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

主要研究人员建议研究多维(三维或更多维)保体积和辛映射中混沌的破坏和结构(如稳定轨道和环面)的同时产生。将同时使用分析和计算技术。一种方法是基于Aubry在1992年引入的“反可积”(AI)极限。这一原理给出了混沌动力学存在的解析界，并为周期轨道族的连续提供了一种有效的数值方法。通过外推，人们还可以遵循准周期轨道和异宿轨道。混沌的毁灭是通过系统中的第一次分叉和通过产生稳定的轨道和环面的分叉来创造秩序来表示的。第二个项目是对二次Henon映射的多维版本的异宿轨道及其分支进行分类。分类将通过构造“主交流形”和确定它们在“基本环”上的同调来给出。动力系统中结构的拓扑分类在数据分析和数值实验的解释中都是重要的。众所周知，混沌系统往往具有具有各种拓扑性质的分形不变量集。在这个方案中，我们将通过紧集的“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“空隙”的定义和这些集合的依赖于分辨率的近似的Betti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.