonic parameter estimation methods by using chaotic syn-chronization which the model and parameter of chaotic system are known ,in this part,the estimation of chaotic attractor of unification chaos is also given . The main researchcontent are: (1) we proved the relationship between chaotic synchronization error and interference which proof the method of this paper. Simulations show the method is notsensitively to the initial conditions. (2) designing the chaotic synchronization controllerof linear feedback by using estimation of chaotic attractor, emulation show it is betterthan variable substitute chaotic synchronization method.(3) we adopt nonlinear feedbackcontroller to synchronize chaotic system which need not estimation of chaotic attractorand can get a more exact feedback gain.(4) feedback gain adaptive control chaotic syn-chronization is a robust method, which can be used in strong chaotic noise and otherstrong noise.(5)using part of spectrum of chaotic signal ,the response chaotic system canalso synchronize the driving chaotic system, in this section, the filtered driving signaland the filtered error signal are both used to chaotic synchronization , the two methodsare suit for case that background noise is mixture of chaotic noise and wideband noise,and they can detect lower SNR harmonic signal.
3. In practice engineering ,chaotic parameters will vary slightly because of compo-nents aging or changing of environment ,so we present a chaotic synchronization methodof parameters variable .In order to design controller ,we estimate the attractor of chaoticsystem with parameters vary. Base on theorem of chaotic synchronization with parame-ter vary, both the linear feedback and nonlinear feedback chaotic synchronization methodare used to estimation harmonic parameter .Since the parameters of driving and responsesystem are different ,the SNR is higher than the case parameter are known. In the end,a ffexible feedback idea is present to compensate the drawback of parameter vary.
4. we present a harmonic parameter estimation method based on chaotic synchro-nization with chaotic system parameter fully unknown .In fact , most of application inengineering has no idea of parameter . A technique is introduced for estimating unknownparameters when a time series of only one variable from a multivariate nonlinear dynam-ical system is given. The technique is based on dynamic minimization of synchronizationerror and it loosen conditions of harmonic parameter estimation . In this section twodifferent control methods, linear feedback and adaptive control for synchronizing systemvariables are employed .Though the noise effect the estimation of parameter and syn-chronization ,the adaptive control gain method still work will on estimation parameterfrom chaos interference and other noise.
5.In the natural science research ,some chaotic models are so complicated that it ishard to say the two system are same or almost same, such as laser array and life sciences.The research on harmonic parameter estimation by using chaotic synchronization withdifferent models has important value. The controller of chaotic synchronization consti-tute two part: one part is used to compensate the effect of response system variables which are the production of difference of driving-response system ,another is feedbackof error . The thesis only research some representative chaotic system because of thecomplication of chaotic synchronization between different system. Simulation show themethod is also effective.
Summarily, in this paper the harmonic signal parameter estimation based on chaoticsynchronization approaches in different conditions is proposed . The method not only canestimation parameters in chaotic noise , but also do in mix noise (chaotic noise and otherkind of noise which can be correlated Gaussian noise and all kinds of uncorrelated noise) .It have many prominent virtues such as strong noise suppression ability, high estimationaccuracy and low computational cost. Both theoretical analysis and simulation resultsdemonstrate the effectiveness of the new methods. Keywords: chaotic noise background, harmonic signal, parameter estimation, chaotic synchronization , cross-higher-ordercumulant.
本文首先介绍了混沌理论和混沌信号处理的相关知识,在此基础上提出了基于单变量驱动混沌同步的谐波频率估计方法,并利用互四阶累积量参数估计方法提高频率估计精度。具体研究内容包括:基于确定系统混沌同步的谐波频率估计方法;基于系统模型已知但参数摄动混沌同步的谐波频率估计方法;基于系统模型已知但系统参数完全未知混沌同步的谐波频率估计方法;基于结构互异混沌同步的谐波频率估计方法。本文方法能实现强混沌噪声和强随机噪声共存背景下谐波频率估计;随机噪声可以为相关的高斯(白色或有色)噪声,也可为互不相关的(白色或有色、高斯或非高斯)噪声。理论分析和仿真实验表明,本文提出的混沌噪声背景下谐波信号频率估计方法,具有噪声抑制能力强、估计精度高的优点。
Chaotic signal is come from a determinate system which looks like ruleless andseemingly stochastic behavior , it is a ubiquitous complicated dynamics and naturalphenomena. the determinate system mean the chaotic function is known and unchanged.The stochastic character of chaos is that the evolvement of chaos function depend oninitial conditions sensitively and is unpredictable for long time . Chaos theory mainlystudy on nonlinear dynamics system , the purpose of chaotic theory is discovering pos-sible basic rules which hide in the seemingly random phenomena .In fact ,many naturephenomena is nonlinear and traditional linear theory can not explain the phenomena.Chaotic theory study has achieve many fruits since 1970’s,including mechanism researchof chaos ,how to produce chaos ,the judgement of chaos and phase space reconstruc-tion etc. Today ,the new trend of chaotic research is change from abstract mathematicsproblem to engineering field.
Chaotic signal processing is a fire-new branch of non-linear signal processing. It hasabundant research tasks in natural and engineering field,such as ocean, geology ,biologyand chemistry etc, it also can be used in chaotic secure communication , digital water-mark and electronic warfare. At present, chaotic signal processing becomes a popularscience research field. As an important branch of chaotic signal processing, harmonicsignal parameter estimation from chaotic background has important theory and appli-cation value. For example, the target signal character parameter estimation from seaclutter(which has been proved to be chaotic),fetal electrocardiogram extracted from ma-trix in biomedicine ,weak signal extracted from visual evoked potential and resistingchaotic interference all belong to signal extraction character parameter from chaoticbackground. The theory of harmonic parameter estimation in the chaotic noise back-ground is a new research field in recent ten years ,thus it has no systemic theory. Themethod of harmonic parameter estimation in chaotic background must closely dependon the character and attribute of chaotic signal which is different from other signal pro-cess method .The attribute of chaotic signal including predictable for short time, phase space reconstruction, controllable, fraction etc. Now, most of chaotic signal processingtheory are based on predictable for short time and phase space reconstruction, a fewresearcher use chaotic fraction to study signal detection. While the paper put forwarda new method to chaotic signal processing based on chaotic controllable character (infact , chaotic synchronization belong to chaotic control theory ). In this thesis , manynew theoretical methods ,which are used to harmonic parameter estimation from chaoticbackground under different conditions such as signal to noise , frequency ranges ,havebeen put forward by application of chaotic synchronization and modern signals pro-cessing theories ( cross power spectrum dense analysis , cross-higher-order cumulant ).Imitations by computer have proved effectiveness , robustness and usefulness of thesemethods.
The following are the creative works of this thesis:
1. This paper presents a frequency parameter estimation method for weak harmonicsignal embedded in chaotic’noise’based on chaotic synchronization. First of all ,wechanged the chaotic synchronization problem to stabilization of chaotic synchronizationerror system .second , we consider the chaotic synchronization driving by mixed signals(chaotic signal ,harmonic signal and other noise) as the stabilization of interfered system,and give some proof of the theory. Last, when chaotic synchronization occurred , theerror data series between drive and response signal have the harmonic signal , using signalprocessing method ,it’s easy to estimate the harmonic frequency. The main characteristicof the method can be generalized as: (1) the method is mix of controlling theory andsignal processing theory , because chaotic synchronization actually belongs to chaoticcontrol theory. (2)the method only study the chaotic synchronization driving by onevariable.(3) three methods are presented to analyzing the synchronization error whichare cross spectrum ,MUSIC and SV D ff MN. simulation results show that the twoalgorithms can realize higher accurate and higher reliable frequency parameter estimationthan power spectrum used by other chaotic signal processing method ,another meritof these methods is that they can process more background noise such as Gaussiancolor noise and non-gaussian color noise.(4) The method is suit for the situation wherebackground interference signals is mixture of strong chaotic signal and strong other kindof noise, which is advantage of other kind of chaotic signal methods .
