Bouncing Bifurcations: A comparison of the bifurcation and chaotic behaviors between two dynamical systems
- Michael Frantz, Chair, Mathematics/Physics/Computer Science Department, University of La Verne
- David Chappell, Physics Program Chair and Associate Professor of Physics, University of La Verne
A bouncing ball on a sinusoidally vibrating plate is a classic physics model that has been thoroughly studied over the last few decades with the results being utilized for various applications. Similarly, a droplet bouncing on a bath of the same viscous fluid that oscillates vertically has been a popular subject of study in recent years to gain further insight into applications of fluid dynamics. In this presentation, we analyze the bouncing ball model and compare our findings with results obtained from our laboratory experiments with bouncing droplets of silicon oil to investigate whether the two systems exhibit similar bifurcation and chaotic behavior. For the bouncing ball model, we identify the equations of motion and a coefficient of restitution and construct a system of iterated equations with dimensionless parameters while employing the high-bounce approximation to simplify the analysis of the behavior of the bouncing ball. We then determine the fixed points of the system and mathematically analyze where the threshold of bifurcation occurs and when chaotic regimes materialize. In a computer simulation, we vary the values of the dimensionless parameters to plot the dynamics of the bouncing ball model and create bifurcation diagrams to aid our analysis. In our laboratory experiments, a shallow bath of silicon oil of a particular viscosity was placed on an electromagnetic shaker where it was driven by a constant frequency just below the threshold of Faraday instability. A small droplet of the same viscosity was then manually created on the surface of the vibrating bath such that the droplet was sufficient in size so as not to coalesce with the silicon bath. The bouncing behavior of the droplet was observed and recorded with the aid of a high speed camera. Computer software was utilized to process the images and plot the dynamics of the droplet.