基于随机相位对几类动力系统的混沌控制
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摘要
混沌现象是20世纪科学史上最重要的发现之一,与相对论、量子力学一起成为20世纪物理学的三次重大革命。混沌控制是研究混沌现象的一个重要的领域,它在混沌应用中起着至关重要的作用。本文研究了阻尼单摆系统、二维细胞神经网络系统和一类Mathieu方程的混沌现象,对系统进行混沌控制。
目前混沌控制的方法主要包括两大类:反馈控制和非反馈控制。本文采用了随机相位控制方法,此方法是非反馈控制法方法的一种,通过在混沌方程加上一个随机相位来实现混沌系统的控制。这里采用高斯白噪声作为系统的随机相位。通过数值仿真可以发现,当高斯白噪声的强度大于某一临界值时,系统可以被控制成非混沌系统。本文采用最大Lyapunov指数作为判定系统是否混沌的指标。通过平均最大Lyapunov指数的变化,我们可以看出混沌系统被控制。其中,平均最大Lyapunov指数是根据Khasminskii球面坐标变换方法计算得到。除此之外,做出系统的相图、庞加莱截面和时间遍历图来验证所得结果,可以看出,两种方法结果是完全一致的。
由于本文所控制的三个方程在工程和实验中均有广泛的应用,因此,文章所得结果不但具有理论意义,而且有着重要的工程实用价值。
Chaos is one of the most important discoveries in the 20~(th) century. It is one of the three major revolutions in physics. Chaos control is an important field in explorations of chaos motions, and it is crucial in application of chaos. In this paper, we consider three chaos systems: chaos pendulum system, two-dimensional cellar neural network and the Mathieu equation.
At present, there are mainly two sorts of the chaos control's method: feedback control and non-feedback control. In this paper, we control these chaos systems by random phase, it's one of the non-feedback chaos control methods. By adding a random phase we can make chaotic portrait stable. We use Gaussian white noise as the random phase. We can find the chaotic systems dynamical behavior will be suppressed as the noise intensity increases slightly. In this paper, we show that random phase can suppress chaos by the average of the top Lyapunov exponent, which is computed based on Khasminskii's spherical coordinate formulation for linear stochastic systems. In addition, phase portraits, Poincarésurface of section and time evolution are studied to confirm the obtained results. Both methods lead to fully consistent results.
In this paper, the three systems have extensive applications, therefore, the results are valuable not only in theory but in project.
目前混沌控制的方法主要包括两大类:反馈控制和非反馈控制。本文采用了随机相位控制方法,此方法是非反馈控制法方法的一种,通过在混沌方程加上一个随机相位来实现混沌系统的控制。这里采用高斯白噪声作为系统的随机相位。通过数值仿真可以发现,当高斯白噪声的强度大于某一临界值时,系统可以被控制成非混沌系统。本文采用最大Lyapunov指数作为判定系统是否混沌的指标。通过平均最大Lyapunov指数的变化,我们可以看出混沌系统被控制。其中,平均最大Lyapunov指数是根据Khasminskii球面坐标变换方法计算得到。除此之外,做出系统的相图、庞加莱截面和时间遍历图来验证所得结果,可以看出,两种方法结果是完全一致的。
由于本文所控制的三个方程在工程和实验中均有广泛的应用,因此,文章所得结果不但具有理论意义,而且有着重要的工程实用价值。
Chaos is one of the most important discoveries in the 20~(th) century. It is one of the three major revolutions in physics. Chaos control is an important field in explorations of chaos motions, and it is crucial in application of chaos. In this paper, we consider three chaos systems: chaos pendulum system, two-dimensional cellar neural network and the Mathieu equation.
At present, there are mainly two sorts of the chaos control's method: feedback control and non-feedback control. In this paper, we control these chaos systems by random phase, it's one of the non-feedback chaos control methods. By adding a random phase we can make chaotic portrait stable. We use Gaussian white noise as the random phase. We can find the chaotic systems dynamical behavior will be suppressed as the noise intensity increases slightly. In this paper, we show that random phase can suppress chaos by the average of the top Lyapunov exponent, which is computed based on Khasminskii's spherical coordinate formulation for linear stochastic systems. In addition, phase portraits, Poincarésurface of section and time evolution are studied to confirm the obtained results. Both methods lead to fully consistent results.
In this paper, the three systems have extensive applications, therefore, the results are valuable not only in theory but in project.
