对几类系统的动力学分析及混沌控制
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摘要
混沌是一种特殊的自然现象,它揭示了自然界中有序无规则的运动特性。它在诸多领域中都有着十分广阔的应用前景,是近年来学科研究领域的前沿。混沌系统的最大特点就在于系统的演化对初始条件十分敏感。因此,从长期意义上讲,系统的行为是不可预测的。
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。本文研究了卫星系统、约瑟夫森结系统和电力系统的混沌现象,对这几类系统进行混沌控制。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。除此之外,做出系统的相图、庞加莱截面和时间历程图来验证所得结果,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 a special phenomenon which reflects the irregular rand highly complex structures in tine and in space that follow deterministic laws and equations. It has been studied extensively in many areas in recent years. The most important characteristic of a chaotic system is that its evolution has highly sensitive dependence on initial conditions. So, from a long-term perspective, future behaviors of a chaotic system are unpredictable.
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 satellite system, Josephson junction systems and power systems.
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 suppress 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. 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.
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。本文研究了卫星系统、约瑟夫森结系统和电力系统的混沌现象,对这几类系统进行混沌控制。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。除此之外,做出系统的相图、庞加莱截面和时间历程图来验证所得结果,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 a special phenomenon which reflects the irregular rand highly complex structures in tine and in space that follow deterministic laws and equations. It has been studied extensively in many areas in recent years. The most important characteristic of a chaotic system is that its evolution has highly sensitive dependence on initial conditions. So, from a long-term perspective, future behaviors of a chaotic system are unpredictable.
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 satellite system, Josephson junction systems and power systems.
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 suppress 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. 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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13 H. Bergman. Synchronized Chaotic Bursts in a Delayed Feedback System. Trends Neurosci. 1990, 21(1): 32~50
14 G. Chen, X. Dong. From Chaos to Order: Methodologies, Perspectives and Applications. Word Scentific. 1998, 3(14):1298~1303
15胡海岩.应用非线性动力学.航空工业出版社, 2000:25~42
16 B. Irina, R. Lev. Sensitivity and Chaos Control for the Forced Nonlinear Oscilla. Chaos, Solitons & Fractals. 2005, 25(3):1437~1451
17龙运佳.混沌振动研究.清华大学出版社, 1997:13~16
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30 N. F. Rulkov. Regularization of Synchronized Chaotic Bursts. Phys. Rev. Lett.. 2001, 86(1):183~186
31 Y. Zhang, Q. Zhang, L. Zhao, C.Yang. Dynamical Behaviors and Chaos Control in a Discrete Functional Response Model. Chaos, Solitons & Fractals. 2006, 8(21):132~136
32 F. J. Romeiras, C. Grebogi, E. Ott. Controlling Chaotic Dynamieal Systems. Physica D. 1990:165~92
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34 E. Ott. Chaos in Dynamical System. Cambridge University Press. 1992, 58:77~94
