Multi-Frequency Oscillations and Applications in Communications Systems

本计画主要研究几乎周期性受迫振子及自激系统之多频振荡。要考虑的主要问题是由几个频率的相互作用，特别是当这些频率接近共振的复杂动力学的性质。 通过研究诸如湍流的平均运动和准周期分叉等问题，研究者希望解决这些系统中几乎自守振荡所起的重要作用。本计画之研究成果将应用于晶体振荡器之电路设计。本计画致力于研究呈现多相振荡之物理系统之数学模型（例如：机械装置和电路）或经受季节变化的系统（例如，生物振荡器和人口模型）。由于这些物理系统的复杂性，涉及到大的维度，多参数和许多振荡频率，（借助于数值计算）对于指导物理系统的实际设计，对于预测有价值的设计参数和解释新的振荡现象等将是极其重要的。该项目的结果将对通信系统的各种应用中出现的晶体振荡器的电路设计产生重大影响。

This project concerns the study of multi-frequency oscillations arising in almost periodically forced oscillators and self-excited systems. Main problems to be considered are the nature of complicated dynamics resulting from the interactions of several frequencies especially when these frequencies are close to resonance. By studying issues such as mean motions of toral flows and quasi-periodic bifurcations, the investigator would like to address the important role played by almost automorphic oscillations in these systems. The results of this project will have significant applications to electric circuit designs of crystal oscillators.This project is devoted to the study of mathematical models of physical systems that present multi-phase oscillations (e.g., mechanical devices and electric circuits) or systems that are subjected to seasonal variations (e.g., biological oscillators and population models). Due to the complexity of these physical systems involving large dimension, multi-parameters and many oscillating frequencies, qualitative analysis (with help of numerical computations) will be extremely important to guide practical designs of the physical systems with respect to the prediction of valuable design parameters and the explanation of new oscillatory phenomena etc. In particular, the results of this project will be of a great impact on electric circuit designs of crystal oscillators arising in a large variety of applications of communication systems.