Nonlinear systems are everywhere, from the grandfather clock in your living room to the electrical circuits powering your computer to the motion of the cosmos. Under careful analysis, the response of even simple systems is much more complicated than it often appears. Because of this complexity, the study of chaotic systems appears overwhelming. We are interested in putting some order back into chaos and our work centers on developing techniques to predict the behavior of chaotic systems. The research involves comparing simulated chaotic behavior with experimental results. We are searching for invariant measures in system dynamics; one might describe them as "finger prints" of the physics. These measures can involve spatial information (e.g. fractal dimensions in phase space) as well as temporal information (e.g. Lyapunov exponents of diverging trajectories).
Precision Chaotic Pendulum:
We have developed a chaotic pendulum with well understood mechanical characteristics (torque, friction, damping, inertia). Simulations of chaotic behavior using parameters measured from the apparatus exhibit a high degree of agreement with our experimental results and serve as an excellent test bed for developing invariant measures.
Building on our pendulum we have added a magnetic potential to the system by placing a magnet on the pendulum and having it interact with an external magnetic field. The interaction potential includes both magnetic and gravitational terms and can be varied to study a variety of potential functions. The variable potentials provide additional tests for our invariant measures.
By controlling our external magnetic field with a chaotic circuit we are developing a system that studies the synchronization of electro-mechanical chaos. Synchronization studies are typically conducted on systems with much faster response times and the slow nature of the mechanical response presents special challenges.
Sonoluminescence, the transformation of sound into light is a poorly understood, highly nonlinear process. Bubbles are trapped by ultrasonic waves in a liquid. Under the right conditions, the bubble collapse and produce a very short and intense pulse of light. Most research in the area focuses on the spectrum of the resulting light, but we are taking a slightly different path and looking for chaotic structures in the timing of the light following the bubbles collapse.
This research depends on the talented work of undergraduate research assistants and support from a variety of sources including: Minnesota Space Grant, NSF and the University of St. Thomas.