9501239 Haller Resonances are regions in the phase space of a dynamical system in which the frequencies of some angular variables become nearly commensurate. This proposal is aimed toward the development and application of new geometric methods for the study of complicated dynamics near resonances. The main goal is to capture invariant structures in the phase space with a strong impact on the general evolution of the system. Ideal candidates for such structures are hyperbolic partially slow manifolds that arise in many resonance problems and carry physically important motions with different time scales. A subtle combination of regular and singular perturbation techniques can describe this effect in detail by detecting remarkable families of solutions connecting neighborhoods of these manifolds. Such solutions describe repeated, complicated transitions between physically distinguished motions Many evolutionary processes in nature as solutions of sets of differential equations. Sets of these solutions can be visualized as surfaces in some higher dimensional phase space and irregular transitions between different states of the evolutionary system appear as chaotic orbits connecting these surfaces. This proposal deals with the detection of such transitions in the important and frequent case when the solution surfaces are composed of solutions with several different time scales. The proposed mathematical techniques can be used to study yet unexplained details of molecular vibrations, patterns of surface waves in ocean dynamics, gravitational interactions of planetary systems, and chaotic oscillations of engineering structures.
