扭转映射定理是KAM理论的一种具体形式, 早在1962年, Moser就对具有周期性小扰动的平面扭转映射的不变曲线存在性进行了研究, 并得到了肯定的结果,这一结果可以用以研究许多非线性常微分方程解的存在性和有界性.随后人们开始考虑, 在扰动函数为拟周期函数的情形下, 是否具有类似的不变曲线存在定理. 在对扭转映射做出不同合理假设的前提下,例如假设其满足所谓的``恰当辛条件"或``时逆条件", 这一问题也已经得到了肯定的答案.最近一些学者开始考虑扰动函数为概周期函数的情形, 即一种具有可数无穷多个频率的, 具有某些回复性质的函数, 并得到了一些肯定的结果.受到这些工作的启发, 本文研究了当概周期扰动函数具有一种快速收敛的傅里叶级数表示时, 不变曲线的存在性. 加以一些其他合理的假设时,本文对于不变曲线的存在性, 在这种情形下同样得到了肯定的结果, 并对上述条件中傅里叶级数的收敛速度做出了较为量化的描述.作为应用本文还利用不变曲线的存在性, 研究了概周期摆型方程, 得到了存在无穷多个概周期解以及所有解有界的充分必要条件.
本文内容分为四章. 第一章介绍了所研究课题的相关背景及本文的主要结果. 第二章又分为三个部分: 首先, 我们列出了一些关于概周期函数的重要定义和性质; 随后, 我们证明了本文的一些主要的引理; 最后, 利用这些引理我们得到了不变曲线的存在性定理, 即本文的主定理.在第三章, 作为应用, 我们研究了概周期摆型方程所有解的有界性和无穷多个概周期解的存在性. 最后我们对本文的工作进行了总结, 并思考了不足之处和对下一步工作的展望.

The twist mapping theorem is a specific form for the KAM theory. In 1962, Moser had studied the existence of invariant curves of planar twist mappings with periodic perturbations and obtained the existence when perturbations are small enough.This result can be applied to the research of existence and boundedness of solutions for many nonlinear ordinary differential equations. When the perturbations are quasi-periodic, whether there exists a similar invariant curve theorem, this question had been considered by many scholars later. Under some different but reasonable conditions for the twist mapping, for example, the ``exact symplectic condition" or ``reversible condition", rich results had been obtained. Recently, some scholars had also considered this problem when the perturbations are almost periodic, generally speaking, functions possessing some kind of recurrent properties with countable infinite frequencies. Inspired by predecessors' works, this paper considers the invariant curve theorem when the almost periodic perturbations admit fast convergent Fourier series. Together with some reasonable assumptions, the existence of invariant curves is proved. As an application, we study the sufficient and necessary conditions that all solutions of the almost periodic pendulum-type equation are bounded and there exist infinitely many almost periodic solutions.
The content of this thesis is divided into three chapters. In Chapter one we introduce the related background of the research and the main results. Chapter two is divided into three parts: First, we list some important definitions and properties which are used later.
Then, some useful lemmas are proved. Finally, the invariant curve theorem which is the main theorem is obtained. In Chapter three, as an application, almost periodic pendulum-type equations are studied. In the last Chapter four, we give a summarization of this thesis, and discuss the limitations and the expectations for future work.