Week One- Time Series

Most of the time, it can be difficult to find direction in mathematics research. Trying to phrase a meaningful question that could lead to meaningful answers is a great challenge in initiating research. To accomplish this step, I spent my first week of research reading various books on Chaos and time-series. I became familiar with terminology that pervades the area of interest. I found it helpful to write a list of definitions, which will be useful when I write about my research.

To learn what I need to know for my research on time-series analysis, I have been working through examples of topics pertaining to my research found in the books, Chaos: An Introduction to Dynamical Systems by Alligood, Sauer, and Yorke, Introductory Time Series with R by Cowpertwait and Metcalfe, and Nonlinear Dynamics and Chaos by Strogatz. From reading with a group of fellow researchers in biweekly meetings and from Professor Sarah Day’s lessons, I learned the three defining characteristics of Chaos. They are as follows:

1) Periodic points are dense. Which is to say, that every point is approached by periodic points very closely.

2) Sensitive dependence. In other words, the outcome is highly sensitive to the initial conditions.

3) Topological transitivity. A point in a dynamical system will move from one arbitrarily open set to any other. Therefore, the dynamical system cannot be decomposed into two disjoint sets with nonempty interiors.

I also practiced extracting a time-series from a tent function, which is defined as the following piecewise function where r is a parameter:

f(x) = { r x ,  x < 0.5

                   { r (1-x) ,  x > 0.5

I used python under the software, Canopy, to accomplish this task. Using the concept of a binary tree, I will attempt to create a symbolic representation of the time-series.

This week has been a fantastic learning experience. I am excited to learn more and begin forming concrete questions.