扭转性与轨道的有序结构

The KAM theory studies the persistence of invariant tori of nearly-integrable Hamiltonian systems. Quasi-periodic motion remains when KAM tori persist. When the KAM tori breakdown, what can be said about the quasi-periodic motion? The Aubry-Mather theory succeeded in answering this question for area-preserving twist mappings of the annulus or exact symplectic twist mappings on the cylinder. Now The Aubry-Mather theory has been generalized to high-dimensional positive definite Tonelli systems.. The KAM theory also applies to volum-preserving mappings, reversible systems, dissipative (conformally symplectic) systems, and some systems with quasi-periodic driving or potential. Our main concern is that could the quasi-periodic (or ordered) motion be preserved partially whenever the invariant tori constructed by KAM method are destroyed? The main topic of this project is to apply differential equations theory, in particular the monotone dynamical systems theory, together with variational methods and topological techniques, to study the existence of ordered orbits and their geometrical properties, the criterion for the existence of invariant circles, the relationship between topological entropy and rotation set, and other dynamical properties connected closely with the twist condition, for conformally symplectic systems (in particular dissipative twist mappings) and the Frenkel-Kontorova model with quasi-periodic potential.

KAM 理论研究近可积哈密顿系统不变环面的保持。KAM环面上的运动是拟周期运动。当KAM环面破裂后，是否会有部分拟周期运动保留？Aubry-Mather理论在环域上的保面积扭转映射（或柱面上恰当辛扭转映射）时成功回答了这一问题。Aubry-Mather理论现推广至高维正定Tonelli 系统。.KAM理论已推广至保体积映射、翻转系统、耗散系统（共形辛系统），以及一些拟周期驱动的系统。本项目我们主要研究耗散系统（或共形辛系统）和拟周期势能系统当用KAM理论构造的不变环面破裂时，是否会有部分拟周期运动（或有序运动）被保留？我们将综合运用微分方程， 特别是单调动力系统理论，变分法，以及拓扑的工具，研究共形辛系统（以耗散扭转映射为范例）和拟周期势能的Frenkel-Kontorova 模型中有序轨道的存在性及其几何性质，不变曲线的存在性判据，拓扑熵与旋转数集合的关系等与扭

本项目利用扭转性条件研究系统在拟周期驱动下有序轨道的存在性及其结构, 单参数族系统在参数变化时旋转集的变化以及动力学行为与旋转集的关系. 得到了如下结论:对于由单调回复关系诱导的高维柱面上的单调扭转映射, 包括耗散和保守情形, 它们是二维曲面上扭转映射的多自由度推广, 旋转集是闭集, 对于每一个旋转集中的元素, 存在一条有序轨道以此为旋转数. 同时, 如果有一条轨道不存在旋转数, 或存在一个稳定的无理旋转数, 则系统有正拓扑熵. 同时,我们将这些成果应用于粒子链行波的脱钉相变研究, 得到了一系列有意义的结果.