This research encompasses three separate lines of study: a) Asymptotics beyond all orders; b) Kinetic theory and resonant triads, and c) Models for motion of curves and surfaces in space. The results have potential for important applications in nonlinear dynamics and applications of the KAM theory. The KAM theorem asserts that under appropriate circumstances most of the trajectories of the system remain regular at the onset of chaos; only a few trajectories actually become irregular, or chaotic. So far there is no available theory that can identify which trajectories become chaotic under small perturbations. If successful, the work proposed here will enable scientists to work with dynamical systems containing both chaotic and regular trajectories. This issue is important in many fields, one of which is the design of plasma fusion devices like tokomaks.