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36 Y. Braiman, I. Goldhirsch. Taming Chaotic Dynamics with Weak Periodic Perturbations, Phys. Rev. Lett.. 1991, 66:2545~2548
37 H. K. Qammar, F. Mossagch, et al. Dyamieal Complexity Arising in the Adaptive Control of a Chaotic System. Phys. Rev. Lett.. 1993, 178(3):279~283
38 P. M. Alsing, A. Garielldes. Using Neural Networks for Controlling Chaos. Phys.Rev. E. 1994, 49:1225~1231
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48 A. Basios, S. Ruffo. Finite-Size Effects in a Population of Interacting Oscillators. Phys. Rev. E.. 1999, 59(2):1633~1636
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51 Y. Xu, W. Xu, G. M. Mahmoud. On a Complex Beam–beam Interaction Model with Random Forcing. Chaos, Solitons & Fractals. 2005, 23:265~271
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2黄润生.混沌及其应用.武汉大学出版社, 2000:21~55
3 E. Lorenz. Deterministic Non-periodic Flows. J. Atmos. SCI. 1963, 20:130~141
4 D. Ruelle, F. Takens. On the Nature of Turbulence. Communication in Mathematical Physics. 1971, 20:167~192
5 T. Li, J. Yorke. Period Three Implies Chaos. AMS. Monthly. 1975, 82:985~992
6 M. R. May. Simple Mathematical Models with very Complicated Dynamics. Nature. 1976, 261:459~467
7 M. J. Feigenbaum. Quantitative Universality for a Class of Non-linear Transformations. J. Stat. Phys.. 1978, 19:25~52
8 F. Takens. Determining Strange Attractors in Turbulence. Lecture Notes in Math.. 1981, 898:361~381
9 B. D. Greenspan, P. J. Holmes. Homoclinic Orbits, Subharmonics and Global Bifurcation in a Periodically Forced Family of Oscillators. Nonlinear Dynamics and Turbulence. 1983, 6:172~214
10 P. Grassberger, I. Procaccia. Measuring the Strangeness of Strange Attraetors. Physica D. 1983, 9:189~208
11 A. Wolf, J. B. Swift, H. L. Swinney. Determining Lyapunov Exponents from a Time Series. Physica D. 1985, 5(16):285~317
12 E. Ott, C. Grebogi, J. Yorke. Controlling Chaos. Phys. Rev. Lett.. 1990, 11:1196~1199
13 H. Bergman. Synchronized Chaotic Bursts in a Delayed Feedback System. Trends Neurosci. 1990, 21(1): 32~50
14 G. Chen, X. Dong. From Chaos to Order: Methodologies, Perspectives and Applications. Word Scentific. 1998, 3(14):1298~1303
15胡海岩.应用非线性动力学.航空工业出版社, 2000:25~42
16 B. Irina, R. Lev. Sensitivity and Chaos Control for the Forced Nonlinear Oscilla. Chaos, Solitons & Fractals. 2005, 25(3):1437~1451
17龙运佳.混沌振动研究.清华大学出版社, 1997:13~16
18舒仲周,张继业,曹登庆.运动稳定性.中国铁道出版社, 2000:22~39
19刘曾荣.混沌的微扰判据.上海科技教育出版社, 1994:31~40
20 B. Hassard, N. Kazarinoff, Y. Wan. Theory and Applications of Hopf Bifurcation. Cambridge. Cambridge University Press. 1981:132~144
21 K. Pyragas. Continuous Control of Chaos by Self-controlling Feedback, Phys. Rev. Lett.. 1992, 170(6):421~428
22 P. Manneville, Y. Pomeau. Intermittency and the Lorenz Model. Phys. Lett.. 1979, 75(1):l~2
23郑伟谋,郝柏林.实用符号动力学.上海科技教育出版社, 1994:51~76
24廖晓昕.稳定性的理论、方法和应用.华中科技大学出版社, 1999:70~79
25 J. Alvarez-Ramirez, P. G. Espinosa, H. Puebla. Chaos Control Using Small-amplitude Damping Signals. Physics Letters A. 2003, 316:196~205
26 H. K. Qammar, F. Mossayebi, L.Murphy. Dynamical Complexity Arising in the Adaptive Control of a Chaotic System. Physics Letter A. 1993, 178(3):279~283
27 J. Milton, P. Jung. Epilepsy as a Dynamic Disease. Springer Berlin. 2003:1~14
