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

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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. ***