The Existence Of Quasi-Periodic Solutions Of Appropriately Degenerate Systems

In this paper, we mainly apply the finite dimensional KAM theory to prove that there exists a quasi-periodic solution for a class of pendulum equation with quasi-periodic terms, and there exists quasi-periodic solutions for the van der Pol-Mathieu-Duffing equation with quasi-periodic terms and a FHE term. Meanwhile, we discuss the existence of quasi-periodic solutions for a three-dimensional quasi-periodic system with hyperbolic-type degenerate equilibrium point. This paper is divided into four chapters and an appendix.In chapter 1, we mainly introduce the historical background and significance of the pendulum equation, van der Pol-Mathieu-Duffing equation and KAM the-ory. Meanwhile, we give the main results and significance of this paper.In chapter 2, by the finite dimensional KAM theory, it is proven that the existence of quasi-periodic solutions for pendulum equation with quasi-periodic terms. Firstly, we outline the background and the research status of the pendulum model, then we give a pendulum equation which will be considered in this paper, and we also show four KAM theorems. Firstly, we provide an iterative lemma. Then we prove the main results by the iterative lemma.In chapter 3, we discuss the existence of quasi-periodic solutions for van der Pol-Mathieu-Duffing equation with quasi-periodic terms and a FHE term. We first summarize the background of the van der Pol-Mathieu-Duffing equation and give a van der Pol-Mathieu-Duffing equation which will be considered in this paper, and we also show a KAM theorem. It is proven that there exists a quasi-periodic solution in 4-dimensional phase space for the above equation. In order to discuss the existence of the quasi-periodic solution in 4-dimensional phase space, we need to make a series of transformations such as translations, stretches, polar coordinates, re-scale, and so on, then the above equation is transformed into a normal-form. By KAM theorems, it is proven that there exists a quasi-periodic solution for the normal-form. Furthermore, we obtain that there is a quasi-periodic solution for the original equation.In chapter 4, we discuss the existence of quasi-periodic solutions for a three-dimensional quasi-periodic system with hyperbolic-type degenerate equilibrium point under small perturbation. For non-degenerate systems, we usually consider the linear part as the main terms to reduce the system. However, since the appearance of zero characteristic roots, we can not directly consider the linear part as the main terms to reduce degenerate systems. We can consider a high order term as the main item and consider the low order terms as the perturbed terms. By KAM theorems, we obtain that there is a quasi-periodic solution for the degenerate systems.In the appendix, we list some lemmas which are used in this paper.