The Application Of KAM Theory In Differential Equations

In this paper, we mainly study the application of KAM theory in Lotka-Volterra. system and Ginzburg-Landau equation. Firstly, we prove the existence of positive quasi-periodic solutions in a Cantor family for the3-dimensional Lotka-Volterra system then prove the stability of the solutions based on the above conclusion with the help of Lya-punov function. Secondly, it is shown that there exist3-dimensional invariant tori for1D complex Ginzburg-Landau equation in a rigorous analytic way. This paper is divided into three parts.In chapter one, we mainly introduce the historical background and our main work in this paper.In chapter two, it is shown there exist positive quasi-periodic solutions in a Cantor family for the3-dimensional Lotka-Volterra system by the KAM theory and Newton iteration, and then shown the stability of the solutions based on the above conclusion with the help of Lyapunov function.In chapter three, we prove that there exist3-dimensional invariant tori for1D com-plex Ginzburg-Landau equation in a rigorous analytic way, which is based on degenerate infinite-dimensional KAM theory and normal form technique.