The investigators propose to continue their research on singular perturbations, systems of conservation laws and related systems not in conservation form, and homoclinic and heteroclinic bifurcation phenomena. These areas are related: sharp fronts in singular perturbation problems, and shock waves in systems of conservation laws and related systems not in conservation form, can be viewed as traveling waves, which correspond to heteroclinic (or sometimes homoclinic) solutions of an associated equation. The proposed research falls into four areas: (1) systems of conservation laws (failure of strict hyperbolicity, geometric singular perturbation theory and the Dafermos regularization); (2) shock solutions of initial-value problems for systems not in conservation form; (3) chaos in a perturbed Rayleigh-Benard convection model that models the dynamo mechanism responsible for the earth's magnetic field; (4) other applied problems involving singular perturbation or homoclinic bifurcation (forced Chua's circuit, multiple-stage cancer models, plasma and sheath models).
Sharp wave fronts occur in many areas of science. At the first level of approximation, they are simple traveling waves that keep their shape and move at a constant speed. Mathematically, traveling waves correspond to "heteroclinic solutions of ordinary differential equations." These can be thought of as moving particles that start near an unstable equilibrium and move toward another unstable equilibrium. Cycles of such solutions often give rise to chaos. The investigators propose to continue their research in these related areas. For example, they will study certain unusual shock waves that arise in model equations for oil recovery. They will study shock waves for systems "not in conservation form," which arise in fluid mechanics and elasticity problems when the original systems are simplified to make them more tractable. They will study chaos in a model for the earth's magnetic field, which is believed to be related to irregular reversals of the magnetic field that have occurred in the past. They will try to improve the "plasma-sheath" models used to model various industrial situations, such as fluorescent light fixtures; the solutions include a sharp front.
