Multi-Frequency Oscillations and Related Perturbation Problems

项目编号：0204119项目负责人：易英飞项目单位：格鲁吉亚项目名称：多频振荡及相关扰动问题摘要本项目主要研究物理系统在规则或奇异扰动下的多频振荡，包括准周期、准周期和准自守振荡。 拟周期振荡将主要研究起源于流体力学、固体物理和生物学的哈密顿和广义哈密顿系统。 拟周期受迫振动系统和弱耦合拟周期振子网络的概周期和概自守振动将被研究。 我们的目标是更好地理解复杂的动力学所造成的相互作用的几个频率和不稳定现象，特别是当频率接近共振。 对于奇摄动振荡系统，多频振荡的研究将集中在松弛或快速类型，特别关注生态，生物医学和通信系统。该项目的结果将对某些生物、电和机械振荡器的分析、计算和设计有重要的应用。该项目涉及生物、电和机械系统中出现的多相振荡的动力学数学模型。 多相振荡是物理系统中的普遍现象，涉及大量的自由度，多参数和许多振荡频率。 它们的研究对于理解自然界中的复杂现象非常重要，例如流体中的湍流，电信中的噪声信号，DNA链的运输以及传染病的传播动力学。 该项目的结果也将对机械和电气系统和设备的实际工程设计产生很大影响，特别是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.