The bouncing ball simulation system is a program for the Apple Macintosh computer that provides a physically accurate rendering of the motions of a ball impacting with a sinusoidally vibrating table. The program adheres to all the Macintosh interface guidelines, thus making simulations easy to run and examine. The program accompanies the text book An Experimental Approach to Nonlinear Dynamics and Chaos by Nicholas B. Tufillaro, Tyler Abbott, and Jerermiah Reilly (Addison-Wesley, 1992), and it is an excellent tool for learning about the behavior of chaotic systems. Chapter one of the book provides a good introduction to the basic periodic and chaotic dynamics of the bouncing ball system. The book also comes with an extensive User’s Manual. In addition to its pedagogical value, the system is also of great practical interest in several engineering applications, as well as in basic research.
The bouncing ball simulation system allows the user to see and hear (long experimental sounds file mp3) the periodic and chaotic impacts between the ball and table. The impacts can be mapped to various melodic and harmonic schemes, giving rise to a kind of chaotic music, or just noise. The motions can be viewed and analyzed by a multitude of methods. Here several windows on the ball’s motion are shown ranging from a simple plot of the trajectory of the ball and table (upper left), an animation of the ball’s motion (lower left), to the more abstract Poincare maps (right) clearly indicating the balls chaotic nature. Numerical data (upper right) can also be saved from a simulation for further analysis.
A closer examination of the strange attractor is possible by examining the Poincare (Impact) Map window in more detail:
or, a window on the phase space dynamics:
In addition to single simulations, the user can also investigate how the dynamics changes as the parameters are varied by constructing a bifurcation diagram.
The bifurcation diagram above clearly shows that the bouncing ball system can approach a chaotic motion by the now classic period doubling route to chaos. Indeed, as described in the book mentioned above, it is quite easy to set up an experimental apparatus (using a loud-speaker, a glass lens, a function generator, and a ball bearing) in which students can measure Feigenbaum’s delta for themselves.
Here are two pictures from the first bouncing ball experiment I set up in the basement of the Physics building at Bryn Mawr College (circa 1985).
Colorful and informative pictures are also provided by a basin of attraction diagram which shows the complex (fractal) dependence of the asymptotic solutions (multiple attractors) on the initial conditions.
A demonstration copy (or an updated version for registered users) of the program for evaluation by teachers who are considering using the text and software in the classroom is available by electronic transfer from the first author. Internet: email@example.com
The most recent research paper on the bouncing ball is here.
Here are some additional pictures of the experiment and experimental results showing the strange attactor using the electronics to construct a peak detector map: