Mathematical Sciences: Nonlinear Oscillations in Functional Differential Equations

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This project concerns the development of numerical methods designed to assist the computer-aided analysis of nonlinear oscillation problems. The investigator undertakes work on the computation of normal forms for certain infinite dimensional systems, and a study of the relationship between that and a Lyapunov-Schmidt based technique developed by the investigator for the analysis of Hopf bifurcation problems. The use of symbolic manipulation software is expected to play an important role in this local analysis. Simultaneously and complementarily, the investigator will undertake significant modifications and improvements of existing simulation and numerical curve tracking software that he has developed. The issues addressed and techniques proposed are selected to promote the analysis of problems typified by systems possessing "time-delay" effects. Such so-called functional differential equations are important in their own right, and provide a transition towards analyzing analogous questions in problems from fluid dynamics. Specific problems include models of chugging in liquid propellant fuel rockets (where the time delay corresponds to the time it takes to vaporize fuel), oscillations in coupled (Josephson Junction) semiconductors, and pattern formation in diffusive chemical reactions. Both the analytic and computational aspects of this research are motivated by the goal of making the mathematical analysis of certain dynamical systems more accessible to nonspecialists in the field.