拟周期驱动系统在Liouville频率下响应解的存在性

我们考虑了几种拟周期驱动模型(包括高维常微分方程和非适定偏微分方程) 响应解(与驱动频率一致的拟周期解)的存在性。本项目试图在已有工作的基础上进一步探索使KAM或者不动点定理成立的较弱非退化条件(驱动频率的Liouville程度)。 . 本项目主要有三个研究课题：(i)用修正的KAM定理研究非适定Boussinesq方程在有限维 Liouville驱动频率下响应解的存在性，扩充了已有文献中针对适定方程或者驱动频率是Diophantine的结论。(ii)用不动点定理证明椭圆方程在Liouville频率下响应解的存在性，扩充了解析解的结论。(iii)用参数化方法并结合不动点定理研究双曲不变流形扰动理论中导致混沌现象的横截相交，给出新的Melnikov function表达式。拓展了平面系统的结果且新的Melnikov function具有快速的收敛性且易于验证和应用。

We consider several quasi-periodically forced models (including both multidimensional ordinary differential equations and partial differential equations, possibly ill-posed). We are interested in studying response solutions (i.e., quasi-periodic solutions with the same frequency as the forcing). Our goal in this program is to explore the conditions for the establishment of KAM or fixed point theorem, especially the weaker non-degenerate conditions (i.e. degree of Liouville for the forcing frequencies).. This project contains three subjects: (i) We study the existence of quasi-periodic solutions for the ill-posed Boussinesq equation with finite-dimensional Liouvillean forcing frequencies by the modified KAM theorem. The result obtained in this subject strengthens the existing results in the literature where the system is well-posed or the forcing frequency is assumed to be Diophantine. (ii) The existence of quasi-periodic solutions for elliptic equations with Liouvillean frequencies is proved by the fixed point theorem. The obtained results extend the existing results in the literature where the system is analytic. (iii) We study the effect of perturbation for normally hyperbolic manifolds. Of particular interest are the transversal intersections leading to chaotic behavior. Our method is based on the parameterization method and the fixed point theorem. We give the new formulas of Melnikov function. Our results extend the existing results in the literature where the system is planar and they have fast convergence and can be easily checked and used.