Multi-Frequency Oscillations and Related Perturbation Problems

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Proposal #0204119PI: Yingfei YiInstitution: Georgia TechTitle: Multi-frequency oscillations and related perturbation problemsABSTRACTThis project concerns the study of multi-frequency oscillations, including quasi-periodic, almost-periodic and almost-automorphic oscillations, in physical systems that are either regularly or singularly perturbed. Quasi-periodic oscillations will be studied mainly for Hamiltonian and generalized Hamiltonian systems originating in fluid mechanics, solid state physics and biology. Almost-periodic and almost-automorphic oscillations will be studied for quasi-periodically forced oscillatory systems and weakly coupled networks of quasi-periodic oscillators. The goal is to better understand the complicated dynamics resulting from the interaction of several frequencies and the intermittency phenomena that occur especially when the frequencies are close to resonance. For singularly perturbed oscillatory systems, the study of multi-frequency oscillations will focus on those of either relaxation or fast type, with particular attention to ecological, bio-medical and communication systems. The results of this project will have significant applications to the analysis, computation and design of certain biological, electrical and mechanical oscillators.This project is concerned with mathematical models of dynamics arising in biological, electrical and mechanical systems that present multi-phase oscillations. Multi-phase oscillations are universal phenomena in physical systems that involve a large number of degrees of freedom, multi-parameters, and many oscillating frequencies. Their study is of great importance for the understanding of complicated phenomena in nature, such as turbulence in fluids, noisy signals in telecommunication, transport of DNA chains, and transmission dynamics of infectious diseases. The results of this project will also have a large impact on practical engineering designs of mechanical and electrical systems and devices, in particular the identification of useful design parameters and the prediction of singular oscillatory behavior.