PhD Comprehensive Exam | Kevin Church, Toward a bifurcation theory of impulsive delay differential equations

MC 5501

Candidate

Kevin Church , Applied Mathematics, University of Waterloo

Title

Toward a bifurcation theory of impulsive delay differential equations

Abstract

Impulsive differential equations have become increasingly more relevant in applications in recent years. Optimal control, mathematical biology and chemical kinetics applications abound, this subfield of discontinuous dynamics has grown wide and has been the subject of a great deal of theoretical research. However, research into bifurcation theory of impulsive systems has been surprisingly modest. For periodic, finite-dimensional systems, the problem can be reduced to studying a particular discrete-time system, and there are many applications that utilize this correspondence to study bifurcations. However, for nonautonomous systems that are not necessarily periodic, there are very few results concerning bifurcations and most are only suitable for scalar equations of a very specific form. For systems with impulses and delays, there are to our knowledge no general results concerning bifurcations in such systems, and there are few to no analytically sound investigations of bifurcations in particular systems.

During this comprehensive exam presentation, we will motivate the need for research into bifurcation theory of impulsive delay differential equations.  We will comment on current literature and then outline several approaches we intend to pursue during the course of our research in order to develop the theory.