Strange Attractors: Description and Visualization

Drexel University, Philadelphia PA

Investigators

Abstract

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 quantum numbers 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 embeddings b. determine the spectrum of quantum numbers that distinguish diffeomorphic but topologically inequivalent attractors c. describe the duality between embeddings and quantization of strange attractors d. identify useful connections between Nonlinear Dynamics and String Theory.