Strange Attractors: Description and Visualization

该项目有两个组成部分。一个涉及到当地高中生的推广，另一个是科学的，涉及到有利于德雷克塞尔物理本科生和研究生以及非线性动力学领域的研究。推广部分：PI将培养制作奇怪吸引子的三维模型的能力，并将这些模型用于高中生的演讲中，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。在一个年轻的领域工作的兴奋（非线性动力学和混沌），以及通过施加周期性边界条件来创建由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。科学部分：混沌数据分析的第一步是将数据拓扑嵌入到适当维数的空间中。在三维空间中，一个成功的嵌入开辟了使用一些强大的最近开发的工具来确定吸引子的拓扑结构的可能性。现在可以理解的是，吸引子的拓扑结构可以依赖于嵌入，但产生混沌行为的机制是独立的嵌入。由不同嵌入产生的吸引子由量子数区分，量子数保证满足一定的周期性边界条件。整数与吸引子的分解有关，使用的工具类似于弦论中的工具。这些吸引子也可以通过某些新引入的、容易计算的、真实的测度的值来识别。许多这些结果是有效的奇怪吸引子的维数大于3。计划：A。为embeddingsb创建拓扑测试。确定量子数的频谱，区分双纯拓扑不等价吸引子。描述奇异吸引子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.