几类非线性动力系统的Hopf分岔研究

发布时间：2018-07-27 13:37

【摘要】：Hopf分岔是一类重要的动态分岔,Hopf分岔控制作为一个前沿研究课题,极具挑战性。本文研究几类非线性动力系统的Hopf分岔以及相关分析和控制问题,进而丰富和完善分岔的理论结果。讨论系统平衡点的动力学行为,给出系统产生Hopf分岔的条件,分析系统产生分岔的特性,并设计分岔控制器,提出控制方法,使系统产生所期望的动力学行为。重点研究系统Hopf分岔极限环振幅的控制,给出幅控关系,能够较为准确地预测幅值,实现了系统的Hopf分岔延迟控制和稳定性控制。设计了几种控制策略,每种控制方法都有其各自的特点,都能达到预期的控制目标。选取了几类典型的非线性动力系统作为例子进行讨论。首先综述了非线性控制理论、分岔控制、Hopf分岔控制以及混沌控制等的研究现状。介绍了非线性动力系统Hopf分岔的一些基本概念和分类,给出Hopf分岔定理和几种分岔控制方法,并引入几个常用的稳定性理论以及动力系统理论,为本文的研究作准备。利用一种改进的多尺度法求出了广义Van der Pol型强非线性振动系统的极限环振幅表达式。构造了几类线性、非线性反馈控制器,获得了其反馈系数与极限环振幅的近似解析关系。通过选择适当的反馈系数,可对极限环的振幅进行控制,讨论并比较了不同控制器的控制效果。数值模拟的结果验证了幅值预测的正确性与控制的有效性,且对较大的参数?,仍具有很高的精确度。讨论一类具有多个未知参数的混沌Van der Pol-Duffing系统的分岔与控制。利用Routh-Hurwitz判据分析了平衡点的稳定性,得到Hopf分岔的参数临界值。利用中心流形定理及规范型理论给出了分岔解的稳定性指标。在不改变分岔解的稳定性下,设计Washout filter线性控制器用于改变分岔值。在不改变分岔值的情形下,设计Washout filter非线性控制器用于控制系统极限环幅值。利用中心流形定理和规范型理论所得到的极限环振幅与控制增益之间的近似解析关系具有较高的精确度,预测可靠。数值模拟的结果验证了理论分析的正确性、控制的有效性以及幅值预测的可靠性。通过数值模拟出的最大李雅普诺夫指数图显示了一个新的混沌系统存在混沌吸引子。利用特征方程给出了系统产生Hopf分岔的条件;通过详细计算,得到系统的第一李雅普诺夫系数,由此来分析所产生的分岔解的稳定性。结果表明,新混沌系统的两个平衡点都能够发生非退化的超临界Hopf分岔,因此在平衡点能够分岔出稳定的周期解。数值模拟的结果与理论推导一致。讨论了Lü系统平衡点的非线性动力学性质,利用Routh-Hurwitz判据分析了平衡点的稳定性,得到Hopf分岔的参数临界值。利用中心流形定理及规范型理论给出了分岔解的稳定性指标。分别设计线性、非线性控制器从理论上实现了Hopf分岔的延迟控制及稳定性控制。数值模拟的结果进一步验证了理论分析的正确性与可行性。研究一个改进的超混沌Lü系统的Hopf分岔控制。提出一个状态反馈联合参数控制的混合控制策略,该控制策略不仅保持了原系统的平衡点结构不变,也没有增加原系统的维数。通过选择合适的控制参数,实现了系统Hopf分岔的延迟控制。通过规范型理论,分岔解的稳定性指标也进一步被求得。最后,给出两组参数进行数值模拟,验证了该控制策略的有效性。高维非线性系统的分岔控制比低维系统更加复杂,而本文提出的方法简单有效,因此,该方法对高维非线性系统的分岔控制是非常有意义的。
[Abstract]:Hopf bifurcation is a kind of important dynamic bifurcation, and Hopf bifurcation control is a challenging topic. This paper studies the Hopf bifurcation and related analysis and control problems of several nonlinear dynamical systems, and then enriches and improves the theoretical results of the bifurcation, discusses the dynamic behavior of the system's balance point, and gives the Hopf score of the system. The characteristic of the bifurcation is analyzed. The bifurcation controller is produced and the bifurcation controller is designed. The control method is proposed to make the system produce the desired dynamic behavior. The control of the amplitude of the Hopf bifurcation limit loop is studied. The amplitude control relation is given. The amplitude is predicted more accurately, and the system's Hopf bifurcation delay control and stability control are realized. Several control strategies are designed. Each control method has its own characteristics and can reach the desired control goal. Several typical nonlinear dynamic systems are selected as examples. Firstly, the research status of nonlinear control theory, bifurcation control, Hopf bifurcation control and chaos control are summarized. The nonlinear dynamics are introduced. Some basic concepts and classifications of Hopf bifurcation of force system are given, the Hopf bifurcation theorem and several bifurcation control methods are given, and several commonly used stability theories and dynamic system theories are introduced to prepare the study of this paper. The limit cycle amplitude table of the generalized Van der Pol type strong nonlinear vibration system is obtained by an improved multiscale method. A number of linear and nonlinear feedback controllers are constructed and the approximate analytic relationship between the feedback coefficient and the amplitude of the limit cycle is obtained. By selecting appropriate feedback coefficients, the amplitude of the limit cycle can be controlled and the control effect of the different