2. the thesis present harmonic parameter estimation methods by using chaotic syn-chronization which the model and parameter of chaotic system are known ,in this part,the estimation of chaotic attractor of unification chaos is also given . The main researchcontent are: (1) we proved the relationship between chaotic synchronization error and interference which proof the method of this paper. Simulations show the method is notsensitively to the initial conditions. (2) designing the chaotic synchronization controllerof linear feedback by using estimation of chaotic attractor, emulation show it is betterthan variable substitute chaotic synchronization method.(3) we adopt nonlinear feedbackcontroller to synchronize chaotic system which need not estimation of chaotic attractorand can get a more exact feedback gain.(4) feedback gain adaptive control chaotic syn-chronization is a robust method, which can be used in strong chaotic noise and otherstrong noise.(5)using part of spectrum of chaotic signal ,the response chaotic system canalso synchronize the driving chaotic system, in this section, the filtered driving signaland the filtered error signal are both used to chaotic synchronization , the two methodsare suit for case that background noise is mixture of chaotic noise and wideband noise,and they can detect lower SNR harmonic signal.
3. In practice engineering ,chaotic parameters will vary slightly because of compo-nents aging or changing of environment ,so we present a chaotic synchronization methodof parameters variable .In order to design controller ,we estimate the attractor of chaoticsystem with parameters vary. Base on theorem of chaotic synchronization with parame-ter vary, both the linear feedback and nonlinear feedback chaotic synchronization methodare used to estimation harmonic parameter .Since the parameters of driving and responsesystem are different ,the SNR is higher than the case parameter are known. In the end,a ffexible feedback idea is present to compensate the drawback of parameter vary.
4. we present a harmonic parameter estimation method based on chaotic synchro-nization with chaotic system parameter fully unknown .In fact , most of application inengineering has no idea of parameter . A technique is introduced for estimating unknownparameters when a time series of only one variable from a multivariate nonlinear dynam-ical system is given. The technique is based on dynamic minimization of synchronizationerror and it loosen conditions of harmonic parameter estimation . In this section twodifferent control methods, linear feedback and adaptive control for synchronizing systemvariables are employed .Though the noise effect the estimation of parameter and syn-chronization ,the adaptive control gain method still work will on estimation parameterfrom chaos interference and other noise.
5.In the natural science research ,some chaotic models are so complicated that it ishard to say the two system are same or almost same, such as laser array and life sciences.The research on harmonic parameter estimation by using chaotic synchronization withdifferent models has important value. The controller of chaotic synchronization consti-tute two part: one part is used to compensate the effect of response system variables which are the production of difference of driving-response system ,another is feedbackof error . The thesis only research some representative chaotic system because of thecomplication of chaotic synchronization between different system. Simulation show themethod is also effective.
Summarily, in this paper the harmonic signal parameter estimation based on chaoticsynchronization approaches in different conditions is proposed . The method not only canestimation parameters in chaotic noise , but also do in mix noise (chaotic noise and otherkind of noise which can be correlated Gaussian noise and all kinds of uncorrelated noise) .It have many prominent virtues such as strong noise suppression ability, high estimationaccuracy and low computational cost. Both theoretical analysis and simulation resultsdemonstrate the effectiveness of the new methods. Keywords: chaotic noise background, harmonic signal, parameter estimation, chaotic synchronization , cross-higher-ordercumulant.
引文
[1]詹姆斯·格莱特著,张淑誉译.混沌:开创新科学[M].北京:高等教育出版社, 2004
[2] E. N. Lorenz. Deterministic nonperiodic ?ow[J]. Journal of the Atmospheric Sci-ences. 1963, 20:130–141
[3] Li T.Y., Yorke J .A. Period Three Implies Chaos[J]. AmerMathMonthly. 1975,82:985–992
[4] Takens F. Dectecting strange attractors in turbulence[C]. Lecture Notes in Math-ematics,Springer,Berlin. 1981, 898:366–370
[5] H. Kantz, Schreiber T. Nonlinear Time Series Analysis[J]. Cambridge, UK: Cam-bridge University Press, 1997
[6] T. Sauer, J.A. Yorke, M. Casdagli. Embedology [J]. JStatPhys. 1991, 65(3):579–616
[7] Cao Liangyue. Practical method for determining the minimum embedding dimen-sion of a scalar time series[J]. Physica D. 1997, 110:43–50
[8]郝柏林.自然界中的有序和混沌[M].百科知识. 1984, 1:69–72
[9] Leung H. Experimental modeling of electromagnetic scattering from an oceansurface using chaos theory[J]. Chaos,Solition and Fractals. 1992, 2:25–43
[10] Leung H., Lo T. Chaotic radar signal processing over the sea[J]. IEEE JOceanicEngineering. 1993, 18(3):287–295
[11] Leung H. Applying chaos to radar dectection in an ocean environment:An Exper-imental Study[J]. IEEE JOceanic Engineering. 1995, 20(1):56–64
[12]何建华,杨宗凯.激光雷达探潜背景信号的混沌和分形行为研究[J].华中理工大学学报. 1997, 25(5):67–70
[13] Richer M., T. Sehreiber, Kaplan D.T. Fetal EGG extraction with nonlinear state-space projection[J]. IEEE TransBiomEng. 1998, 45(1):133–137
[14] T. Sehreiber, Kaplan D.T. Signal separation by nonlinear projections:The fetalelectrocardiogram[J]. Physical Review E. 1996, 53(5):4326–4329
[15] Pijn J.P., Neeren J.V., Noeest A. Chaos or noise in EEG signals:dependence onstate and brain site[J]. Electroencephaloqr clin Neurophysiol. 1991, 79:371–381
[16] Regan D. Human Brain Electrophysiology: Evoked Potentials and Evoked Mag-netic Fields in Science and Medicine[C]. Elsevier, Amsterdam. 1989:672
[17] Stark J., Arumugam B V. Extracting slowly varying signals from chaotic back-ground[J]. IntJBif and Chaos. 1992, 2:413–417
