The amount of controversy concerning the concept of ``quantum chaos'' is exceeded only by the amount of interest in potential applications, which range from prediction of the conductivity and response of mesoscopic quantum nanostructures to modeling the excited state spectra of complex nuclei. We are currently focusing on one aspect of quantum chaos which, although not yet fully understood, is very intriguing the difference in statistics of energy levels for chaotic and integrable systems with the dual aims of understanding the relationships among chaos, integrability, and statistics and of using this understanding to interpret and predict transport properties of actual mesoscopic electronic devices.

Iterated nonlinear maps of the unit interval form perhaps the simplest, most instructive, and most extensively studied class on chaotic dynamical systems, as well as an important set of simplified models for applications such as population dynamics. The sequence of sudden changes bifurcations in the behavior of a map as the strength of the nonlinearity is varied is one of the most interesting aspects of the dynamics. Starting from the quadratic ``logistic map,'' we have studied the extent to which this sequence can be reproduced by piecewise linear approximations to the nonlinear map.

The group does various research projects in nonlinear dynamics. The research projects include (1) the formation and dynamics of spatial patterns, (2) Hamiltonian systems with few degrees of freedom, (3) dynamic cell models, and (4) almost periodic and quasiperiodic systems.

I work on research problems related to classical and quantum chaos. In particular, I wish to understand the transition from a quantum system to a classical system where the classical system is chaotic. The project includes the studies of (1) quantum dynamics around KAM tori, (2) semiclassical approximation and trace formulas, (3) classical and quantum billiard systems, and (4) renormalization group transformations for a classical field theory.

The Center for Complex Systems Research is an interdisciplinary group of faculty and students involved in research on complex dynamic processes in a variety of scientific fields. Current studies include: adaptive controls of time-varying systems; dynamics of amplitude equations and weak turbulence; turbulence experiments; quantum relationships with classical chaos; constructing equations of motion from data; forecasting high-dimensional chaotic systems; principles of organization and morphogenesis in organisms; chemical oscillations and chaotic dynamics; coupled cellular systems and neural networks; measurements of evolutionary activity.

Since 1890 there has been a series of changes in our technical knowledge and sources of information in science. These changes are based on mathematical discoveries, on new opportunities generated by digital computers, and on new experimental opportunities. These three bases have fundamentally changed the variety of possible ``scientific methods'' that can be used to validate our understanding of physical phenomena. Moreover, these discoveries have shown that the microreduction/synthesis philosophy of this century has no scientific basis, and that new unify- ing principles for all science needs to be established. This research involves exploring this ongoing evolution of science.

Complex systems often have a large number of dynamic attractors with very different behaviors. The problem of obtaining a mathematical model that can describe the dynamics of such systems is a fundamental challenge in science. A new method of open-plus-closed-loop interactions on general systems of ordinary differential equations developed by Jackson and Grosu, with general entrainment capabilities, is being applied to transfer systems between chaotic attractors, sometimes only using prerecorded data, and will be used in a fundamental resonant-modeling technique to explore for the most accurate global-dynamic model of such systems.

The dynamic-attractor characteristics of small neural networks (modules) are being investigated for neural dynamics with realistic features. These features include absolute and relative refractory periods, arbitrary excitatory and inhibitory connections satisfying Dale's law, variable thresholds, and noisy connectivity influences. The response of these networks to selected input signals and how the multiple-attractor dynamics might relate to information processing and short-term memory are being investigated.

We are studying the question of the existence of chaotic behavior in quantum mechanical systems whose classical analogs are known to be nonintegrable and exhibit chaotic behavior. The system that we use is the interaction of low-frequency, high-power microwave radiation with one-dimensional hydrogen atoms. These atoms are prepared by laser excitation of atomic hydrogen in the presence of strong dc electric fields.