Strange Attractors: Description and Visualization

这个项目有两个组成部分。一个是面向当地高中生的外展，另一个是科学外展，涉及对德雷克塞尔物理学本科生和研究生以及非线性动力学领域有益的研究。外展部分：PI将发展制作奇怪吸引子的三维模型的能力，并将这些模型用于向高中生演示，以激发他们对科学的兴趣。这些模型将在现场生成并留在现场。将强调在年轻领域(非线性动力学和混沌)工作的兴奋，以及通过施加周期边界条件创建用整数量子数描述的奇怪吸引子家族与量子力学的联系，以及非线性动力学的一些基本工具与弦理论中存在的类似工具的联系。科学组成部分：分析混沌数据的第一步是将数据拓扑嵌入到合适的维度空间中。在三维空间中，成功的嵌入打开了使用一些最近开发的强大工具来确定吸引子的拓扑结构的可能性。现在可以理解，吸引子的拓扑可以依赖于嵌入，但由此揭示的产生混沌行为的机制与嵌入无关。通过量子数来区分由不同嵌入产生的吸引子，这些量子数保证满足某些周期边界条件。这些整数与吸引子的分解有关，使用的工具与弦理论中的工具类似。这些吸引子也可以通过某些新引入的、容易计算的真实度量的值来识别。其中许多结果对维度大于3的奇异吸引子是有效的。它被计划为：a.为Embeddingsb创建一个拓扑测试。确定区分微分同胚但拓扑不等价吸引子c的量子数的谱。描述奇怪吸引子d的嵌入和量子化之间的对偶性。找出非线性动力学和弦理论之间的有用联系。

This project has two components. One involves outreach to local area high school students and the other is scientific, involving research that benefits Drexel physics undergraduate and graduate students and the field of Nonlinear Dynamics.Outreach component: The PI will develop the capability of making three dimensional models of strange attractors and use these models in presentations to high school students to stimulate an interest in the sciences. These models will be generated on site and left at site. The excitement of working in a young field (nonlinear dynamics and chaos) will be emphasized, as well as the connection with quantum mechanics through the imposition of periodic boundary conditions to create families of strange attractors described by integer quantum numbers, and the connection of some fundamental tools of nonlinear dynamics with similar tools that exist in string theory.Scientific component: The very first step in the analysis of chaotic data is the topological embedding of the data in a space of appropriate dimension. In three dimensions a successful embedding opens up the possibility of determining the topological structure of the attractor using a number of powerful recently developed tools. It is now understood that the topology of the attractor can depend on the embedding, but that the mechanism for generating chaotic behavior so revealed is independent of the embedding. Attractors created by different embeddings are distinguished by quantumnumbers which guarantee that certain periodic boundary conditions are satisfied. The integers are related to a decomposition of the attractor using tools similar to those found in String Theory. These attractors can also be identified by the values of certain newly introduced, easily computed, real measures. Many of these results are valid for strange attractors of dimension greater than three. It is plannned to:a. create a topological test for embeddingsb. determine the spectrum of quantum numbers that distinguish diffeomorphic buttopologically inequivalent attractorsc. describe the duality between embeddings and quantization of strange attractorsd. identify useful connections between Nonlinear Dynamics and String Theory.