Most natural phenomena are nonlinear; yet even today, theoretical analysis of physical systems is usually based on linear mathematical models or ones having small deviations from linearity. Linear models are still routinely used because they are much easier to solve than the correct nonlinear ones. Within the last two decades, however, both theoretical and experimental investigations of nonlinear phenomena have shown that often behavior which appears to be random or chaotic is actually deterministic in its orgin. Nonlinear deterministic systems under these conditions are predictable only for short times. This paradoxical situation exists because the deterministic solutions depend very sensitively on initial conditions. Such systems are said to exhibit deterministic chaos. It has also been found that several classes of systems show universal behavior at the onset of chaos. Thus, systems as diverse as a dripping faucet and a heart in ventricular fibrillation show many common features in their dynamics.
At Ohio University, the research is an interdisciplinary effort involving faculty from the Department of Physics and Astronomy and the Department of Chemistry. We are particularly interested in improving methods of recognizing and describing deterministic chaos and in studying methods for controlling chaos, using both experiments and computer simulations. Present studies include the development of methods for controlling the stability of various states in chaotic systems by applying small changes in system parameters. The techniques include the analysis of time series data, estimation of Lyapunov exponents, adaptive learning, and feedback control. Some of these techniques were applied to experiments including systems of up to 32 coupled diode resonators showing spatio-temporal chaos and an electro-chemical cell operating in a highly nonlinear regime where spontaneous oscillations and chaos are observed. Recent numerical studies include coupled map systems, models of spatially extended systems described by partial differential equations, and models of thermal pulse combustors that exhibit deterministic chaos.