[18] Broomhead D.S., Huke J.P., Jones R. Signals in chaos:A method for the cancella-tion of deterministic noise from discrete signals[J]. Physica D. 1995, 80:413–432
[19] Broomhead D.S., Huke J.P., Potts M A S. Cancelling deterministic noise by con-structing nonlinear inverses to linear filters[J]. Physica D. 1996, 89:439–458
[20] Tim Sauer. A noise reduction method for signals from nonlinear systems[J]. PhysicD. 1992, 58(1-4):193–201
[21] Robert Cawley. Local-geometric-projection method for noise ruduction in chaoticmaps and ?ows[J]. Physical Review A. 1992, 46(6):3057–3082
[22] Short K.M. Steps toward unmasking secure communications[J]. IntJBifand Chaos.1994, 4(4):959–977
[23] Short K.M. Signal extraction from chaotic communications[J]. IntJBifand Chaos.1997, 7(7):1579–1597
[24] Short K.M, Parker T. Unmasking a hyperchaotic communication scheme [J]. Phys-ical Review E. 1998, 58(1):1159–1163
[25] Leung H., Haykin S. Detection and estimation using an adaptive rational functionfilter[J]. IEEE TransSigProc. 1994, 42(12):3366–3376
[26] Leung H., Huang X P. Parameter estimation in chaotic noise[J]. IEEE Trans SigProc. 1996, 44(10):2456–2463
[27] Leung H. System identification using chaos with application to equalization of achaotic modulation system[J]. IEEE TransCircuits and Systems. 1998, 45(3):314–320
[28] Haykin S, Li X B. Detection of signals in chaos[J]. Proceedings of IEEE. 1995,83(1):94–122
[29] Ajeesh P.Kurian, Henry Leung. Model Based Synchronization for Weak SignalDetection[C]. IEEE International instrumentation and Measurement TechnologyConference,Victoria,Canada. 2008:1219–1222
[30]何建华,杨宗凯,王殊.基于混沌和神经网络的弱信号探测[J].电子学报. 1998,26(10):33–37
[31]程文青,何建华,沈春蕾.一种基于神经网络的激光水下目标探测方法[J].华中理工大学学报. 1999, 27(3):56–58
[32]黄显高,徐健学.利用小波多尺度分解算法实现混沌系统的噪声减缩[J].物理学报. 1999, 48(10):1810–1816
[33] Huang X G., Xu J X. Unmasking chaotic mask by a wavelet multiscale decompo-sition algorithm [J]. IntJBif and Chaos. 2001, 11(2):561–569
[34] Zhu changfang, Hu Guangshu. Combining independent component analysis insingle-trail VEP estimation[J]. Proceeding IEEE. 2000, 48(3):761–763
[35]汪芙平,郭静波.强混沌干扰中的谐波信号提取[J].物理学报. 2001, 50(6):1019–1023
[36]李鸿光,孟光.基于经验模式分解的混沌干扰下谐波信号的提取方法[J].物理学报. 2004, 53(7):2069–2074
[37]杜京义,侯媛彬.混沌背景中微弱谐波信号检测的SVM方法[J].仪器仪表真学报.2007, 28(3):554–559
[38]陈争,曾以成,付志坚.混沌背景中信号参数估计的新方法[J].物理学报. 2008,57(1):46–50
[39]陆振波,蔡志明,姜可宇.基于Volterra滤波器的混沌背景弱信号检测[J].系统仿真学报. 2008, 20(7):1778–1780
[40]徐伟.混沌背景中的微弱信号检测[D].南京:南京信息工程大学硕士论文, 2007
[41]陈颖.湮没在混沌背景下的微弱信号检测[D].苏州:苏州大学硕士论文, 2006
[42]祝开艳.混沌背景中的谐波恢复[D].长春:吉林大学硕士论文, 2005
[43]徐炜.混沌背景中的弱信号检测和提取[D].上海:上海交通大学, 2006
[44]王国光.提取混沌中信号方法的研究[D].长春:吉林大学, 2008
[45] Huang N E, Shen Z., Long S R. The empirical mode decomposition and the Hilbertspectrum for nonlinear and non-station time series analysis [J]. Proceeding of theRoyal Society of London A. 1998, 454:903–995
[46] S. Hein. Exploiting chaos to suppress spurious tone in general double-loopΣ?modulators[J]. IEEE Trans CAS-I. 1993, 40(10):651–659
[47] Lorenz E.N. Atmosspheric predictablity esperiments with a large numerical model[J]. Tellus. 1982, 34:505–513
[48]张学义.混沌同步及其在通信中的应用[D].哈尔滨:哈尔滨工程大学博士论文,2001
[49]陈奉苏.混沌学及其应用[M].北京:中国电力出版社, 1998
[50]王诗斌.混沌理论及混沌振荡器的研究[D].长沙:湖南大学, 2004
[51]关新平等.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]谢红梅.基于混沌理论的信号处理方法研究[D].西安:西北工业大学博士论文,2003
[53]孙海云,曹庆杰.混沌时间序列建模及预测[J].系统工程理论与实践. 2001,5:106–110
[54]赵小梅.混沌预测与混沌优化理论与算法研究[D].杭州:浙江大学, 2002
[55]王凌.智能优化算法及其应用[M].北京:清华大学出版社, 2001
[56] Mahmut Ciftci. Channel Equalization for Chaotic Communication Systems[D].Georgia Institute of Technology ,Ph.D, 2002
[57] Yassen MT. Chaos synchronization between two di?erent chaotic systems usingactive control[J]. Chaos ,solitions and Factals. 2005, 23:131–140
[58] YW Wang, AH Guan, HO Wang. Feedback and adative control for synchronizationof chen system via a single variable[J]. Physics letter A. 2003, 311:389–395
[59] Moez Feki. A adative Chaos synchronization scheme applied to secure communi-cation [J]. Chaos ,solitions and Factals. 2003, 18:141–148
[60]李树钧.数字化混沌密码的分析与设计[D].西安:西安交通大学, 2003
[61] M.S Baptista. Cryptography with Chaos[J]. Phys LettA. 1998, 240:50–54
[62] F. Dachselt, W. Schwarz. Chaos AND Cryptography [J]. Phys LettA. 2001,48(12):1498–1509
[63] T. Yang, C.W. Wu, L.O. Chua. Cryptography based on chaotic systems [J]. IEEETransCAS-I. 1997, 44(5):469–472
[64] R. Dettmer, P. Falcone, T. Keviczky, J. Asgari, D. Hrovat. Chaos and engineer-ing[J]. IEE Review. 1993, 39(5):199–203
[65]龙运佳.混沌工程学[J].中国工程科学. 2001, 3(2):10–15
[66]曹建福,韩崇昭,方洋旺.非线性系统理论及应用[M].西安:西安交通大学出版社,2001
[67]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
[68]方锦清.驾驭混沌与发展高新技术[M].北京:原子能出版社, 2002
[69] A. Wolf, J.B. Swift, H.L. Swinney. Determining Lyapunov exponents from a timeseries[J]. Physical D. 1985, 16:285–317
[70]陶朝海,陆君安,吕金虎.统一混沌系统的反馈同步[J].物理学报. 2002,51(7):1497–1501
[71] Lu J.G. Chaotic dynamics and synchronization of fractional-order Arneodo’s sys-tems[J]. Chaos,Solition and Fractals. 2005, 26(4):1125–1133
[72] Li F., Hu A.H., Xu Z.Y. Robust synchronization of uncertain chaotic systems[J].Chinese Physics. 2006, 15(3):507–512
[73] Wang Y.W., Guan Z.H. Generalized synchronization of continuous chaotic sys-tems[J]. Chaos,Solition and Fractals. 2006, 27(1):97–101
[74] Taherion S., Lai Y.C. Observability of lag synchronization of coupled chaoticoscillators[J]. Physical Review E. 1999, 59(6):6247–6250
[75] Shahverdiev E.M., Sivaprakasam S., Shore K.A. Lag synchronization in time de-layed system[J]. Physical Review A. 2002, 292(6):320–324
[76] Chen J.Y., Wong K.W. Phase synchronization discrete chaotic systems[J]. PhysicalReview E. 2000, 61(3):2559–2562
[77] R. Brown, L. Kocarev. A unifying definition of synchronization for dynamicalsystems [J]. Chaos. 2000, 10(2):344–349
[78] Pecora L.M., Carroll T. L. Synchronization in chaotic systems[J]. PhysRevLett.1990, 64(8):821–825
[79] Kocarev L., Parlitz U. General approach for chaotic synchronization with applca-tions to communication[J]. Physical Review Letters. 1995, 74:5028–5031
[80] Gallego J.A. Nonlinear regulation of a Lorenz system feedback linearization tech-niques[J]. Dynamic Control. 1994, 4:277–298
[81]唐芳,邱琦.混沌系统的辅助参考反馈控制[J].物理学报. 1999, 48(5):802–807
[82] Chen G.R., Dong X. On feedbaek control of chaotic continuous-time systems[J].IEEE Transon Circuits and System. 1993, 40(9):591–601
