几类微分方程系统的动力性分析
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摘要
由于混沌系统的控制有着诱人的应用前景，激发了人们对混沌系统控制方法的浓厚兴趣．本文首先对混沌系统的控制方法做了一些研究，利用随机延迟的方法控制了一类混沌系统．在这个研究过程中，我们发现延迟对于系统的动力学行为起着很重要的作用．紧接着文章对几类微分系统的Hopf分岔问题进行了研究，提出了有关的方法和理论，并给出了相应的数值仿真．作者的主要贡献如下
·考虑到随机因素和延迟对动力系统的影响，及混沌系统本身所具有的一些特殊性质，设计出一种控制混沌系统的随机延迟方法，这种方法是以研究系统的Lyapunov指数为出发点，利用白噪声将混沌系统控制到系统的平衡点，在本文中是控制到原点(对于其它平衡点，可以利用线性变化将其转化为目标态是原点的情形)．同时讨论了延迟为零的特殊情形．
·利用延迟反馈方法将一类超混沌控制到它的周期态，即讨论了带延迟的超混沌系统的Hopf分岔．我们知道，随着系统的维数和延迟个数的增加，分析系统的Hopf分岔的难度就会加大．但是对于系统的每个状态取相同的延迟可能达不到控制的效果．本文利用多延迟对超L(u|¨)系统进行控制，同时指出了利用含有相同的延迟个数的延迟反馈方法不能将超Lorenz系统和超Chen系统控制到它们的周期态．
·在上面的研究的启发下，讨论了两种延迟积分微分方程的Hopf分岔问题．到目前为止，关于延迟积分微分方程的研究主要是集中在数值解的稳定性分析方面，对系统Hopf分岔的研究还不是很多．在讨论延迟微分系统的动力学行为的时候，最关键的问题就是其特征方程的根的分布问题，本文借助数值方法，利用边界点法给出Hopf分岔点的可能取值，再利用仿真确定系统的Hopf分岔点．本文是将一般的延迟积分微分方程和带无限延迟的积分微分方程利用不同的方法分开进行讨论．对于一般的延迟积分微分方程，首先利用敏感性分析来确定系统最合适的Hopf参数，然后利用特征根方法讨论Hopf分岔点的存在性．对于带无限延迟的积分微分方程，我们是将其转化为所熟悉的常微分方程来进行讨论的．
Chaos controlling has shown broad application potentials, hence leading to interests in their methodology development. This thesis investigates control methodologies for chaotic systems, along with the stochastic delay method to control a class of chaotic systems. It is found in this study that delay plays an important role in dynamical system. A dynamical system with delay realistically reveals the characteristics of the nature of certain processes. The thesis further investigates chaotic control problems and the Hopf bifurcation problem in several kinds of difference systems, proposes the relevant theory and methodologies, and presents the results from their corresponding numerical simulations. The primary contributions from this thesis are itemized as follows
·It presents a stochastic delay methodology for controlling chaotic systems, based on realistic effects of stochastic factors and delay on dynamic systems, and the corresponding characteristics of chaotic system. This method, based on the Lyapunov exponent, utilizes white noise to control chaotic systems to reach their equilibrium. The equilibrium refers to the original point in the scenarios discussed in this study. For equilibriums other than the original point, linear transformation is applied to transform objective states into their corresponding original states. A special case, i.e. delay is equal to zero is also discussed.
·It presents a methodology, based on delay feedback to control a class of hyper-chaotic systems to their periodic states; that is, the Hopf bifurcation analysis of hyperchaotic systems with delays is investigated. It is known that an increase of system dimensions and delays complicates the Hopf bifurcation analysis. Nevertheless a uniform delay for each state affects the system control adversely. The approach presented in this study deploys multiple delays to control hyperchaotic Lüsystems. It is further revealed that the feedback method based on the same number of delays cannot be applied to controlling hyperchaotic Lorenz system or hyperchaotic Chen systems to reach to their periodic states.
·This study analyzes Hopf bifurcation of two kinds of integro-differential delay equations. At present, the research on integro-differential delay equations is primarily focused on the stability analysis of numerical solutions, and the Hopf bifurcation is a topic that has yet to be thoroughly studied. When analyzing dynamical behaviors of difference delay systems, the key is to identify the distribution of the roots of system's characteristic equation. This study introduces a systematic approach, using numerical analysis and based on boundary points method, to identify the values of Hopf bifurcation points. The method for constant delay integro-differential equations varies from that for integro-differential equations with infinite delay. For constant delay integro-differential equations, the sensitivity analysis is used to identify the most suitable Hopf parameter, followed by identifying the existence of Hopf bifurcation points with characteristic root method. For integro-differential equations with infinite delay, they are transformed into ordinary differential equations prior to their investigation.
