Final Summary

Throughout this summer, I have gained valuable insight into mathematics research. Coming in with little knowledge of dynamical systems, I began by reading and learning all about the topic. Participating in a reading group helped me understand the material quickly and in depth, since other students and I tackled and solved practice problems together.

My project originally stated that it would study the two-dimensional henon map and use Conley Index Theory to uncover its dynamics. However, in practicing with one-dimensional maps, specifically the tent map and logistic map, I found a new approach to time series analysis by using a new technique of partitioning the set. This idea took my project in an unexpected direction. Though, the outcome of the idea should prove to be useful.

I have encountered many unanticipated problems, each of which caused me to second guess the validity of my idea. Instead of destroying my idea, these problems have made it stronger and have given me a more in depth understanding of my project. The end of my summer has been mostly spent on developing the code that implements my idea to a given time series. Once it is completed, which should be soon, I will have the ability to uncover the underlying dynamics of any one-dimensional time series using Conley Index Theory and software developed by Jesse Berwald.

I plan to continue this approach in higher dimensional time series, which present their own unique obstacles. I also plan to prove that my algorithm gives a finite representation equal to the underlying system’s outer approximation. This proof will allow me to compute finite representations without having to also compute the outer approximation each time. The reason I don’t want to compute outer approximations is that I want to uncover underlying dynamics with only the information provided by the time series. Finally, after implementing my approach in higher dimensions, I will use this technique to analyze higher dimensional biological data, such as data of red blood cells.

I am grateful to the Cummings Memorial Fund and the Charles Center for funding my research this summer. It has been a very valuable experience for me. I also am grateful to Dr. Sarah Day and Dr. Jesse Berwald, who both guided me through this project, answered all of my questions, and taught me so much math despite their busy schedules.