SIR Physics Investigation Abstract
ERGODIC PROPERTIES OF NONLINEAR BILLIARDS
Presenters:
Timothy Credo, Illinois Mathematics and Science Academy, 1500 West Sullivan Road, Aurora, IL, 60506; tfcredo@imsa.edu
William Hahm, Illinois Mathematics and Science Academy, 1500 West Sullivan Road, Aurora, IL, 60506; whahm@imsa.edu
Mentor:
Dr. Yau Wah, University of Chicago, High Energy Physics Department, 5640 South Ellis Avenue, Chicago, IL, 60632; 312-702-7592; 312-702-1914; wah@uchepd.uchicago.edu
Abstract:
The effects of nonlinear behavior reach across science, from astronomy to economics, cardiology to quantum dynamics. Yet despite the tremendous variety of nonlinear or chaotic systems, some properties of their behavior appear to be universal. Our research investigates these properties in a two-dimensional billiard system, a closed planar region in which a particle propagates according to Newtonian laws. While ergodic effects have been thoroughly explored for one-dimensional mapping procedures, their role in higher dimensions is still unclear. By writing programs using Matlab software, we were able to explore the chaotic properties of the billiard system both graphically and analytically. Using techniques like Poincaré sections and Lyapunov exponents enabled us to examine the transition between chaos and classical behavior despite the challenges posed by a multidimensional phase space. Ultimately, our work hopes to verify the universality of scaling in chaotic systems with the calculation of a scaling constant for the fractal structures present in higher dimensions.