The purpose of this project is to develop new numerical techniques for the study of periodic solutions of nonlinear differential equations with time delay. The theory of such equations has been developed in the last twenty years and applied to study of various dynamical systems in engineering, biological sciences, and economics. These equations are also called functional differential equations. One of the phenomena of practical importance and theoretical interest is the existence of oscillations, mathematically described as periodic solutions. Three sub-projects will be pursued by this investigator. First, he will translate earlier known results on the so called Hopf bifurcation (a change from non-periodic to periodic solutions due to a small change of a parameter) into new computational methods. The investigator has recently developed a technique to detect the existence of small periodic solutions which avoids the analysis of equations in the infinite dimensional state space. His method is based on the analysis of a scalar "bifurcation function", which can be computed in a very direct manner. He now proposes to extend the use of this technique to more general equations such as Volterra integral equations, functional differential equations of neutral type, and abstract functional differential equations. Second, the investigator will develop algorithms to locate numerically the values of parameters for which the Hopf bifurcation occurs. He will extend the existing methods to compute such bifurcation points by analyzing the characteristic equation of a linearized system, and by performing a numerical minimization of the norm of the left hand side. He will also use curve tracking procedures, similar to those developed for ordinary differential equations. He will also develop methods for computing the Floquet multipliers for the equations under study, by using collocation methods. Much of this work is technically difficult and requires substantial computational skill and resources. The investigator will use a Cray-2 supercomputer at the main campus of the University of Minnesota in Minneapolis. Third, the investigator will study a number of concrete examples equations arising in applications, and will test his numerical methods on these examples. The results of this work are expected to provide new interpretations for dynamical phenomena encountered in a number of other disciplines. The University of Minnesota at Duluth is a primarily undergraduate institution which has good traditions of student involvment in research. In this project students will be employed to assist with programming and computations. Their involvment in this research will have a strong positive impact on their education, especially because of various possible interpretations of the results on the grounds of applications to other sciences. The project will be funded jointly with the Air Force Office of Scientific Research.