28吴祥兴,陈忠等.混沌学导论.上海科学技术出版社, 1997:45~66
29 H. Chen, C. Lee. Anti-control of Chaos in Rigid Body Motion. Chaos, Solitons & Fractals. 2004, 21:957~965
30 N. F. Rulkov. Regularization of Synchronized Chaotic Bursts. Phys. Rev. Lett.. 2001, 86(1):183~186
31 Y. Zhang, Q. Zhang, L. Zhao, C.Yang. Dynamical Behaviors and Chaos Control in a Discrete Functional Response Model. Chaos, Solitons & Fractals. 2006, 8(21):132~136
32 F. J. Romeiras, C. Grebogi, E. Ott. Controlling Chaotic Dynamieal Systems. Physica D. 1990:165~92
33 D. Auerbach, C. Grebogi, E.Ott. Controlling Chaos in High Dimensional Systems. Phys. Rev. Lett.. 1992, 69:3479~3482
34 E. Ott. Chaos in Dynamical System. Cambridge University Press. 1992, 58:77~94
35 K. Pragas, W. Gadomski. Experimental Control of Chaos by Means of Weak Parametric Perturbations. Phys. Rev. E. 1994, 49(4):2528~2531
36 Y. Braiman, I. Goldhirsch. Taming Chaotic Dynamics with Weak Periodic Perturbations, Phys. Rev. Lett.. 1991, 66:2545~2548
37 H. K. Qammar, F. Mossagch, et al. Dyamieal Complexity Arising in the Adaptive Control of a Chaotic System. Phys. Rev. Lett.. 1993, 178(3):279~283
38 P. M. Alsing, A. Garielldes. Using Neural Networks for Controlling Chaos. Phys.Rev. E. 1994, 49:1225~1231
39 K. Otawara, L. T. Fan. Controlling Chaos with an Artificial Neural Networks. Proc. IEEE Int. Joint. Conf. Fuzzy Syst. 1995, 4:1943~1948
40郝柏林.分岔、混沌、奇怪吸引子、湍流及其他.物理学进展. 1983, 3(3):329~416
41刘延柱,陈文良,陈立群.振动力学.高等教育出版社, 1998:306~323
42 C. T. Lin. Controlling Chaos by GA-based Reinforcement Learning Neural Network. IEEE Transactions on Neural Networks. 1999, 4(10):846~859
43 M . Ramesh, S. Narayanan. Chaos, Solitons & Fractals. 1999, 3(10):1473~1484
44 K. Pyragas. Continuous Control of Chaos by Self-Controlling Feedback. Phys. Lett.. 1992, 170(6):421~428
45 L. Tsimring, A. Pikovsky. Noise-Induced Dynamics in Bistable System with Delay. Phys. Rev. Lett.. 2001, 87(1):25~37
46 J. G. Wei. G. Leng. Lyapunov Exponent and Chaos of Duffing’s Equation Perturbed by White Noise. Appl. Math. Comput. 1997:77~90
47 T. Y. Li, J. Yorke. A Periodic Three Implies Chaos. Am. Math. Monthly. 1975, 82:985~992
48 A. Basios, S. Ruffo. Finite-Size Effects in a Population of Interacting Oscillators. Phys. Rev. E.. 1999, 59(2):1633~1636
49 W. Y. Liu, W. Q. Zhu, Z. L. Huang. Effect of Bounded Noise on Chaotic Motion of a Duffing Oscillator Under Parametric Excitation. Chaos, Solitons & Fractals. 2001, 12:527~541
50 Y. Lei, W. Xu, Y. Xu. T. Fang. Chaos Control by Harmonic Excitation with Proper Random Phase. Chaos, Solitons & Fractals. 2004, 21:1175~1190
51 Y. Xu, W. Xu, G. M. Mahmoud. On a Complex Beam–beam Interaction Model with Random Forcing. Chaos, Solitons & Fractals. 2005, 23:265~271
52刘延柱,陈立群,彭建华.航天器混沌姿态运动研究进展综述[J].自然科学进展. 1998, 4:386~390
53刘延柱,陈立群,成功,戈新生.航天器姿态动力学中的稳定性、分岔和混沌.力学进展. 2000, 30(3):351~357
54于洪杰,刘延柱.复杂力场中磁性刚体航天器混沌姿态运动的控制.上海交通大学学报. 2004, 8:1409~1411
55成功,刘延柱.近地球赤道面轨道上磁性刚体卫星的混沌运动.空间科学学报. 2000, 20(1):69~73
56 L. Q. Chen, C. Liu. Chaotic Attitude Motion of a Class of Spacecraft on an Elliptic Orbit. Tech Mech. 1998, 18(1):41~44
57陈立群,刘延柱.一类非自旋航天器清浅姿态运动及基参数开闭环控制.力学学报. 1998, 30(3):363~369
58陈立群,刘延柱.控制清浅振动的逆系统方法.上海力学. 1999, 20(2):95~98
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