This project is concerned with singularly perturbed systems of differential equations with turning points and their applications. The goal is to establish a comprehensive geometric singular perturbation theory for such systems. Singularly perturbed problems typically involve multiple-scale features that result in non-uniform behavior. The presence of turning points causes a loss of stability and creates complications in the dynamical structure. The investigator will use a dynamical-systems approach to systematically investigate turning-point behavior. The dynamical behavior depends heavily on the topological structure of the set of turning points and its relation to the vector field. The investigator will classify the main structures and apply analytical and geometric tools of modern dynamical systems theory to their study.
Multiple time and space scales make real-world systems rich and exciting, and understanding real-life problems is the driving force for the development of mathematical theory. This project focuses on the study of problems arising from many areas of science and engineering--such as fluid dynamics, population dynamics, neural networks, and biochemical processes--and involving multiple time and space scales. The theoretical study will identify critical parameters responsible for the complicated structure of such systems and examine mathematical forms of interactions in the processes. As an important component of this proposal, two specific fields of applications will be examined: (1) relaxation oscillations in biological systems such as predator-prey models and epidemic disease models, and (2) engineered biochemical process of wastewater treatment. These systems involve vastly different scales, both in space and in time, and demonstrate complicated behaviors. This project, if successful, will significantly improve our understanding of physical phenomena with multiple scales and provide insight for better engineering designs for control purposes. The proposed activity will also greatly enhance education and training programs in this important area for students and non-experts because of the intuitive formulation of the approach as well as the results.
