Boundary Layer Phenomena and Periodic Solutions for Functional Differential Equations

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ABSTRACT DMS-9706891 Nussbaum Nussbaum proposes research to gain insight into the solutions of nonlinear functional differential equations or "FDE's". Functional differential equations are equations in which the derivative of a function x(t) is a specified function of the history of x up to time t. An example which plays a central role in this proposal is ax'(t)=f(x(t),x(t-r)), r=r(x(t)). Here f and r are specified functions, and a is a positive real. Under natural assumptions on f and r and for a small, there exists a slowly oscillating periodic solution x(t), dependent on a. Nussbaum will seek to describe precisely the limiting shape of the graph of this solution x(t) as a approaches zero. Nonlinear functional differential equations (sometimes called differential-delay equations) have been suggested as models in a variety of areas, particularly in biology. For example, A. Longtin and J. Milton propose such equations in studying the human eye's pupil-light reflex; and M. Mackey and J. Milton are led to consider nonlinear FDE's in their studies of oscillations in the human body's red blood cell population. Considered from a pure mathematician's viewpoint, the equations which arise are highly nontrivial. In this proposal, Nussbaum studies classes of nonlinear FDE's which are, in part, motivated by examples from biology; and we seek to understand the behaviour of solutions of such equations.