Strange Attractors: Description and Visualization

这个项目有两个组成部分。其中一个项目涉及到当地的高中生，另一个项目是科学的，涉及到有利于德雷克塞尔大学物理系本科生和研究生以及非线性动力学领域的研究。拓展部分：PI将开发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学生演示，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。将强调在一个年轻的领域（非线性动力学和混沌）工作的兴奋，以及通过强加周期边界条件来创建由整数量子数描述的奇异吸引子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。科学成分：混沌数据分析的第一步是在适当维数的空间中对数据进行拓扑嵌入。在三维空间中，一个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拓扑结构的可能性。现在我们知道吸引子的拓扑结构可以依赖于嵌入，但是产生混沌行为的机制是独立于嵌入的。由不同嵌入产生的吸引子用量子数来区分，保证满足一定的周期边界条件。整数是用类似于弦理论的工具来分解吸引子的。这些吸引子也可以通过某些新引入的、易于计算的实际度量值来识别。这些结果中的许多对于大于3维的奇异吸引子是有效的。规划如下：a。为embeddingsb创建拓扑测试。确定区分微分同构拓扑不等价吸引子的量子数谱。描述奇异吸引子的嵌入和量化之间的对偶性。识别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.