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

[308009]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。这些问题是由科学和工程中的特定应用引起的。主要应用在激光不稳定性和数学生物学领域。激光问题集中在典型的B类激光器的振荡响应上。本课题发展了描述时间周期解及其分岔的新渐近方法。生物学问题集中在(1)在化学和生化系统中观察到的爆发振荡和(2)弱耦合细胞系统中繁殖失败的条件。许多激光器是表现出脉动响应的不稳定装置。这些激光器包括实用激光器，如半导体或固态激光器。为了控制这些不稳定性，研究者分析了数学上特定的激光模型。传播失败（电压或传导阻滞）是已知的神经传导问题，具有重要的医学后果。大多数关于破坏机制的研究都是数值模拟。研究者详细分析了这些模型，以获得简单的失效条件。＊＊＊

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. ***