Mathematical Sciences: Asymptotic and Singular Perturbation Methods for Bifurcation Problems with Applications

9308009 埃尔讷 研究者研究了一系列与奇异摄动和分岔问题有关的数学问题。 这些问题的动机是在科学和工程中的具体应用。主要的应用出现在激光不稳定性和数学生物学领域。激光问题集中在B类激光器的典型振荡响应上。该项目开发了新的渐近方法来描述时间周期解及其分叉。生物学问题集中在（1）在化学和生物化学系统中观察到的爆发振荡和（2）弱耦合细胞系统中传播失败的条件。 许多激光器是不稳定的装置，表现出脉动响应。 这些激光器包括诸如半导体或固态激光器的实用激光器。为了控制这些不稳定性，研究人员在数学上分析了特定的激光模型。传播故障（电压或传导阻滞）是神经传导中的已知问题，具有重要的医疗后果。目前，对破坏机理的研究主要是数值模拟。研究者详细分析了这些模型，以获得简单的失败条件。 ***

9308009 Erneux The investigator studies a series of mathematical problems connected with singular perturbation and bifurcation problems. These problems are motivated by specific applications in science and engineering. The principal applications appear in the area of laser instabilities and in mathematical biology. The laser problems concentrate on oscillatory responses typical to class B lasers. The project develops new asymptotic methods to describe time-periodic solutions and their bifurcations. The biological problems concentrate on (1) the bursting oscillations observed in chemical and biochemical systems and (2) the conditions for propagation failure in a system of weakly coupled cells. Many lasers are unstable devices that exhibit a pulsating response. These lasers include practical lasers such as semiconductor or solid state lasers. In order to control these instabilities, the investigator analyzes mathematically specific laser models. Propagation failure (voltage or conduction block) is a known problem in nerve conduction that has important medical consequences. Most studies on the mechanisms leading to failure have been numerical simulations. The investigator analyzes these models in detail to obtain simple conditions for failure. ***