This project supports mathematical research on problems arising in the study of dynamical systems and differential equations. The equations are functional differential equations of retarded type. In certain equations occurring in biological models, the retardation of memory effect is a natural feature. Work will focus on four main topics. The first concerns monotone flows on monotone dynamical systems. Efforts to modify standard systems theory to present a unified treatment of differential equations for which the set of equilibrium points contains all (phase space) constant functions will be made. The second topic relates to instability of physical systems. Applications of invariance principles of finite delay equations will be made to obtain instability results for finite and infinite delay functional differential equations. Instability is characterized by the property that all solutions passing near an unstable point must leave every neighborhood of the point. Neutral functional differential equations make up the third element of this project. Work will be done in establishing comparison-convergence techniques for approximating solutions. The final goal of the research is that of using monotone iteration techniques to study periodic boundary value problems for functional differential equations. Both the existence of periodic solutions and techniques for constructing them will be considered.
