Nonlinear phenomena introduce new behavior that is both counterintuitive and qualitatively different from linear phenomena which had been the focus of most past plasma research. The unpredictability is manifested by complex problems unexpectedly becoming simple and vice versa, by new states of matter that appear, by new methods for controlling or engineering plasmas unexpectedly becoming possible, or by the plasma spontaneously self- organizing itself into a coherent structure. The present research concentrates on two areas of nonlinear dynamics where few results are currently available: (a) the interaction of intrinsic stochasticity with self-consistent properties; and (b) stochasticity in multi- dimensional systems. Some progress has been made on the first topic in applying the ideas to sheath heating in r.f. discharges. This has both led to new fundamental understanding and practical possibilities of improving r.f. plasma processing, to new understanding of plasma loss in electron cyclotron heated plasmas, and to the containment of electromagnetic radiation in periodic guiding systems. The present work is concentrating on investigation of multi-degree-of-freedom systems using digital phase locked loops (DPLL). A single loop has already been studied and shown how it can be designed to optimize locking ability. Interconnected systems of oscillators have many applications including achieving higher power from microwave devices by combining the output of several synchronized oscillators have also been used as models in neurophysiology. Interconnected systems of DPLLs offer a convenient way to experimentally and numerically study the phenomenology of these systems since DPLLs are probably the simplest system with locking behavior. Since digital systems are described by mappings, it is computationally very efficient to study these systems as opposed to a system of ordinary differential equations.
