| Citation: | ZHANG Wei, YAO Minghui, ZHANG Junhua, LI Shangbao. Study of global bifurcations and chaotic dynamics for high-dimensional nonlinear systems[J]. Advances in Mechanics, 2013, 43(1): 63-90. doi: 10.6052/1000-0992-12-053
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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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