Destruction of Chaos and Detection of Order in Multi-dimensional Dynamical Systems

9971760 Meiss主要研究者提出研究混沌的破坏和多维（三维或多维）保体积和辛映射中结构（如稳定轨道和环面）的同时创建。将采用分析和计算技术。一种方法是基于Aubry在1992年提出的“反可积”（AI）极限。这一原则产生的混沌动力学的存在的分析界限，以及一个有效的数值技术的延续家庭的周期轨道。通过外推，人们也可以遵循准周期和异宿轨道。混沌的破坏是由系统中的第一个分叉和创建秩序的分叉，创建稳定的轨道和环面的信号。第二个项目是分类异宿轨道和他们的分歧的多维版本的二次Henon地图。分类将通过“主要交叉流形”的建设，并通过确定他们的“基本环上的同调。”“动力系统结构的拓扑分类在数据分析和数值实验的解释中都很重要。众所周知，混沌系统往往具有分形不变集，这些分形不变集具有不同的拓扑性质。本提案将通过紧集的“不连通性”和“离散性”来研究这些问题。PI将开发计算同源性的技术，产生一个定义为“空隙度”和计算方法的贝蒂数的分辨率依赖近似这些集。

9971760MeissThe principal investigator proposes to study the destruction of chaos and the concurrent creation of structures (such as stable orbits and tori) in multi-dimensional (three or more dimensions) volume-preserving and symplectic maps. Both analytical and computational techniques will be employed. One approach is based on the "anti-integrable" (AI) limit, introduced by Aubry in 1992. This principle yields analytical bounds for the existence of chaotic dynamics, as well as an efficient numerical technique for continuation of families of periodic orbits. By extrapolation one can also follow quasiperiodic and heteroclinic orbits as well. The destruction of chaos is signaled by the first bifurcations in the system and the creation of order by the bifurcations that create stable orbits and tori. A second project is to classify the heteroclinic orbits and their bifurcations for a multi-dimensional version of the quadratic Henon map. Classification will be given through construction of "primary intersection manifolds," and by determining their homology on the "fundamental annuli." A topological classification of structures in dynamical systems is important both in the analysis of data, and in the interpretation of numerical experiments. It is well known that chaotic systems often have fractal invariant sets with various topological properties. These will be studied in this proposal through the "disconnectedness" and "discreteness" of compact sets. The PI will develop techniques for computational homology, yielding a definition for "lacunarity" and computational methods for Betti numbers of resolution dependent approximations to these sets.