Limiting Profiles for Solutions of Differential-Delay Equations

Abstract

This expository paper discusses the ”limiting profile” as ε approaches zero of the graph of x _ε , where x _ε is a periodic solution either of ε x′ _ε (t) = f(x _ε (t), x _ε (t - 1)) or of ε x′ _ε (t) = f(x _ε (t), x _ε (t - r)), r = r(x _ε (t)), and f and r are given functions. A variety of theorems from the literature are summarized and proofs of some of the simpler results are sketched. The paper is divided into four sections. The first section gives some background material on fixed point theory, degree theory and the fixed point index. The second section discusses the equation ε x′(t) = f(x(t), x(t - 1)) and treats the family of examples f(x, y) = -x -μ y + y ³ , μ > 1. The final two sections give a sampling of theorems concerning ε x′(t) = f(x(t), x(t - r)), r = r(x(t)), and attempt to give some indication of the techniques involved in the analysis.