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

多维动力系统中混沌的破坏和秩序的检测

• 批准号: 9971760
• 负责人: James Meiss
• 金额: $ 8.03万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-01 至 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971760&HistoricalAwards=false
• 关键词: Destruction; Chaos; Detection; Order; Multi

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.

9971760MeissThe主要研究者建议研究混乱的破坏，并在多维（三个或以上）体积预测和符号图中同时创建结构（例如稳定的轨道和Tori）。将采用分析技术和计算技术。一种方法是基于Aubry于1992年引入的“反积分”（AI）极限。该原理为混乱动力学的存在而产生分析界限，以及一种有效的数值技术，用于继续定期轨道的家族。通过外推也可以遵循准碘和杂斜轨道。混乱的破坏是由系统中的第一个分叉发出的，并通过产生稳定的轨道和托里的分叉创建秩序。第二个项目是将杂斜轨道及其分叉分类为二维亨逊地图的多维版本。分类将通过构建“主要交叉歧管”，并确定其在“基本环体”上的同源性。动态系统中结构的拓扑分类在数据分析以及数值实验的解释中都很重要。众所周知，混沌系统通常具有具有各种拓扑特性的分形不变套件。这些将在该提案中通过紧凑型集的“断开连接”和“离散性”进行研究。 PI将开发用于计算同源性的技术，从而为“漏洞”和betti数量的分辨率近似值的计算方法产生定义。