高阶微分方程的周期解及多重性

The aim of this project is to study the existence, multiplicity and stability of higher order differential equations and the systems of differential equations. There are a lot of results about the existence, multiplicity and stability of second order differential equations. Could we give some results for higher order differential equations and the systems of differential equations for a class of nonlinearity? There are many researches for the spectrum for second order differential equations, such as Floquet theory and rotation numbers, periodic and anti-periodic eigenvalues. However, the research for the spectrum for higher order differential equations and the systems of differential equations is small. The spectrum for higher order differential equations will be studied, and then multiplicity and stability of higher order differential equations and the systems of differential equations will be studied.

本项目主要研究高阶微分方程与方程组周期解的多重性与稳定性。二阶常微分方程周期解的存在性、多重性与稳定性已有广泛研究。而有关高阶微分方程周期解的多重性与稳定性的研究成果还很少。部分原因是高阶微分方程或方程组的近似方程与二阶非线性方程的近似方程有明显区别，后者的谱分析结果丰富，如Floquet理论以及Hill方程的旋转数刻画方法，而前者的谱分析结果较少。本项目将分析一类线性化算子的谱，进而研究高阶微分方程与方程组的周期解的多重性以及稳定性。

本项目按照研究计划开展了微分方程周期解的存在性，多重性与稳定性研究。证明工具主要包含连续性方法，度理论，Hill方程的Floquet理论及周期特征值与反周期特征值的估计与分支理论的证明方法。项目针对采用p范数刻画参数函数的回复力集合，讨论Duffing方程的周期解的存在性，多重性与精确的个数分析，以及周期解的稳定性问题；与无穷范数刻画回复力集合的参数集合进行比较；在方程具有唯一稳定周期解情形，能否估计解的衰减速度，能否说明所得衰减速度的最优性；当回复力显式依赖参数时，能否分析周期解的存在性，精确个数与稳定性对参数的依赖性。主要的科学意义是讨论了Duffing方程在使用无穷范数以及p范数下的回复力集合上，周期解的相关结论，也就是说即使参数集合不属于经典的无穷范数集合时，周期解的相关结论依然可能成立，这也说明了采用不同的刻画方法刻画参数集合的差异。该方程是一个经典方程，在电子工程的设计中有重要角色。