[83] Pyragas K. Continous control of chaos by self-controlling systems[J]. Physicalletters A. 1992, 170:421–428
[84] Pyragas K., Tamasevicius A. Experimental control of chaos by delayed self-controlling feedback[J]. Physical letters A. 1993, 180:99–102
[85]王光瑞,于煦龄,陈式刚.混沌的控制、同步和应用[M].北京:国防工业出版社,2001
[86]邹恩,李详飞,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
[87] Jiang J.P., Tang K.S. A global synchronization criterion for coupled chaotic systemvia unidirectional linear error feedback approach[J]. IntJBifurcation chaos. 2002,12(10):2239–2253
[88] Bu S.L., Wang S.Q. A algorithm based on variable feedback to synchronize chaoticand hyperchaotic systems[J]. Physica D. 2002, 164:45–52
[89]高金峰,罗先觉,马西奎.控制与同步连续时间混沌系统的非线性反馈方法[J].物理学报. 1999, 48(9):1618–1627
[90]刘扬正,费树眠. Genesio一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
[91]陶朝海,陆君安.混沌系统的速度反馈同步[J].物理学报. 2005, 54(11):5058–5061
[92]卢俊国,汪小帆,王执栓.连续时间混沌系统控制与同步的状态反馈方法[J].控制与决策. 2001, 16(4):476–479
[93] Bai E.W., Lonngren K.E. Sequential synchronization of two Lorenz systems usingactive control[J]. Chaos,Solitons and Fraetals. 2000, 11:1041–1044
[94] Kouomou Y.C., Woafo P. Stability and optimization of chaos synchronizationthrough feedback coupling with delay[J]. Physics Letters A. 2002, 298:181–28
[95]陈凤祥.混沌控制与同步的鲁棒控制器研究[D].上海:上海交通大学博士论文,2008
[96] E. Huberman, B.A.and Lumer. Dynamics of adaptive systems[J]. IEEE Transac-tions on CAS. 1990, 37:547–550
[97] John J.K., Amritkar R.E. Synchronization of unstable orbits daptive control[J].Physical Review E. 1994, 49(6):4843–4848
[98] John J.K., Amritkar R.E. Synchronization by Feedback and Adaptive Control[J].Bifurcation and Chaos. 1994, 4(6):1687–1695
[99] Yassen M.T. Adaptive chaos control and synchronization for uncertain new chaoticdynamical system[J]. Physics Letters A. 2006, 350(1-2):36–43
[100] Shihua Chen, Jinhu Lü. Parameters identification and synchronization of chaoticsystems based upon adaptive control[J]. Physics Letters A. 2002, 299(4,8):353–358
[101]吴立刚,王常虹,曾庆双. Terminal滑模自适应控制实现一类不确定混沌系统的同步[J].控制与决策. 2006, 21(2):229–231
[102] Yongguang Yu, Suochun Zhang. Adaptive backstepping synchronization of uncer-tain chaotic system[J]. Chaos,Solitons and Fractals. 2004, 21(3):643–649
[103] Samuel Bowong, F.M. Moukam Kakmeni. Synchronization of uncertain chaotic sys-tems via backstepping approach[J]. Chaos,Solitons and Fractals. 2004, 21(4):999–1011
[104] Wang C., Ge S.S. Adaptive synchronization of uncertain chaotic systems via back-stepping design[J]. Chaos,Solitons and Fractals. 2001, 12:1199–2006
[105] Zhenya Yan. Chaos Q–S synchronization between R¨ossler system and the newunified chaotic system[J]. Physics Letters A. 2005, 334(5-6):406–412
[106] Tan Xiaohui, Jiye Zhang, Yiren Yang. Synchronizing chaotic systems using back-stepping design[J]. Chaos,Solitons and Fractals. 2003, 16(1):37–45
[107] Fang J Q, Hong Y G, Chen G R. Switching manifold approach to chaos synchro-nization[J]. Physics Letters E. 1999, 59(3):2523–2526
[108] Liao T. L., Huang N.S. Control and synchronization of discrete time chaotic sys-tems via variable structure control technique[J]. Physics Letters A. 1997, 234:262–268
[109] Lilian Huang, Rupeng Feng, Mao Wang. Synchronization of uncertain chaoticsystems with perturbation based on variable structure control[J]. Physics LettersA. 2006, 350(3-4):197–200
[110]黄丽莲,王茂,冯汝鹏.参数不确定混沌系统的反步变结构控制[J].系统工程与电子技术. 2005, 27(4):704–707
[111]张洪,陈天麒.参数未知的时延混沌系统滑模变结构同步控制[J].物理学报. 2005,21(12):1937–1941
[112] Zhi Li, Songjiao Shi. Robust adaptive synchronization of Rossler and Chen chaoticsystems via slide technique[J]. Physics Letters A. 2003, 331(4-5):389–395
[113]单梁.混沌系统的若干同步方法研究[D].南京:南京理工大学博士论文, 2006
[114]高铁杠,陈增强,袁著社,顾巧论.基于观测器的混沌系统的同步研究[J].物理学报. 2004, 53(5):1305–1308
[115] Morgul O., Solak E. On the synchronization of chaotic systems by using ststeobservations[J]. IntJBifurcation and chaos. 1997, 7(6):1307–1322
[116] Nijmeijer H., Mareeis I.M. An Observer looks at synchronization [J]. IEEE TransonCircuits and ssystems-I. 1997, 44(10):882–890
[117]关新平,范正平,陈彩莲.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, 1993
[118]杨涛,邵惠鹤.一类混沌系统的同步方法[J].物理学报. 2002, 51(4):742–748
[119] Boutayeb M., Darouach M., Rafaralyahy H. Generalized state-space observersfor chaotic synchronization and secure communication [J]. IEETransCircuits andsystem-I. 2002, 49(3):345–349
[120]姚利娜,高金峰,廖旎焕.实现混沌系统同步的非线性状态观测器方法[J].物理学报. 2006, 55(1):35–41
[121] Li G.H., Zhou S.P., Xu D.M. Chaos synchronization based on intermittent stateobserver [J]. Chinese Physics. 2004, 13(2):168–172
[122]黄玮,张化光.一类混沌系统的脉冲控制同步[J].东北大学学报(自然科学版).2004, 25(11):1027一1029
[123] Xie W., Wen C.Y. Impulsive control for the stabilization and synehronization ofLorenz systems[J]. Physics Letters A. 2000, 275:67–72
[124] Wang Y.W., Wen C.Y., Xiao J.W. ImPulsive synchronization of Chua’s oscillatorsvia a single variable[J]. Chaos,Solitons and Fractals. 2006, 29:198–201
[125] Lu J.A., Xie J., Lu J.H. Control chaos in transition using sampled-data feedback[J].Applied Methmatics and Mechanics. 2003, 24(11):1309–1316
[126] Yu D.C., Wu A.G., Wang D.Q. A simple asymptotic trajectory control of full stateof a unified chaotic system[J]. Chinese Physics. 2006, 15(2):306–309
[127]修春波,刘向东,张宇河.混沌优化与模糊控制在混沌控制中的应用[J].控制理论与应用. 2005, 22(1):63–67
[128] W.H. Deng, C.P. Li. Chaos synchronization of the fractional Lüsystem [J]. PhysicaA. 2005, 353(1):61–72
[129] Jianping Yan, Changpin Li. On chaos synchronization of fractional di?erentialequations [J]. Chaos,Solitons and Fractals. 2007, 32(2):725–735
[130] Junwei Wang, Xiaohua Xiong, Yanbin Zhang. Extending synchronization schemeto chaotic fractional-order Chen systems[J]. Physica A. 2006, 370(2):279–285
[131] Mohammad Saleh Tavazoei, Mohammad Haeri. Determination of active slid-ing mode controller parameters in synchronizing di?erent chaotic systems[J].Chaos,Solitons and Fractals. 2007, 32(2):583–591
[132] Ricardo Aguilar-Lo′pez, Rafael Martinez-Guerra. Partial synchronization of dif-ferent chaotic oscillators using robust PID feedback[J]. Chaos,Solitons and Fractals.2007, 32(2):572–581
[133] Guo-Hui Li. Generalized projective synchronization between Lorenz system andChen’s system[J]. Chaos,Solitons and Fractals. 2007, 32(4):1454–1458