controllers is discussed and compared. The results of the numerical simulation verify the correctness and control of the amplitude prediction. The effectiveness of the system and the larger parameters still have high accuracy. The bifurcation and control of a class of chaotic Van der Pol-Duffing systems with multiple unknown parameters are discussed. The stability of the equilibrium point is analyzed by using the Routh-Hurwitz criterion and the critical value of the parameter of the Hopf bifurcation is obtained. The central manifold theorem and the standard type theory are used. The stability index of the bifurcation solution. Without changing the stability of the bifurcation solution, the Washout filter linear controller is designed to change the bifurcation value. Without changing the bifurcation value, the Washout filter nonlinear controller is designed to control the limit cycle amplitude of the system. The limit cycle amplitude obtained by the central manifold and the canonical theory is used. The approximate analytical relationship between the control gain and the control gain has high accuracy and reliable prediction. The results of the numerical simulation verify the correctness of the theoretical analysis, the effectiveness of the control and the reliability of the amplitude prediction. The chaotic attractor in a new chaotic system has been shown by the maximum Li Yap Andrianof exponent of the numerical simulation. The characteristic equation gives the condition of the Hopf bifurcation of the system. By the detailed calculation, the first Lyapunov coefficient of the system is obtained, and the stability of the bifurcation solution is analyzed. The results show that the two equilibrium points of the new chaotic system can have a non degenerate supercritical Hopf bifurcation, so that the bifurcation can be bifurcated at the equilibrium point. The results of the numerical simulation are in agreement with the theoretical derivation. The nonlinear dynamic properties of the equilibrium point of the L u system are discussed. The stability of the equilibrium point is analyzed by the Routh-Hurwitz criterion and the critical value of the parameter of the Hopf bifurcation is obtained. The nonlinear controller realizes the delay control and stability control of the Hopf bifurcation theoretically. The results of the numerical simulation further verify the correctness and feasibility of the theoretical analysis. The Hopf bifurcation control of an improved hyperchaotic L u system is studied. A hybrid control strategy of state feedback joint parameter control is proposed, and the control strategy is proposed. It not only keeps the balance point structure of the original system, but also does not increase the dimension of the original system. By selecting the appropriate control parameters, the delay control of the Hopf bifurcation is realized. Through the standard theory, the stability index of the bifurcation solution is further obtained. Finally, two sets of parameters are given for numerical simulation, and the control strategy is verified. The bifurcation control of the high dimensional nonlinear system is more complex than the low dimensional system, and the method proposed in this paper is simple and effective. Therefore, this method is very meaningful for the bifurcation control of a high dimensional nonlinear system.
【学位授予单位】：湖南大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O19

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