星号流的奇异双曲性研究

Modern differentiable dynamical systems began at the early sixties of the last century. Within three decades or so, a remarkable theory of structural stability and hyperbolicity was established in dynamical systems. While most study of dynamical systems in the west focus on discrete systems using geometrical and functional analysis methods, Liao Shantao in China focuses on flows directly. He proposed two fundamental notions of "standard systems of differential equations" and "obstruction sets", and made a systematic study around them. His method is very powerful and fits particularly the continuous time systems, especially vector fields with singularities. Since the theory for structural stable systems has been quite mature and the structurally stable systems were realized to be less universal, the main focus of research nowadays is shifting to systems that are "beyond hyperbolicity". As the famous Lorenz-attractor shows, a highly non-trivial phenomenon in vector fields is that periodic orbits robustly accumulate on singularities, hence forcing the system to be non-hyperbolic. On the other hand,the strange attractor itself exists robustly under small perturbations. How to characterize the non-hyperbolic star flows with singularities or the robustly transitive singular sets are all related to this phenomenon. Unfortunately, the methods used for discrete systems can not always carry over to vector fields with singularites. This makes Liao's theory particularly important in the studies of singular vector fields and singular hyperbolic systems. In the present project, we will study, via Liao's theory, the singular hyperbolicity of singular star flows as well as heteroclinic and homoclinic bifurcation theory of vector fields.

微分动力系统兴起于上世纪六十年代初,是常微分方程定性理论的延伸。国外学派林立，硕果累累，侧重研究离散系统，多采用几何和泛函分析的方法。国内廖山涛先生则侧重研究连续系统，创立了典范方程组和阻碍集的方法和学派，特别在处理向量场奇点上有深刻的独到之处。上世纪末稳定性猜测解决之后，微分动力系统的研究重心转移到双曲之外的系统。对于向量场，一个典型的、高度非平凡的非双曲现象就是周期轨道逼近奇点，甚至在扰动下持续地逼近奇点。著名的持续传递问题，以及目前深受关注的非双曲有奇星号流的特征刻画问题，其症结都与此相关。这类问题中奇点的特殊地位决定了无法用离散系统的方法处理，从而凸显了倚重廖山涛方法的必要性。本项目将运用廖理论和向量场的同宿、异宿分岔理论方法，研究有奇星号流的奇异双曲性问题。

微分动力系统兴起于上世纪六十年代初, 是常微分方程定性理论的延伸。上世纪末结构稳定性猜测解决之后，微分动力系统的研究重心转移到双曲之外的系统。对于向量场，一个典型的、高度非平凡的非双曲现象就是周期轨道逼近奇点，甚至在扰动下持续地逼近奇点。其中深受关注的有奇星号流的特征刻画问题，其症结就与此相关。这类问题中奇点的特殊地位决定了无法用离散系统的方法处理。国内廖山涛先生创立了典范方程组和阻碍集的方法和学派，特别在处理向量场奇点上有深刻的独到之处。本项目运用廖理论和向量场的同宿、异宿分岔理论方法，研究了有奇星号流的奇异双曲性问题。主要证明了对二维流形上的星号流，其链回复集分解为有限个链分支的并。并且，当系统没有非平凡的回复轨道时，这些链分支是奇异双曲的。对于三维的星号系统，证明了其只有有限个吸引子，并且每个含奇点的吸引子都是奇异双曲的。对于高维的有奇星号系统，证明了不存在倾斜翻转的同宿轨分支和Shilnikov型的同宿轨分支。这些结果对有奇星号流的特征做了部分刻画，有助于理解一致双曲之外的动力系统的复杂性特征。