An impact oscillator is a system that has some forcing potential driving a particle to repeatedly impact a barrier. The impact oscillator has long been an interesting system of study for both engineers and physicists. For engineers it acts as a nice model of systems such as earthquake scenarios , articulated mooring towers , and engine rattling . Fermi investigated the same system when dealing with cosmic rays impacting the atmosphere . This system is also ideal for studying the dynamics of a seemingly chaotic system. It’s conceptually easy to grasp, yet there is still much to learn about it. One particular generalization of the impact oscillator is to consider a particle bouncing on a two dimensional, infinitely long, infinitely dense table with some known periodic curvature. The particle is being pulled along at some velocity, v, and the forcing potential is due to a combination of a vertical and horizontal spring acting on the particle. The ball has some coefficient of restitution that does not conserve energy at the impact. There is also a kinetic friction term that will resist the motion of the ball on the surface. One particular physical interpretation would be an atomic force microscope being drug along a surface at some velocity v.
The goal of this project is to study the dynamics of this system. It is believed that at certain velocities, the particle will bounce chaotically along the surface. This project will include an investigation of what velocities the chaotic bouncing will occur, both theoretically and numerically by using tools such as bifurcation diagrams with the varied parameter being velocity. This system will be recast into a billiard system and will then readily admit a 3-dimensional Jacobian and also giving a 3-dimensional Poincare’ map. This project will also include an investigation of the system using periodic orbit theory. A few additional things that we will play with is using a Lorentz map of the maximum height the ball reaches relative to the surface verses the previous maximum height. We will also investigate using the technique of the PIM-triple method . The ultimate, and therefore ideal, goal of this project is then to understand this system well enough to make a prediction of the coefficient of kinetic friction just by knowing when the ball begins to bounce chaotically.  J. M. T. Thompson and R. Ghaffari. Phys. Lett. A 91 5 (1982)  J.M.T. Thompson, A.R. Bokaian & R. Ghaffari. J. Energy Resources Technology (Trans ASME), 106, 191-198 (1984).  S. W. Shaw and P. J. Holmes. Phys Rev Lett 51 623 (1983)  E. Fermi. Phys. Rev. 75, 1169 (1949)  H. E. Nusse and J. A. Yorke. Physica D 36, 137 (1989).