Mathematical Sciences: Topics in Theory and Application of Hopf Bifurcation

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This is a computational project aimed at developing scientific software for analysis of Hopf bifurcation. Work in three distinct areas within Hopf bifurcation theory is proposed. 1. Location and analysis of Hopf bifurcations in delay differential equations, and the general eigenvalue problem for these equations. 2. Numerical algorithms for continuation of periodic solutions of delay differential systems. 3. Unfolding of Hopf bifurcations for ordinary differential equations. The first project includes new methods for testing linear stability of systems of delay differential equations and will build on the existing software for the ordinary differential equations. Adaptation of this software to the case of delay differential equations presents a number of difficulties, one of them being the numerical computation of eigenvalues. The second project aims at developing computational methods for finding families of periodic solutions corresponding to various values of the parameter in the equation. Methods such as piecewise polynomial approximation with collocation at Gauss- Legendre points and a variant of LU decomposition will be used to solve the problem. The third project involves calculating higher order expansion terms of a germ fuction by possibly combining symbolic and numerical computing. All three projects have some analytic components and applications to specific equations arising in various other areas of research. One of the models included in the investigation is the Hodgkin-Huxley current-clamped nerve conduction model. These projects are intended to yield computer codes of wider applicability than the specific systems studied. This research falls into the general area of computational methods for mathematical analysis of problems described by ordinary and functional differential equations. Such equations are often used to describe dynamical phenomena in physics, engineering and biology.