·考虑到随机因素和延迟对动力系统的影响，及混沌系统本身所具有的一些特殊性质，设计出一种控制混沌系统的随机延迟方法，这种方法是以研究系统的Lyapunov指数为出发点，利用白噪声将混沌系统控制到系统的平衡点，在本文中是控制到原点(对于其它平衡点，可以利用线性变化将其转化为目标态是原点的情形)．同时讨论了延迟为零的特殊情形．
·利用延迟反馈方法将一类超混沌控制到它的周期态，即讨论了带延迟的超混沌系统的Hopf分岔．我们知道，随着系统的维数和延迟个数的增加，分析系统的Hopf分岔的难度就会加大．但是对于系统的每个状态取相同的延迟可能达不到控制的效果．本文利用多延迟对超L(u|¨)系统进行控制，同时指出了利用含有相同的延迟个数的延迟反馈方法不能将超Lorenz系统和超Chen系统控制到它们的周期态．
·在上面的研究的启发下，讨论了两种延迟积分微分方程的Hopf分岔问题．到目前为止，关于延迟积分微分方程的研究主要是集中在数值解的稳定性分析方面，对系统Hopf分岔的研究还不是很多．在讨论延迟微分系统的动力学行为的时候，最关键的问题就是其特征方程的根的分布问题，本文借助数值方法，利用边界点法给出Hopf分岔点的可能取值，再利用仿真确定系统的Hopf分岔点．本文是将一般的延迟积分微分方程和带无限延迟的积分微分方程利用不同的方法分开进行讨论．对于一般的延迟积分微分方程，首先利用敏感性分析来确定系统最合适的Hopf参数，然后利用特征根方法讨论Hopf分岔点的存在性．对于带无限延迟的积分微分方程，我们是将其转化为所熟悉的常微分方程来进行讨论的．
Chaos controlling has shown broad application potentials, hence leading to interests in their methodology development. This thesis investigates control methodologies for chaotic systems, along with the stochastic delay method to control a class of chaotic systems. It is found in this study that delay plays an important role in dynamical system. A dynamical system with delay realistically reveals the characteristics of the nature of certain processes. The thesis further investigates chaotic control problems and the Hopf bifurcation problem in several kinds of difference systems, proposes the relevant theory and methodologies, and presents the results from their corresponding numerical simulations. The primary contributions from this thesis are itemized as follows
·It presents a stochastic delay methodology for controlling chaotic systems, based on realistic effects of stochastic factors and delay on dynamic systems, and the corresponding characteristics of chaotic system. This method, based on the Lyapunov exponent, utilizes white noise to control chaotic systems to reach their equilibrium. The equilibrium refers to the original point in the scenarios discussed in this study. For equilibriums other than the original point, linear transformation is applied to transform objective states into their corresponding original states. A special case, i.e. delay is equal to zero is also discussed.
·It presents a methodology, based on delay feedback to control a class of hyper-chaotic systems to their periodic states; that is, the Hopf bifurcation analysis of hyperchaotic systems with delays is investigated. It is known that an increase of system dimensions and delays complicates the Hopf bifurcation analysis. Nevertheless a uniform delay for each state affects the system control adversely. The approach presented in this study deploys multiple delays to control hyperchaotic Lüsystems. It is further revealed that the feedback method based on the same number of delays cannot be applied to controlling hyperchaotic Lorenz system or hyperchaotic Chen systems to reach to their periodic states.
·This study analyzes Hopf bifurcation of two kinds of integro-differential delay equations. At present, the research on integro-differential delay equations is primarily focused on the stability analysis of numerical solutions, and the Hopf bifurcation is a topic that has yet to be thoroughly studied. When analyzing dynamical behaviors of difference delay systems, the key is to identify the distribution of the roots of system's characteristic equation. This study introduces a systematic approach, using numerical analysis and based on boundary points method, to identify the values of Hopf bifurcation points. The method for constant delay integro-differential equations varies from that for integro-differential equations with infinite delay. For constant delay integro-differential equations, the sensitivity analysis is used to identify the most suitable Hopf parameter, followed by identifying the existence of Hopf bifurcation points with characteristic root method. For integro-differential equations with infinite delay, they are transformed into ordinary differential equations prior to their investigation.