[134]程丽,张入元,彭建华.用单一驱动变量同步混沌与超混沌的一种方法[J].物理学报. 2003, 52(3):536–541
[135]胡岗,肖井华,郑志刚.混沌控制[M].上海:上海科技出版社, 2000
[136] LüJ.H., Zhou T.S., Zhou S.C. Chaos Synchronization Between Two Di?erentChaotic Systems[J]. Chaos,Solitions and Fractals. 2002, 14(4):529–541
[137] Qiang Jia. Chaos control and synchronization of the Newton–Leipnik chaoticsystem[J]. Chaos,Solitions and Fractals. 2008, 35(4):814–824
[138] Park J.H. Stability Criterion for Synchronilzation Linearly Coupled the UnifiedChaotic Systems[J]. Chaos,Solitions and Fractals. 2005, 23(1):79–85
[139]蒲兴成.混沌同步控制及应用研究[D].重庆:重庆大学博士论文, 2006
[140] H.K. Khalil.非线性系统[M].北京:电子工业出版社, 2005
[141] J. P. LaSalle. The extent of asymptotic stability[J]. Proceedings of the NationalAcademy of Sciences. 1960, 46(3):363–365
[142] J. P. LaSalle. Some extensions of Liapunov’s second method[J]. IRE Trans CircuitTheory. 1960, 7:520–527
[143] H. M. Rodrigues. Abstract methods for synchronization and applications[J]. Ap-plicable Anal. 1996, 62:263–296
[144] V. S. Afraimovich, S.N. Chow, J. K. Hale. Synchronization in lattices of coupledoscillators[J]. Physica D. 1997, 103:442–451
[145] L.F.C. Rodrigues, H. M.and Alberto. On the Invariance Principle: Generalizationsand Applications to Synchronization[J]. IEEE Trans on Circuits and Systems.2000, 47(5):730–739
[146] Y. Orlov. Extended Invariance Principle for nonautonous Swittch Systems[J]. IEEETrans on Automatic Control. 2003, 48(8):1448–1452
[147] A. Swami, J.M. Mendel. Cumulant-based approach to the harmonic retrieval prob-lem[C]. International Conference on Acoustics, Speech, and Signal Processing.1998, 4:2264–2267
[148] A. Swami, J.M. Mendel. Cumulant-based approach to harmonic retrieval andrelated problems[J]. IEEE Transactions on Signal Processing. 1991, 39(5):1099–1109
[149]石要武,戴逸松.测量噪声背景下微弱正弦信号参数估计的互功率谱方法[J].电子学报. 1994, 22(1):1–8
[150]石要武,戴逸松,丁宏.有色噪声背景下正弦信号频率估计的互谱Pisarenko和MUSIC方法[J].电子学报. 1996, 24(10):46–50
[151] Ma Yan, Shi Yaowu, Dai Yisong. Harmonic Retrieval Embedded in Hybrid ColoredNoise : a Novel Pisarenko Method Based on Cross-high-order Cumulant[J]. HighTechnology Letters. 2000, 6(3):23–28
[152]于晓辉.噪声背景下chirp信号参数估计理论与方法研究[D].长春:吉林大学博士论文, 2007
[153] Wax M., Kailath T. Detection of signals by information theoretic criteria [J]. IEEETransactions on Acoustics, Speech, and Signal Processing. 1985, 32(2):387–392
[154]余秋星.基于混沌和分型的信号检测[D].西安:西北工业大学硕士论文, 2001
[155] Chen G.R., Ueta T. Yet another chaotic attractor[J]. Int Journal of Bifurcationand Chaos. 1999, 9:1465–11466
[156] Carroll T L. synchronizing chaotic systems using filtered signals[J]. Phys Rev E.1994, 50:2580–2588
[157] Rodrigues H. M., Alberto L. F. C., Bretas N. C. Uniform Invariance Principle andSynchronization. Robustness with respect to parameter variation[J]. Journal ofDi?erential Equations. 2001, 169(1):228–254
[158]陈明杰,李殿璞,张爱筠.失配混沌系统的同步研究[J].哈尔滨工程大学学报.2005, 26(5):633–638
[159] Gameiro M.F., Rodrigues H.M. Applications of Robust Synchronization to Com-munication Systems[J]. Applicable Analysis. 2001, 79:21–45
[160] Park J.H. Adpative synchronization of R¨ossler system with uncertain Par-maeters[J]. Chaos,Solitons and Fractals. 2005, 25(2):333–338
[161] Wenwu Yu, Guanrong Huang, Jinde Chen, Jinhu Lü, Ulrich Parlitz. Parame-ter identification of dynamical systems from time series[J]. Phys Rev E. 2007,75:067201
[162] Ming-Chung Ho, Yao-Chen Hung, I-Min Jiang. On the synchronization of uncer-tain chaotic systems[J]. Chaos,Solitons and Fractals. 2007, 33:540–546
[163]李农,李建芬,刘宇平,马健. Parameter identification ba sed on linear feedbackcontrol for uncertain chaotic system[J].物理学报. 2008, 57(3):1404–1408
[164] J.J. Yan, M.L. Hung, T.L. Liao. Adaptive sliding mode control for synchroniza-tion of chaotic gyros with fully unknown parameters[J]. Sound Vibration. 2006,298:298–306
[165] M.S. Tavazoei, M. Haeri. Determination of active sliding mode controller param-eters in synchronizing di?erent chaotic systems[J]. Chaos Solitons and Fractals.2007, 32:583–591
[166] Xuyang Lou, Baotong Cui. Synchronization of neural networks based on parameteridentification and via output or state coupling[J]. Chaos Solitons and Fractals.2007, 32:583–591
[167] Yassen M.T. Adaptive synchronization of Rossler and systems with fully uncertainParameters[J]. Chaos,Solitons and Fractals. 2005, 23:1527一1536
[168]褚衍东,常迎香,张莉,张建刚. Parameters Identification and IFSHPS in Chaoticand Hyper-chaotic Systems with Di?erent Structure[J].系统仿真学报. 2008,20(19):5350–5354
[169]王东风.基于遗传算法的统一混沌系统比例一积分一微分控制[J].物理学报.2005, 54(4):1495–1499
[170]关新平,彭海朋,李丽香. Lorenz混沌系统的参数辨识与控制[J].物理学报. 2001,50(1):26–29
[171] Mohammad Haeri, Mahsa Dehghani. Impulsive synchronization of di?erent hyper-chaotic(chaotic) systems[J]. Chaos Solitons and Fractals. 2008, 38:120–131
[172] Lingling Zhang, Lihong Huang, Zhizhou Zhang, Zengyun Wang. Fuzzy adap-tive synchronization of uncertain chaotic systems via delayed feedback control[J].Physics Letters A. 2008, 372:6082–6086
[173] A. Maybhate, R. E. Amritkar. Use of synchronization and adaptive control inparameter estimation from a time series[J]. Phys Rev E. 1999, 59(1):284–293
[174] Chen J., Liu Z.R. A Method of Controlling Synchronizationin Diferent Systems[J].Chinese Physical Letters. 2003, 20(9):1441–1446
[2] E. N. Lorenz. Deterministic nonperiodic ?ow[J]. Journal of the Atmospheric Sci-ences. 1963, 20:130–141
[3] Li T.Y., Yorke J .A. Period Three Implies Chaos[J]. AmerMathMonthly. 1975,82:985–992
[4] Takens F. Dectecting strange attractors in turbulence[C]. Lecture Notes in Math-ematics,Springer,Berlin. 1981, 898:366–370
[5] H. Kantz, Schreiber T. Nonlinear Time Series Analysis[J]. Cambridge, UK: Cam-bridge University Press, 1997
[6] T. Sauer, J.A. Yorke, M. Casdagli. Embedology [J]. JStatPhys. 1991, 65(3):579–616
[7] Cao Liangyue. Practical method for determining the minimum embedding dimen-sion of a scalar time series[J]. Physica D. 1997, 110:43–50
[8]郝柏林.自然界中的有序和混沌[M].百科知识. 1984, 1:69–72
[9] Leung H. Experimental modeling of electromagnetic scattering from an oceansurface using chaos theory[J]. Chaos,Solition and Fractals. 1992, 2:25–43
[10] Leung H., Lo T. Chaotic radar signal processing over the sea[J]. IEEE JOceanicEngineering. 1993, 18(3):287–295
