Asymptotic and Singular Perturbation Methods for Bifurcation Problems with Applications

分岔问题的渐近奇异摄动方法及其应用

• 批准号: 9973203
• 金额: $ 7.86万
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2003-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9973203&HistoricalAwards=false
• 关键词: Asymptotic; Singular; Perturbation; Methods; Bifurcation

Erneux9973203 The investigator and his colleagues study a series of newmathematical problems appearing in the area of laser stabilityproblems and in the area of biochemical oscillators. The laserproblems are singular perturbation problems that result from thedifferent time scales of the laser dynamical variables or thatappear in the study of delay-differential equations modelingsemiconductor lasers. All these problems are motivated byexperiments that are done in collaboration. The biochemicaloscillator problems are also singular perturbation problems thatappear in models of cascading enzyme reactions describing, forexample, mitosis. An important part of the research activities concentrates onlaser instabilities. Lasers are used in many applications(communication, surgery) and are part of our every-day life(laser printers, bar code reading at the supermarket, CDplayers). But these lasers are often unstable devices.Mathematical difficulties in analyzing the responses of theselasers have prevented their control. The investigator and hiscolleagues propose to use more advanced mathematical techniquesin order to solve these problems.

Erneux 9973203 研究者和他的同事们研究了激光稳定性问题和生化振荡器领域出现的一系列新的数学问题。 激光问题是由于激光动力学变量的时间尺度不同而引起的奇异摄动问题，或者是在半导体激光器的延迟微分方程研究中出现的奇异摄动问题。 所有这些问题都是由合作完成的实验引起的。 生化振荡器问题也是奇异摄动问题，出现在描述级联酶反应的模型中，例如，有丝分裂。 研究活动的一个重要部分集中在激光不稳定性。 激光被用于许多应用（通信、外科手术），并且是我们日常生活的一部分（激光打印机、超市的条形码阅读、CD播放机）。 但是这些激光器通常是不稳定的装置，在分析这些激光器的响应时，数学上的困难阻碍了它们的控制。 研究人员和他的同事们建议使用更先进的数学技术来解决这些问题。