引文
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[2] Devancy R L. An introduction to chaotic dynamical systems. Massachusetts: Addison-Wesley, 1989.
[3] Marotto F R. Snap-back repellers imply chaos in R~n. J. Math Anal. Appl., 1978, 63: 199-223.
[4] Lorenz E N. Deterministic non-periodic flows. J. Atoms. Sci., 1963, 20:130-141.
[5] Huber A, Luscher E. Resonant stimulation and control of nonlinear oscillators. Naturforsch., 1989, 76(1): 76-81.
[6] Otte E, Grebogi C, Yorke J A. Controlling chaos. Phys. Rev. Lett., 1990, 64:1196-1199.
[7] Pecora L M, Carrol T L. Synchronization in chaotic systems. Phys. Rev. Lett., 1990,64(8): 821-824.
[8] Ditto W L, Rauseo S N, Spano M L. Experimental control of chaos. Phys. Rew. Lett..1990. 65:3211-3214.
[9] Andrievskii B R, Fradkov A L. Control of chaos: methods and applications. I. methods. Automat. Rem. Contr.. 2003, 64(5): 673-713.
[10] Andrievskii B R, Fradkov A L. Control of chaos: methods and applications. Ⅱ. application. Automat. Rem. Contr.. 2004, 65(4): 505-533.
[11] Petrov V, Peng B, Showalter K. A map-based algorithm for controlling low dimensional chaos. J. Chem. Phys.. 1992, 96:7506-7513.
[12] Hunt E R. Stabilizing high-period orbits in a chaotic system-the diode resonator.Phys. Rev. Lett., 1991, 67:1953-1955.
[13] Chen G. Dong X. On feedback control of chaotic continuous-time systems. IEEE. T.Circ. Syst., 1993, 40:591-601.
[14] Huberman B A, Lumer E. Dynamics of adaptive systems. IEEE. T. Circ. Syst., 1990,37(4): 547-550.
[15] Sinha S, Ramaswamy R, Rao J S. Adaptive control in nonlinear dynamics. Physica D, 1990, 43: 118-128.
[16] Guemez J, Matias M A. Control of chaos in unidimensional maps, Phys. Lett. A., 1993, 181:29-32.
[17] Matias M A, Guemez J. Stabilization of chaos by proportional pulses in the system variables. Phys. Rev. Lett., 1994, 72:1455-1462.
[18] Matias M A, Guemez J. Chaos suppression in flows using proportional pulses in the system variables. Phys. Rev. E., 1996, 54:198-209.
[19] Gutierrez J M, Iglesias A, Guemez J, et al. Suppression of chaos through changes in the system variables through Poincare and Lorenz return Inaps. Int. J. Bifurcat. Chaos, 1996, 6: 1351-1362.
[20] Pyragas K, Tamasevicius A. Experimental control of chaos by delayed self controlling feedback. Phys. Lett. A., 1993, 180:99-102.
[21] Pyragas K. Continuous control of chaos by self-controlling feedback. Phys. Lett. A., 1992, 170(9): 421-428.
[22] 杨忠华．非线性分歧：理论和计算．北京：科学出版社，2007．
[23] Hassard B D, kazarinoff N D, Wan Y H. Theory and Application of Hopf Bifuration. Cambridge: Cambridge Univ. Press, 1981.
[24] Hale J K. Theory of functional differential equations. Applied mathematical sciences. Berlin: Springer-Verlag, 1997.
[25] Luzyanina T. Roose D. Numerical stability analysis and computation of Hopf bifurcation points for delay differential equations. J. Comput. Appl. Math., 1996, 72:379-392.
[26] Hout K I, Lubich C. Periodic orbits of delay differential equations under discretization. BIT, 1998, 38(1): 72-91.
[27] Koto T. Naimark-sacker bifurcation in the Euler method for a delay differential equations. BIT, 1998, 39(1): 110-115.
[28] Ford N J, Wulf V. The use of boundary locus plots in the identification of bifurcation points in numerical approximation of delay differential equations. J. Comput. Appl. Math., 1999, 111:153-162.
[29] Wulf V, Ford N J. Numerical Hopf bifurcation for a class of delay differential equations. J. Comput. Appl. Math., 2000, 115:601-616.
[30] 张春蕊．延迟微分方程的数值Hopf分支分析：[博士毕业论文]．哈尔滨：哈尔滨工业大学，2007．
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