[11] Leung H. Applying chaos to radar dectection in an ocean environment:An Exper-imental Study[J]. IEEE JOceanic Engineering. 1995, 20(1):56–64
[12]何建华,杨宗凯.激光雷达探潜背景信号的混沌和分形行为研究[J].华中理工大学学报. 1997, 25(5):67–70
[13] Richer M., T. Sehreiber, Kaplan D.T. Fetal EGG extraction with nonlinear state-space projection[J]. IEEE TransBiomEng. 1998, 45(1):133–137
[14] T. Sehreiber, Kaplan D.T. Signal separation by nonlinear projections:The fetalelectrocardiogram[J]. Physical Review E. 1996, 53(5):4326–4329
[15] Pijn J.P., Neeren J.V., Noeest A. Chaos or noise in EEG signals:dependence onstate and brain site[J]. Electroencephaloqr clin Neurophysiol. 1991, 79:371–381
[16] Regan D. Human Brain Electrophysiology: Evoked Potentials and Evoked Mag-netic Fields in Science and Medicine[C]. Elsevier, Amsterdam. 1989:672
[17] Stark J., Arumugam B V. Extracting slowly varying signals from chaotic back-ground[J]. IntJBif and Chaos. 1992, 2:413–417
[18] Broomhead D.S., Huke J.P., Jones R. Signals in chaos:A method for the cancella-tion of deterministic noise from discrete signals[J]. Physica D. 1995, 80:413–432
[19] Broomhead D.S., Huke J.P., Potts M A S. Cancelling deterministic noise by con-structing nonlinear inverses to linear filters[J]. Physica D. 1996, 89:439–458
[20] Tim Sauer. A noise reduction method for signals from nonlinear systems[J]. PhysicD. 1992, 58(1-4):193–201
[21] Robert Cawley. Local-geometric-projection method for noise ruduction in chaoticmaps and ?ows[J]. Physical Review A. 1992, 46(6):3057–3082
[22] Short K.M. Steps toward unmasking secure communications[J]. IntJBifand Chaos.1994, 4(4):959–977
[23] Short K.M. Signal extraction from chaotic communications[J]. IntJBifand Chaos.1997, 7(7):1579–1597
[24] Short K.M, Parker T. Unmasking a hyperchaotic communication scheme [J]. Phys-ical Review E. 1998, 58(1):1159–1163
[25] Leung H., Haykin S. Detection and estimation using an adaptive rational functionfilter[J]. IEEE TransSigProc. 1994, 42(12):3366–3376
[26] Leung H., Huang X P. Parameter estimation in chaotic noise[J]. IEEE Trans SigProc. 1996, 44(10):2456–2463
[27] Leung H. System identification using chaos with application to equalization of achaotic modulation system[J]. IEEE TransCircuits and Systems. 1998, 45(3):314–320
[28] Haykin S, Li X B. Detection of signals in chaos[J]. Proceedings of IEEE. 1995,83(1):94–122
[29] Ajeesh P.Kurian, Henry Leung. Model Based Synchronization for Weak SignalDetection[C]. IEEE International instrumentation and Measurement TechnologyConference,Victoria,Canada. 2008:1219–1222
[30]何建华,杨宗凯,王殊.基于混沌和神经网络的弱信号探测[J].电子学报. 1998,26(10):33–37
[31]程文青,何建华,沈春蕾.一种基于神经网络的激光水下目标探测方法[J].华中理工大学学报. 1999, 27(3):56–58
[32]黄显高,徐健学.利用小波多尺度分解算法实现混沌系统的噪声减缩[J].物理学报. 1999, 48(10):1810–1816
[33] Huang X G., Xu J X. Unmasking chaotic mask by a wavelet multiscale decompo-sition algorithm [J]. IntJBif and Chaos. 2001, 11(2):561–569
[34] Zhu changfang, Hu Guangshu. Combining independent component analysis insingle-trail VEP estimation[J]. Proceeding IEEE. 2000, 48(3):761–763
[35]汪芙平,郭静波.强混沌干扰中的谐波信号提取[J].物理学报. 2001, 50(6):1019–1023
[36]李鸿光,孟光.基于经验模式分解的混沌干扰下谐波信号的提取方法[J].物理学报. 2004, 53(7):2069–2074
[37]杜京义,侯媛彬.混沌背景中微弱谐波信号检测的SVM方法[J].仪器仪表真学报.2007, 28(3):554–559
[38]陈争,曾以成,付志坚.混沌背景中信号参数估计的新方法[J].物理学报. 2008,57(1):46–50
[39]陆振波,蔡志明,姜可宇.基于Volterra滤波器的混沌背景弱信号检测[J].系统仿真学报. 2008, 20(7):1778–1780
[40]徐伟.混沌背景中的微弱信号检测[D].南京:南京信息工程大学硕士论文, 2007
[41]陈颖.湮没在混沌背景下的微弱信号检测[D].苏州:苏州大学硕士论文, 2006
[42]祝开艳.混沌背景中的谐波恢复[D].长春:吉林大学硕士论文, 2005
[43]徐炜.混沌背景中的弱信号检测和提取[D].上海:上海交通大学, 2006
[44]王国光.提取混沌中信号方法的研究[D].长春:吉林大学, 2008
[45] Huang N E, Shen Z., Long S R. The empirical mode decomposition and the Hilbertspectrum for nonlinear and non-station time series analysis [J]. Proceeding of theRoyal Society of London A. 1998, 454:903–995
[46] S. Hein. Exploiting chaos to suppress spurious tone in general double-loopΣ?modulators[J]. IEEE Trans CAS-I. 1993, 40(10):651–659
[47] Lorenz E.N. Atmosspheric predictablity esperiments with a large numerical model[J]. Tellus. 1982, 34:505–513
[48]张学义.混沌同步及其在通信中的应用[D].哈尔滨:哈尔滨工程大学博士论文,2001
[49]陈奉苏.混沌学及其应用[M].北京:中国电力出版社, 1998
[50]王诗斌.混沌理论及混沌振荡器的研究[D].长沙:湖南大学, 2004
[51]关新平等.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]谢红梅.基于混沌理论的信号处理方法研究[D].西安:西北工业大学博士论文,2003
[53]孙海云,曹庆杰.混沌时间序列建模及预测[J].系统工程理论与实践. 2001,5:106–110
[54]赵小梅.混沌预测与混沌优化理论与算法研究[D].杭州:浙江大学, 2002
[55]王凌.智能优化算法及其应用[M].北京:清华大学出版社, 2001
[56] Mahmut Ciftci. Channel Equalization for Chaotic Communication Systems[D].Georgia Institute of Technology ,Ph.D, 2002
[57] Yassen MT. Chaos synchronization between two di?erent chaotic systems usingactive control[J]. Chaos ,solitions and Factals. 2005, 23:131–140
[58] YW Wang, AH Guan, HO Wang. Feedback and adative control for synchronizationof chen system via a single variable[J]. Physics letter A. 2003, 311:389–395
[59] Moez Feki. A adative Chaos synchronization scheme applied to secure communi-cation [J]. Chaos ,solitions and Factals. 2003, 18:141–148
[60]李树钧.数字化混沌密码的分析与设计[D].西安:西安交通大学, 2003
[61] M.S Baptista. Cryptography with Chaos[J]. Phys LettA. 1998, 240:50–54
[62] F. Dachselt, W. Schwarz. Chaos AND Cryptography [J]. Phys LettA. 2001,48(12):1498–1509
[63] T. Yang, C.W. Wu, L.O. Chua. Cryptography based on chaotic systems [J]. IEEETransCAS-I. 1997, 44(5):469–472
[64] R. Dettmer, P. Falcone, T. Keviczky, J. Asgari, D. Hrovat. Chaos and engineer-ing[J]. IEE Review. 1993, 39(5):199–203
[65]龙运佳.混沌工程学[J].中国工程科学. 2001, 3(2):10–15
[66]曹建福,韩崇昭,方洋旺.非线性系统理论及应用[M].西安:西安交通大学出版社,2001
[67]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
[68]方锦清.驾驭混沌与发展高新技术[M].北京:原子能出版社, 2002
[69] A. Wolf, J.B. Swift, H.L. Swinney. Determining Lyapunov exponents from a timeseries[J]. Physical D. 1985, 16:285–317
[70]陶朝海,陆君安,吕金虎.统一混沌系统的反馈同步[J].物理学报. 2002,51(7):1497–1501
[71] Lu J.G. Chaotic dynamics and synchronization of fractional-order Arneodo’s sys-tems[J]. Chaos,Solition and Fractals. 2005, 26(4):1125–1133
[72] Li F., Hu A.H., Xu Z.Y. Robust synchronization of uncertain chaotic systems[J].Chinese Physics. 2006, 15(3):507–512
[73] Wang Y.W., Guan Z.H. Generalized synchronization of continuous chaotic sys-tems[J]. Chaos,Solition and Fractals. 2006, 27(1):97–101
[74] Taherion S., Lai Y.C. Observability of lag synchronization of coupled chaoticoscillators[J]. Physical Review E. 1999, 59(6):6247–6250
