Study of global bifurcations and chaotic dynamics for high-dimensional nonlinear systems
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摘要:
综述了Melnikov方法的发展历史, 从1963年苏联学者Melnikov提出该方法开始, 一直到目前广义Melnikov方法的提出和发展. Melnikov方法的发展历程可以概括为3 个阶段, 分别综述了每一个阶段Melnikov方法的扩展和应用, 论述了国内外在该方向的研究现状和所获得的主要结果, 指出了各种Melnikov方法之间的联系、存在的问题和不足. 为了对比两种研究高维非线性系统多脉冲混沌动力学的理论, 本文综述了另外一种全局摄动理论, 即能量相位法, 总结了该方法十几年来的发展历史以及国内外的理论研究成果和工程应用实例, 阐述了能量相位法发展的根源以及与Melnikov方法的内在联系, 比较了能量相位法和广义Melnikov方法两种理论研究对象的差别, 以及各自所存在的不足和问题. 简要论述了能量相位法和广义Melnikov方法的理论体系, 并利用广义Melnikov方法分析了四边简支矩形薄板的多脉冲混沌动力学, 数值模拟进一步验证了理论研究的结果. 最后, 详细综述了两种理论的缺点和不足, 说明今后全局摄动理论的发展方向.
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关键词:
- 高维非线性系统/
- 全局分岔/
- 混沌动力学/
- 能量相位法/
- 广义Melnikov方法
Abstract:In this paper, the history of the Melnikov theory is summarized. In 1963, the classical Melnikov method was presented by Melnikov, a Russian scientist. Until now, the Melnikov theory has been extended and developed. The development of the Melnikov method is divided into three historical periods. The extension and application of Melnikov theory are respectively summed up in each historical period, in which the situation of study and main domestic and abroad results in this research field are enumerated. The relationships, problems and deficiencies are pointed out for a variety of Melnikov theories. In addition, another global perturbation method, i.e., energy phase theory, is set forth in order to compare with two theories which are normally used to investigate multi-pulse chaotic motion in the high-dimensional nonlinear systems. The brief history, the theory and the research achievements and engineering applications of the energy phase theory are elucidated. The origin of the energy phase theory and its inherent relations with the Melnikov theory are illustrated. The subject investigated in the energy phase method is contrast with that in the extended Melnikov method to find the difference between them. Disadvantages and open problems are indicated for both the energy phase method and the extended Melnikov method. Furthermore, theoretical frames of these two methods are stated briefly. The multi-pulse chaotic dynamics for a rectangular thin plate, simply supported at the fore-edge, is analyzed by using both of them. Numerical simulation further verifies the analytical prediction. Finally, deficiencies of these two theories are described in detail. The future development direction of the global perturbation theory is demonstrated too.
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