Multi-Frequency Oscillations and Related Perturbation Problems

提案#0204119PI：英飞一所：佐治亚理工学院题目：多频振荡及相关摄动问题本项目研究规则或奇异扰动的物理系统中的多频振荡，包括准周期、准周期和几乎自同构振荡。准周期振荡主要研究起源于流体力学、固体物理和生物学的哈密顿系统和广义哈密顿系统。本文将研究准周期强迫振动系统和弱耦合准周期振子网络的概周期和几乎自同构振动。其目的是更好地理解由几个频率相互作用产生的复杂动力学以及发生的间歇现象，特别是当频率接近共振时。对于奇异摄动的振动系统，多频振荡的研究将集中在松弛或快型振动，特别关注生态、生物医学和通信系统。这个项目的结果将在某些生物、电气和机械振荡器的分析、计算和设计中具有重要的应用。本项目涉及呈现多相振荡的生物、电气和机械系统中的动力学数学模型。多相振荡是物理系统中普遍存在的现象，涉及的自由度多、参数多、振荡频率多。它们的研究对于理解自然界中的复杂现象，如流体中的湍流、通讯中的噪声信号、DNA链的运输和传染病的传播动力学具有重要意义。该项目的成果也将对机电系统和设备的实际工程设计产生重大影响，特别是对有用的设计参数的识别和奇异振荡行为的预测。

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.