[75] Shahverdiev E.M., Sivaprakasam S., Shore K.A. Lag synchronization in time de-layed system[J]. Physical Review A. 2002, 292(6):320–324
[76] Chen J.Y., Wong K.W. Phase synchronization discrete chaotic systems[J]. PhysicalReview E. 2000, 61(3):2559–2562
[77] R. Brown, L. Kocarev. A unifying definition of synchronization for dynamicalsystems [J]. Chaos. 2000, 10(2):344–349
[78] Pecora L.M., Carroll T. L. Synchronization in chaotic systems[J]. PhysRevLett.1990, 64(8):821–825
[79] Kocarev L., Parlitz U. General approach for chaotic synchronization with applca-tions to communication[J]. Physical Review Letters. 1995, 74:5028–5031
[80] Gallego J.A. Nonlinear regulation of a Lorenz system feedback linearization tech-niques[J]. Dynamic Control. 1994, 4:277–298
[81]唐芳,邱琦.混沌系统的辅助参考反馈控制[J].物理学报. 1999, 48(5):802–807
[82] Chen G.R., Dong X. On feedbaek control of chaotic continuous-time systems[J].IEEE Transon Circuits and System. 1993, 40(9):591–601
[83] Pyragas K. Continous control of chaos by self-controlling systems[J]. Physicalletters A. 1992, 170:421–428
[84] Pyragas K., Tamasevicius A. Experimental control of chaos by delayed self-controlling feedback[J]. Physical letters A. 1993, 180:99–102
[85]王光瑞,于煦龄,陈式刚.混沌的控制、同步和应用[M].北京:国防工业出版社,2001
[86]邹恩,李详飞,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
[87] Jiang J.P., Tang K.S. A global synchronization criterion for coupled chaotic systemvia unidirectional linear error feedback approach[J]. IntJBifurcation chaos. 2002,12(10):2239–2253
[88] Bu S.L., Wang S.Q. A algorithm based on variable feedback to synchronize chaoticand hyperchaotic systems[J]. Physica D. 2002, 164:45–52
[89]高金峰,罗先觉,马西奎.控制与同步连续时间混沌系统的非线性反馈方法[J].物理学报. 1999, 48(9):1618–1627
[90]刘扬正,费树眠. Genesio一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
[91]陶朝海,陆君安.混沌系统的速度反馈同步[J].物理学报. 2005, 54(11):5058–5061
[92]卢俊国,汪小帆,王执栓.连续时间混沌系统控制与同步的状态反馈方法[J].控制与决策. 2001, 16(4):476–479
[93] Bai E.W., Lonngren K.E. Sequential synchronization of two Lorenz systems usingactive control[J]. Chaos,Solitons and Fraetals. 2000, 11:1041–1044
[94] Kouomou Y.C., Woafo P. Stability and optimization of chaos synchronizationthrough feedback coupling with delay[J]. Physics Letters A. 2002, 298:181–28
[95]陈凤祥.混沌控制与同步的鲁棒控制器研究[D].上海:上海交通大学博士论文,2008
[96] E. Huberman, B.A.and Lumer. Dynamics of adaptive systems[J]. IEEE Transac-tions on CAS. 1990, 37:547–550
[97] John J.K., Amritkar R.E. Synchronization of unstable orbits daptive control[J].Physical Review E. 1994, 49(6):4843–4848
[98] John J.K., Amritkar R.E. Synchronization by Feedback and Adaptive Control[J].Bifurcation and Chaos. 1994, 4(6):1687–1695
[99] Yassen M.T. Adaptive chaos control and synchronization for uncertain new chaoticdynamical system[J]. Physics Letters A. 2006, 350(1-2):36–43
[100] Shihua Chen, Jinhu Lü. Parameters identification and synchronization of chaoticsystems based upon adaptive control[J]. Physics Letters A. 2002, 299(4,8):353–358
[101]吴立刚,王常虹,曾庆双. Terminal滑模自适应控制实现一类不确定混沌系统的同步[J].控制与决策. 2006, 21(2):229–231
[102] Yongguang Yu, Suochun Zhang. Adaptive backstepping synchronization of uncer-tain chaotic system[J]. Chaos,Solitons and Fractals. 2004, 21(3):643–649
[103] Samuel Bowong, F.M. Moukam Kakmeni. Synchronization of uncertain chaotic sys-tems via backstepping approach[J]. Chaos,Solitons and Fractals. 2004, 21(4):999–1011
[104] Wang C., Ge S.S. Adaptive synchronization of uncertain chaotic systems via back-stepping design[J]. Chaos,Solitons and Fractals. 2001, 12:1199–2006
[105] Zhenya Yan. Chaos Q–S synchronization between R¨ossler system and the newunified chaotic system[J]. Physics Letters A. 2005, 334(5-6):406–412
[106] Tan Xiaohui, Jiye Zhang, Yiren Yang. Synchronizing chaotic systems using back-stepping design[J]. Chaos,Solitons and Fractals. 2003, 16(1):37–45
[107] Fang J Q, Hong Y G, Chen G R. Switching manifold approach to chaos synchro-nization[J]. Physics Letters E. 1999, 59(3):2523–2526
[108] Liao T. L., Huang N.S. Control and synchronization of discrete time chaotic sys-tems via variable structure control technique[J]. Physics Letters A. 1997, 234:262–268
[109] Lilian Huang, Rupeng Feng, Mao Wang. Synchronization of uncertain chaoticsystems with perturbation based on variable structure control[J]. Physics LettersA. 2006, 350(3-4):197–200
[110]黄丽莲,王茂,冯汝鹏.参数不确定混沌系统的反步变结构控制[J].系统工程与电子技术. 2005, 27(4):704–707
[111]张洪,陈天麒.参数未知的时延混沌系统滑模变结构同步控制[J].物理学报. 2005,21(12):1937–1941
[112] Zhi Li, Songjiao Shi. Robust adaptive synchronization of Rossler and Chen chaoticsystems via slide technique[J]. Physics Letters A. 2003, 331(4-5):389–395
[113]单梁.混沌系统的若干同步方法研究[D].南京:南京理工大学博士论文, 2006
[114]高铁杠,陈增强,袁著社,顾巧论.基于观测器的混沌系统的同步研究[J].物理学报. 2004, 53(5):1305–1308
[115] Morgul O., Solak E. On the synchronization of chaotic systems by using ststeobservations[J]. IntJBifurcation and chaos. 1997, 7(6):1307–1322
[116] Nijmeijer H., Mareeis I.M. An Observer looks at synchronization [J]. IEEE TransonCircuits and ssystems-I. 1997, 44(10):882–890
[117]关新平,范正平,陈彩莲.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, 1993
[118]杨涛,邵惠鹤.一类混沌系统的同步方法[J].物理学报. 2002, 51(4):742–748
[119] Boutayeb M., Darouach M., Rafaralyahy H. Generalized state-space observersfor chaotic synchronization and secure communication [J]. IEETransCircuits andsystem-I. 2002, 49(3):345–349
[120]姚利娜,高金峰,廖旎焕.实现混沌系统同步的非线性状态观测器方法[J].物理学报. 2006, 55(1):35–41
[121] Li G.H., Zhou S.P., Xu D.M. Chaos synchronization based on intermittent stateobserver [J]. Chinese Physics. 2004, 13(2):168–172
[122]黄玮,张化光.一类混沌系统的脉冲控制同步[J].东北大学学报(自然科学版).2004, 25(11):1027一1029
[123] Xie W., Wen C.Y. Impulsive control for the stabilization and synehronization ofLorenz systems[J]. Physics Letters A. 2000, 275:67–72
[124] Wang Y.W., Wen C.Y., Xiao J.W. ImPulsive synchronization of Chua’s oscillatorsvia a single variable[J]. Chaos,Solitons and Fractals. 2006, 29:198–201
[125] Lu J.A., Xie J., Lu J.H. Control chaos in transition using sampled-data feedback[J].Applied Methmatics and Mechanics. 2003, 24(11):1309–1316
[126] Yu D.C., Wu A.G., Wang D.Q. A simple asymptotic trajectory control of full stateof a unified chaotic system[J]. Chinese Physics. 2006, 15(2):306–309
