Quantum Ergodicity and the Dynamics of Complexity.

First, the existence of chaotic behavior in quantum mechanical systems is addressed. It is proved that, unlike their classical counterparts, the global behavior of quantum systems subject to forces that vary periodically in time is not chaotic. Instead, it is recurrent and characterized by discrete power spectra in contrast to the continuous, broadband spectra of classical chaotic behavior. Computer simulations are used to illustrate this result as well as extend it to more general time dependences. Further topics include recurrence time estimates, the issue of resonances and the classical limit. Second, the dynamical behavior of weakly coupled nonlinear oscillators is studied by considering two-dimensional difference equations. Both analytic and numerical methods are used to characterize the resulting dynamics. In addition to the common periodic and chaotic behavior seen in the one-dimensional case, the coupling can produce an unlocked, quasiperiodic phase in which the basic frequencies are irrationally related. The effects of noise are also considered. Finally, the behavior of adaptive parallel computing structures is discussed. Treated as dynamic physical systems with many coupled degrees of freedom, quantitative techniques of measuring their properties have been developed. These properties include self-organization as the system adapts to a sequence of inputs, and subsequent recognition of these inputs even when slightly distorted. Furthermore, these structures can recover from errors because the computation is distributed and there exist stable attractive fixed points in their behavior.