高维非线性系统次谐Melnikov理论的研究及应用

The mathematical models can be described by the high-dimensional nonlinear systems in many engineering systems. Research on periodic solutions of high-dimensional nonlinear dynamical systems and its applications is a hot issue in the field of nonlinear dynamics. The main object of this project aims to improve and extend the subharmonic Melnikov theory. Because the current four dimensional subharmonic Melnikov theory only can be used to study the period solutions of several systems with specific forms, the four dimensional subharmonic Melnikov theory will be improved by means of periodic transformation and Poincaré map to investigate the complex nonlinear systems. Then, the four dimensional subharmonic Melnikov theory will be extended to study the six dimensional system, the theorems for the existence and saddle node bifurcation of periodic solutions will be given. The internal connections of the theorems under different dimensions of the systems will be mainly studied. The different types of periodic solutions and its parameter ranges of the honeycomb sandwich plate will be investigated by using the extended subharmonic Melnikov method. The research results of this project not only can promote the development of the theoretical study on the periodic solution for the high-dimensional nonlinear dynamical system, but also can accelerate the applications of the nonlinear dynamics theory.

机械系统中许多问题的数学模型往往用高维非线性动力系统来描述，高维非线性动力系统周期解理论的研究及应用是非线性动力学领域的一个热点问题。本项目的主要研究内容是改进和推广次谐Melnikov理论。针对目前的四维次谐Melnikov理论只能够研究特定形式系统周期解的问题，我们利用周期变换和Poincaré映射，对次谐Melnikov理论进行改进，使其能够用于研究复杂的四维非线性系统。然后，将四维次谐Melnikov理论推广到六维非线性动力系统中，给出六维次谐Melnikov向量函数的精确表达式，获得六维非线性系统周期解的存在条件及鞍结分叉定理，探讨系统在不同维数下定理之间的内在联系。利用推广的次谐Melnikov理论研究蜂窝夹层板的周期解，确定系统产生不同类型周期解的参数域。本项目的研究成果不仅能够加强高维非线性系统次谐Melnikov理论的研究，还可以促进非线性动力学理论在实际问题中的应用。

周期运动广泛存在于机械系统、电力系统和生态系统等各类系统中，这些系统对应的数学模型通常为高维非线性系统，因此研究高维非线性动力系统的周期解具有十分重要的意义。本项目研究高维非线性系统周期解理论及其应用，研究成果包括推广和完善了四维、六维次谐Melnikov理论，探讨了系统在小参数扰动下产生孤立周期解的机理，获得了高维非线性系统周期解的存在条件及鞍结分叉定理；研究了复杂载荷作用下的四边简支矩形薄板、蜂窝夹层板产生两倍周期运动的参数域；研究了点阵夹芯结构在主共振、亚谐共振、内共振等条件下的动力学行为，分析了结构参数对分叉、混沌运动等各种非线性动力学行为的影响。本课题在国内外重要学术期刊发表和录用论文9篇，其中SCI论文7篇。