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

9971760迈斯首席研究员建议研究在多维（三维或多维）保体和辛映射中混沌的破坏和结构（如稳定轨道和环面）的同步创建。分析和计算技术都将被采用。一种方法是基于奥布里在1992年提出的“反可积”（AI）极限。这一原理为混沌动力学的存在提供了解析界，同时也为周期轨道族的延拓提供了一种有效的数值方法。通过外推，我们也可以推导出准周期轨道和异斜轨道。混沌的毁灭是由系统中的第一个分岔标志的，而秩序的创造是由分岔创造的稳定的轨道和环面。第二个项目是对异斜轨道及其分支进行分类，用于多维版本的二次Henon图。分类将通过构造“初级交叉流形”，并通过确定它们在“基本环空”上的同调性来给出。动力系统结构的拓扑分类在数据分析和数值实验解释中都是重要的。众所周知，混沌系统通常具有具有各种拓扑性质的分形不变集。本文将通过紧集的“不连通性”和“离散性”来研究这些问题。PI将开发计算同调的技术，给出“空隙性”的定义和这些集合的分辨率依赖近似的贝蒂数的计算方法。

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.