高维非线性动力系统的局部分叉和周期解的理论、方法及应用的研究是动力学与控制方向非常重要的课题。高维非线性动力系统局部分叉和周期解的研究主要是应用摄动法和规范型理论计算分叉点及周期解的近似解，以此来研究动力系统的局部分叉动力学特性。共振情况下，数学解析方法主要有Hopf分叉定理，奇异性理论，平均法，规范型理论等。这些理论和方法依赖于原系统的线性化系统，在研究航空航天、力学以及机械工程类似领域的真实非线性系统的局部动力特性时遇到了相当大的困难。因此，本项目以工程实际中具有典型意义的一类高维动力系统为研究对象，考虑工程应用的几何直观性和实用性，综合利用奇异性理论和平均法理论研究此系统的局部分叉问题，并应用上述方法研究一类功能梯度材料结构在复杂载荷下的局部分叉动力学特性，主要给出系统的Hopf分叉周期解及双Hopf分叉点。研究结果丰富了高维非线性动力系统局部动力学的方法。

Studies on the theories, methods and applications of the local bifurcation and periodic solution for high-dimensional nonlinear dynamic systems become a significant topics in the field of dynamics and control. The main methods for studding the local dynamics and periodic solutions for high-dimensional nonlinear systems are to calculate the bifurcation points and the approximate solutions of periodic solutions, by using perturbation method and normal form theory, and then study the local bifurcations and dynamic characteristics. In the case of resonance, the main mathematical analytical methods include Hopf bifurcation theorem, singularity theory, average method, normal form theory, etc. These theories and methods strictly depend on the linear system of the original nonlinear system, hence the studies on the local dynamics of the real systems in the field of aerospace, mechanics and mechanical engineering will encounter great difficulties. So considering the geometric intuitiveness and practicability in the engineering application, we take a class of typical high-dimensional nonlinear dynamic systems in engineering practice as the research object in this project, systematically study the singularity theory and average method for this system. We also employ the singularity theory and average method to study the local bifurcations and and dynamic characteristics for functionally graded material structure under complex loading. Research results enrich the methods of the local dynamics for high-dimensional dynamic systems.