非线性振动和波动系统的分岔及混沌研究

【作者】 彭解华；

【导师】 唐驾时； 于德介；

【作者基本信息】 湖南大学 ， 固体力学， 2003， 硕士

【摘要】 本文在全面分析和总结非线性微分动力系统分岔和混沌研究的现状的基础上，运用广大力学和振动理论研究者熟悉的数学理论对非线性微分动力系统的分岔和混沌的基础理论和应用控制进行了较为系统和深入的研究，取得了较为显著的研究成果。 全文共分七章。第一章为绪论，阐述了分岔和混沌研究的理论价值和工程实际意义，分析和总结了国内外在分岔和混沌领域研究发展历史和研究现状，简要介绍了本文研究的目的意义和主要内容。第二章研究分岔的基本理论和方法，推导了一般一维非线性微分动力系统发生鞍—结分岔、叉式分岔和跨临界分岔等基本静态分岔的条件，改进和简化了高维非线性微分动力系统的降维方法。第三章以多项式理论为基础，创建了计算普适开折的多项式计算法。第四章对非线性振动中最常见的Van der Pol-Duffing-Mathieu复合非线性振子的分岔性质进行了系统的研究，发现了该复合系统在非共振、主共振、超谐共振和非1／2阶次谐共振情况下具有相同的分岔特性。研究了在强非线性情况下该复合振子的分岔情况。第五章用正规摄动法给出了一维非线性微分方程的通解，并以此为基础解析地证明了混沌运动对初值的敏感依赖性和对扰动的敏感依赖性，并以杜芬振子为例子以验证，研究了杜芬振子的混沌运动控制的多谐扰动法。第六章总结归纳了将非线性偏微分方程转化为常微分方程的变换法，研究了一维无界区域上非线性薛定谔方程的分岔和混沌性质，解析地研究了用参数周期扰动法控制屈曲梁的混沌运动的效果。在第七章中，总结了全文的研究工作，展望了分岔、混沌研究的前景。更多还原

【Abstract】 Based on a complete summary and examination of the history and the actuality of the bifurcation and chaos research, a systematic investigation into the fundamental theory and application of the bifurcation and chaos phenomenon is made with the ordinary mathematic theory that is familiar to the researchers in the mechanics and vibration field. Some important achievements are obtained in this dissertation.The whole paper consists of seven chapters. Chapter 1 is an introduction of the dissertation. The importance of the research of the bifurcation and chaos is introduced, the advances and actuality of the research are summarized, and the main contents of the dissertation are reported as well. Chapter 2 deals with the fundamental theory and methods of bifurcation. The conditions giving rise to the static bifurcation are derived at first, then, the improved LS reduction method is studied and the centre manifold method is simplified. In chapter 3, a new method that gives the universal unfolding of the bifurcation functions is established. In chapter 4, the bifurcation of the complex nonlinear oscillator of van der Pol- Duffing-Mathieu’s is studied and the common bifurcation property of the oscillator in the situations of nonresonance, principal resonance, ultraharmonic resonance and non-subharmonic resonance is revealed. The bifurcation of strongly nonlinear oscillator is also explored in this chapter. The research work on the chaos is carried out in chapter 5. The properties of chaos sensitively dependent on the initial conditions and the perturbations are proved and a method of controlling chaos of Duffing oscillator is investigated as well. In chapter 6, the bifurcation and chaos of continuous nonlinear systems are researched. The method of change partial differential equation into ordinary differential equation is classified, the bifurcation and chaos in nonlinear Schrodinger system is detected and the effectiveness of controlling chaos of buckling beam with parametric perturbation method is studied in details. Chapter 7 is the summary and conclusion of the dissertation.更多还原