引文
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[4]汪秉宏,李诩神,汪克林,郭光灿.非线性科学选讲.中国科学技术大学出版社.1994:41-99
[5]吴祥兴,陈忠.混沌学导论.上海科学技术文献出版社,1996:263-269
[6]洛仑兹著,刘式达,刘式适,严中伟译.混沌的本质.气象出版社,1997:198-204
[7]B.N.Lorenz.Deterministic nonperiodic flow.J.Atmos.Sci.,1963,20:130-141
[8]格莱克·詹姆斯著,张淑誉译.混沌开创新科学.上海译文出版社,1991:10-35.168-199
[9]刘秉正,彭建华.非线性动力学.高等教育出版社,2004:21-35
[10]Hubler.Adaptive control of chaotic systems.Helv.Phys.Acta,1989,62:343-346
[11]E.Ott,C.Grebogi and J.Yorke.Controlling chaos.Phys.Rev.Lett.,1990,64:1196-1199
[12]W.L.Ditto,S.N.Rauseo and M.L.Spano.Experimental control of chaos.Phys.Rev.Lett.,1990,65:3211-3214
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[14]L.M.Pecora and T.L.Carroll.Synchronization in chaotic systems.Phys.Rev.Lett.,1990,64:821-824
[15]L.M.Pecora and T.L.Carroll.Driving system with chaotic signals.Phys.Rev.Lett.,1991,44:2374-2383
[16]R.Devany等著,卢侃,孙建华等译.混沌动力学.上海翻译出版公司,1990:35-40
[17]陈绥阳,褚蕾蕾著.动力系统基础及其方法.科学出版社,2003:21-30. 46-54
[18]刘延柱,陈文良,陈立群.振动力学.高等教育出版社,1998:306-323
[19]G..R.Chen.Control and Anti- Control of Chaos.University of Houston and City University of Hong Kong,2003:9-40,49-112
[20]郝柏林.从抛物线谈起—混沌动力学引论.上海科技教育出版社,1993:51-62
[21]郝柏林.分岔、混沌、奇怪吸引子、湍流及其它.理学进展,1983,3(3):329-416
[22]Guekenheimer P.On the Bifurcations of Maps of the Interval,Invent.Math.,1977,39:165-178
[23]Feigenbaum M J.The Universal Metric Properties of Nonlinear Transformations,J.Stat.Phys.,1978,19:25-52
[24]于禄,郝柏林.相变和临界现象,北京:科学出版社,1984:61-72
[25]W.Z.Zeng,B.L.Hao,G.R.Wang.et al.Scaling Property of Period- ntupling Sequences in One-dimensional Mappings.Commun.Theor.Phys.,1984,3:283-295
[26]王光瑞,陈式刚.一维单峰映象中高周期序列的普适常数与普适函数.物理学报,1986,35(1):58-65
[27]胡海岩.应用非线性动力学,航空工业出版社,2004:35-56
[28]Ruelle D,Takens F.On the Nature of Turbulence,Commun.Math.Phys.197120(1):167-192
[29]Manneville P,Pomeau Y.Intermittency and the Lorenz Model.Phys.Lett.,1979,75A:1-2
[30]Pomeau Y,Manneville P.Intermittent Transition to Turbulence in Dissipative Dynamical System,Commun.Math.Phys.,1980,74:189-195
[31]郑伟谋,郝柏林.实用符号动力学,上海科技教育出版社,1994:51-62
[32]LI J N,Hao B L.Bifurcation Spectrum in a Delay-differential System Related To Optical Bistability,Commun.Theor.Phys.,1989,11:265-278.
[33]Ott E,Chaos in Dynamical System.Cambridge:Cambridge University Press,1992:61-70
[34]方锦清.非线性系统中的混沌的控制与同步及应用前景(一).物理学进展,1996,16(1):20-25
[35]王光瑞,于熙龄,陈式刚.混沌的控制、同步与利用.国防工业出版社,2001:190-198
[36]K.Pyragas.Continuous control of chaos by self-controlling feedback.Phys.Rev.Lett.,1992,170:421-428
[37]B.A.Huberman and E.Lumer Dynamics of adaptive system.IEEE Trans.CAS.,1990,37:547-550
[38]S.Sinha,R.Ramaswamy and J.Rao.Adaptive control in nonlinear dynamics.Physica D,1990,43:18-128
[39]E.Jean-Jacques,Slotine and Li WeiPing.Applied nonlinear control.China Machine Press,2004.276-307,311-38.
[40]W.P.Li.麻省理工学院,程代展等译.应用非线性控制.北京:机械工业出版社,2006:186-206,210-263
[41]M.A.Matias and J.Guemez.Stabilization of chaos by proportional pulses in the system variables.Phys.Rev.Lett.,1994,72:1455-1458
[42]J.Guemez and M.A.Matias.Control of chaos in unidimensional maps.Phys.Rev.Lett.A,1993,181:29-32
[43]Ramesh M,Narayanan S,Chaos,Solitons & Fractals,199910:1473-1484
[44]Liu W Y,Zhu W Q,Huang Z L,2001,Chaos,Solitons&Fractals,12:527-540
[45]Wei J G 1997 Appl.Math.Comput,88:77-90
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