Mathematical Sciences: Homoclinic and Heteroclinic Bifurcations: Theory and Applications

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9404070 Deng The traditional role of the applied mathematician is to take a real system and find out what its properties are. However, the properties of the specific system cannot be discovered unless one knows what the possibilities are, and the possibilities are often revealed only by the general abstract theory. Practice and theory progress best hand-in-hand. Three projects are proposed in this proposal. They all deal with basic mechanisms by which complex and chaotic structures arise in real systems. The theoretical results are useful in giving better understanding to experimental and numerical results from a system of chemical reaction, insulin secretion in pancreatic beta cells, and electrical pulses along nerve axon. They are also important in guiding future experiments and practical applications of these systems. Three projects are proposed. The first project concerns a type of degenerate heteroclinic loop bifurcation with application to the existence of pulsing waves in a two-phase flow through a packed column. The second project concerns the orbit-flip homoclinic bifurcation in the Rinzel-Terman model for the electrical activity of pancreatic beta cells. The bifurcation is related to the transition from continuous spiking to intermittent spiking of the membrane potential. The third project concerns strategy and numerical technique for locating bifurcation values of a diffusion parameter in a system of bistable reaction diffusion equations. These values are such so that a heteroclinic loop corresponding to traveling front and back waves changes its twist type, which often signals the generation of chaotic dynamics.