若干高维强非线性振动系统的高精度解析近似解研究

许多实际工程问题都可用高维强非线性振动系统来描述，它们的多解共存、拟周期解和极小（大）极限环等研究是目前国际上非线性动力学领域的前沿课题之一，受到相关人员的广泛关注。由于利用高维强非线性系统可以更为准确地模拟工程问题，因此研究这类系统的动力学行为具有重要的理论意义和工程实用价值。本项目拟通过理论分析和高精度数值技术相结合的方法来研究高维强非线性振动系统，希望能够对实际工程问题中这类系统的设计和振动控制有一定的指导意义，主要内容将以同伦分析方法为基本工具，结合多尺度和重整化群等方法，对强非线性振动系统的多解、拟周期解、极小（大）极限环及其稳定性问题进行研究，促进非线性振动理论在实际工程中的应用。

High-dimensional strongly nonlinear oscillation systems can be used to model many engineering phenomena. Their relative problems, such as co-existing solutions, quasi-periodic solutions and limit cycles, are all hot topics in the field of nonlinear dynamics. Specifically, high-dimensional strongly nonlinear systems can simulate practical engineering problems more accurately. Therefore, the investigation of the dynamic behavior of such systems has theoretical significance and high practical value. In this project, we will conduct theoretical analysis and high-precision numerical simulation to the high-dimensional strongly nonlinear oscillation systems. In detail, the homotopy analysis method (HAM) is used as a basic method, and other techniques such as multi-scale and renormalization group are combined with it to obtain the multiple solutions, quasi-periodic solutions, limit cycles and analyze the stability of this systems. Our work is expected to shed some light on the practical application of the nonlinear oscillation theory in engineering field.