[127]修春波,刘向东,张宇河.混沌优化与模糊控制在混沌控制中的应用[J].控制理论与应用. 2005, 22(1):63–67
[128] W.H. Deng, C.P. Li. Chaos synchronization of the fractional Lüsystem [J]. PhysicaA. 2005, 353(1):61–72
[129] Jianping Yan, Changpin Li. On chaos synchronization of fractional di?erentialequations [J]. Chaos,Solitons and Fractals. 2007, 32(2):725–735
[130] Junwei Wang, Xiaohua Xiong, Yanbin Zhang. Extending synchronization schemeto chaotic fractional-order Chen systems[J]. Physica A. 2006, 370(2):279–285
[131] Mohammad Saleh Tavazoei, Mohammad Haeri. Determination of active slid-ing mode controller parameters in synchronizing di?erent chaotic systems[J].Chaos,Solitons and Fractals. 2007, 32(2):583–591
[132] Ricardo Aguilar-Lo′pez, Rafael Martinez-Guerra. Partial synchronization of dif-ferent chaotic oscillators using robust PID feedback[J]. Chaos,Solitons and Fractals.2007, 32(2):572–581
[133] Guo-Hui Li. Generalized projective synchronization between Lorenz system andChen’s system[J]. Chaos,Solitons and Fractals. 2007, 32(4):1454–1458
[134]程丽,张入元,彭建华.用单一驱动变量同步混沌与超混沌的一种方法[J].物理学报. 2003, 52(3):536–541
[135]胡岗,肖井华,郑志刚.混沌控制[M].上海:上海科技出版社, 2000
[136] LüJ.H., Zhou T.S., Zhou S.C. Chaos Synchronization Between Two Di?erentChaotic Systems[J]. Chaos,Solitions and Fractals. 2002, 14(4):529–541
[137] Qiang Jia. Chaos control and synchronization of the Newton–Leipnik chaoticsystem[J]. Chaos,Solitions and Fractals. 2008, 35(4):814–824
[138] Park J.H. Stability Criterion for Synchronilzation Linearly Coupled the UnifiedChaotic Systems[J]. Chaos,Solitions and Fractals. 2005, 23(1):79–85
[139]蒲兴成.混沌同步控制及应用研究[D].重庆:重庆大学博士论文, 2006
[140] H.K. Khalil.非线性系统[M].北京:电子工业出版社, 2005
[141] J. P. LaSalle. The extent of asymptotic stability[J]. Proceedings of the NationalAcademy of Sciences. 1960, 46(3):363–365
[142] J. P. LaSalle. Some extensions of Liapunov’s second method[J]. IRE Trans CircuitTheory. 1960, 7:520–527
[143] H. M. Rodrigues. Abstract methods for synchronization and applications[J]. Ap-plicable Anal. 1996, 62:263–296
[144] V. S. Afraimovich, S.N. Chow, J. K. Hale. Synchronization in lattices of coupledoscillators[J]. Physica D. 1997, 103:442–451
[145] L.F.C. Rodrigues, H. M.and Alberto. On the Invariance Principle: Generalizationsand Applications to Synchronization[J]. IEEE Trans on Circuits and Systems.2000, 47(5):730–739
[146] Y. Orlov. Extended Invariance Principle for nonautonous Swittch Systems[J]. IEEETrans on Automatic Control. 2003, 48(8):1448–1452
[147] A. Swami, J.M. Mendel. Cumulant-based approach to the harmonic retrieval prob-lem[C]. International Conference on Acoustics, Speech, and Signal Processing.1998, 4:2264–2267
[148] A. Swami, J.M. Mendel. Cumulant-based approach to harmonic retrieval andrelated problems[J]. IEEE Transactions on Signal Processing. 1991, 39(5):1099–1109
[149]石要武,戴逸松.测量噪声背景下微弱正弦信号参数估计的互功率谱方法[J].电子学报. 1994, 22(1):1–8
[150]石要武,戴逸松,丁宏.有色噪声背景下正弦信号频率估计的互谱Pisarenko和MUSIC方法[J].电子学报. 1996, 24(10):46–50
[151] Ma Yan, Shi Yaowu, Dai Yisong. Harmonic Retrieval Embedded in Hybrid ColoredNoise : a Novel Pisarenko Method Based on Cross-high-order Cumulant[J]. HighTechnology Letters. 2000, 6(3):23–28
[152]于晓辉.噪声背景下chirp信号参数估计理论与方法研究[D].长春:吉林大学博士论文, 2007
[153] Wax M., Kailath T. Detection of signals by information theoretic criteria [J]. IEEETransactions on Acoustics, Speech, and Signal Processing. 1985, 32(2):387–392
[154]余秋星.基于混沌和分型的信号检测[D].西安:西北工业大学硕士论文, 2001
[155] Chen G.R., Ueta T. Yet another chaotic attractor[J]. Int Journal of Bifurcationand Chaos. 1999, 9:1465–11466
[156] Carroll T L. synchronizing chaotic systems using filtered signals[J]. Phys Rev E.1994, 50:2580–2588
[157] Rodrigues H. M., Alberto L. F. C., Bretas N. C. Uniform Invariance Principle andSynchronization. Robustness with respect to parameter variation[J]. Journal ofDi?erential Equations. 2001, 169(1):228–254
[158]陈明杰,李殿璞,张爱筠.失配混沌系统的同步研究[J].哈尔滨工程大学学报.2005, 26(5):633–638
[159] Gameiro M.F., Rodrigues H.M. Applications of Robust Synchronization to Com-munication Systems[J]. Applicable Analysis. 2001, 79:21–45
[160] Park J.H. Adpative synchronization of R¨ossler system with uncertain Par-maeters[J]. Chaos,Solitons and Fractals. 2005, 25(2):333–338
[161] Wenwu Yu, Guanrong Huang, Jinde Chen, Jinhu Lü, Ulrich Parlitz. Parame-ter identification of dynamical systems from time series[J]. Phys Rev E. 2007,75:067201
[162] Ming-Chung Ho, Yao-Chen Hung, I-Min Jiang. On the synchronization of uncer-tain chaotic systems[J]. Chaos,Solitons and Fractals. 2007, 33:540–546
[163]李农,李建芬,刘宇平,马健. Parameter identification ba sed on linear feedbackcontrol for uncertain chaotic system[J].物理学报. 2008, 57(3):1404–1408
[164] J.J. Yan, M.L. Hung, T.L. Liao. Adaptive sliding mode control for synchroniza-tion of chaotic gyros with fully unknown parameters[J]. Sound Vibration. 2006,298:298–306
[165] M.S. Tavazoei, M. Haeri. Determination of active sliding mode controller param-eters in synchronizing di?erent chaotic systems[J]. Chaos Solitons and Fractals.2007, 32:583–591
[166] Xuyang Lou, Baotong Cui. Synchronization of neural networks based on parameteridentification and via output or state coupling[J]. Chaos Solitons and Fractals.2007, 32:583–591
[167] Yassen M.T. Adaptive synchronization of Rossler and systems with fully uncertainParameters[J]. Chaos,Solitons and Fractals. 2005, 23:1527一1536
[168]褚衍东,常迎香,张莉,张建刚. Parameters Identification and IFSHPS in Chaoticand Hyper-chaotic Systems with Di?erent Structure[J].系统仿真学报. 2008,20(19):5350–5354
[169]王东风.基于遗传算法的统一混沌系统比例一积分一微分控制[J].物理学报.2005, 54(4):1495–1499
[170]关新平,彭海朋,李丽香. Lorenz混沌系统的参数辨识与控制[J].物理学报. 2001,50(1):26–29
[171] Mohammad Haeri, Mahsa Dehghani. Impulsive synchronization of di?erent hyper-chaotic(chaotic) systems[J]. Chaos Solitons and Fractals. 2008, 38:120–131
[172] Lingling Zhang, Lihong Huang, Zhizhou Zhang, Zengyun Wang. Fuzzy adap-tive synchronization of uncertain chaotic systems via delayed feedback control[J].Physics Letters A. 2008, 372:6082–6086
[173] A. Maybhate, R. E. Amritkar. Use of synchronization and adaptive control inparameter estimation from a time series[J]. Phys Rev E. 1999, 59(1):284–293
[174] Chen J., Liu Z.R. A Method of Controlling Synchronizationin Diferent Systems[J].Chinese Physical Letters. 2003, 20(9):1441–1446