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1、湖南大学 硕士学位论文 非线性动力系统的两类分岔控制与混沌控制研究 姓名：欧阳克俭 申请学位级别：硕士 专业：固体力学 指导教师：唐驾时 20070428 硕士学位论文 I 摘 要 分岔控制作为非线性科学中的前沿研究课题，极具挑战性。分岔控制的目的 是对给定的非线性动力系统设计一个控制器，用来改变系统的分岔特性，从而去 掉系统中有害的动力学行为，使之产生人们需要的动力学行为。本文在全面分析 和总结非线性动力系统分岔控制研究现状的基础上，基于非线性动力学、非线性 控制理论、分岔理论等非线性科学的现代分析方法，对倍周期分岔、Hopf 分岔等 进行控制，工作具有较大的理论意义和应用价值。研究内容如下
2、： 第一章对非线性控制理论、分岔控制的研究方法、现状和进展进行综述，介 绍本文的研究目的、研究内容和创新点。 第二章介绍动力学研究的一些基本概念，简述发生鞍结分岔、跨临界分岔、 叉形分岔的充分必要条件，以及这三种静态分岔相互转换的条件；介绍分岔控制 器设计及分析的主要方法。 第三章设计了线性和非线性的状态反馈控制器，对 Logistic 模型的倍周期分 岔进行了控制， 得到了系统在控制前和控制后的分岔图,通过设计不同的参数控制 器，改变了动力系统的分岔特性。根据实际应用目的，设计了不同的控制器改变 了存在的分岔点的参数值，并且调整了分岔链的形状。通过优化控制器可以使 Logistic 模型的分
3、岔行为满足一定的要求。 第四章设计了状态反馈控制器和 washout filter 控制器对 van der Pol-Duffing 系统的 Hopf 分岔的极限环幅值进行了控制。通过对控制方程的分析，了解了控 制参数和极限环幅值的影响情况，进而提出控制策略，设计了状态反馈控制器对 系统的 Hopf 分岔进行了控制。 第五章设计了线性反馈控制器对 Lorenz 系统的平衡点和周期轨道进行了控 制，首先利用 Routh-Hurwitz 准则对受控系统进行了稳定性分析，严格证明了达 到控制目标反馈系数的选择原则，最后通过数值计算证明了该方法能够有效地控 制混沌系统到稳定的平衡点同时也能使系统控制到
4、 1P 周期轨道，并且得到了控 制到稳定的 1P 周期轨道的控制参数的选取范围。 本文的主要创新点在于将分岔控制理论应用于非线性振动系统的研究，丰富 了非线性控制理论研究的内容，加深了分岔理论研究的深度。具体表现在：对 Logistic 模型的倍周期分岔进行了反馈控制；首次将 washout filter 技术应用于二 维 van der Pol-Duffing 系统的 Hopf 分岔控制；应用线性反馈控制成功实现了对 Lorenz 系统平衡点的混沌控制和 1P 周期轨道控制。 关键词：分岔控制；非线性动力系统；状态反馈控制；多尺度法；Hopf 分岔 非线性动力系统的两类分岔控制与混沌控制研究
5、 II Abstract As an engineering leading research field, bifurcation control has become more and more challenging. It aims at designing a controller to modify the bifurcation properties of a given nonlinear system, and achieving some desirable behaviors. Through a complete summary and examination of t
6、he history and the actuality of the bifurcation control research, a systematic investigation into the fundamental theory and application of the bifurcation control has been conducted based on the nonlinear vibration theory, the nonlinear dynamics theory. The studies have more profound theoretical si
7、gnificances and important engineering application values, which contribute to the development and application of period-doubling bifurcation control、 Hopf bifurcation et al. The main contents in this paper can be stated as follows: Chapter one outlines some study methods and recent advances about bi
8、furcation control and nonlinear theory first, and then introduces the studys aim, contents and innovations. Chapter two introduces some basic conceptions about dynamics. The occurring conditions of three elementary static bifurcations, including saddle-node bifurcation, transcrytical bifurcation and
9、 pitchfork bifurcation, and transfer condition of each other are introduced. Then some design and analysis methods about bifurcation controller are presented. In chapter three, linear controller and nonlinear controller are designed to period-doubling bifurcation of logistic model. The bifurcation m
10、aps of logistic model are achieved and the bifurcation characteristic of the dynamical system is changed by using variables parameterized controller. In order to change the parameter value of all existing bifurcation point and modify the shape of type of a bifurcation chain, we can design variables
11、controller in view of practical aims. Moreover, the bifurcation map may get more effective by using controller in an advisable way. In chapter four, state feedback controller and washout filter controller are applied in controlling the amplitude of Hopf bifurcation of van der Pol-Duffing system. The
12、 effects of controlling parameters to the amplitudes of limit cycles are gained by analyzing the controlling equations, so the control strategy is advanced. The desirable state feedback controller is designed and the Hopf bifurcation is controlled well. 硕士学位论文 III In chapter five, linear state feedb
13、ack controllers are designed to control the equilibrium points and period trajectories of Lorenz system. Firstly, we apply Routh-Hurwitz criterion analyzing the stability of the controlled system. The choice principle of feedback coefficients to attain control objective is proved strictly, then nume
14、rical simulation results is indicated that the method can effectively guide the system trajectories to equilibrium, specially, the method can direct the controlled system to 1 periodic trajectories also, we obtain the reigion of controlling parameter of 1 periodic trajectories. In this paper, the in
15、novative is that the bifurcation control theory is used to investigate the nonlinear dynamical systems, which enriches the nonlinear dynamics theory and expands the nonlinear control theory. The creative things are as follows: apply the feedback control technology to the period-doubling bifurcation
16、in logistic model; control the Hopf bifurcation of van der Pol-Duffing system by washout filter controller for the first time; mobilize linear feedback control to control the chaos of equilibrium points and 1 period trajectories of Lorenz system. Key Words：Bifurcation control; Nonlinear dynamical systems; State feedback control; Multiple scales method; Hopf bifurcation 湖湖 南南 大大 学学 学位论文原创性声明学位论文原创性声明 本人郑重声明：所呈交的论文是本人在导师的指导下独立进行研究所 取得的研究成果。除了文中特别加以标注引用的内容外，本论文不包含任 何其他个人或集体已经发表或撰写的成果作品。对本文的研究做出重
