Faculty Research Perspectives

Nonlinear Differential-Delay Equations: Examples, Theorems and Questions

Roger Nussbaum

Monday, March 9, 3:30 PM in Hill 705



Abstract.

Differential-delay equations are, roughly speaking, differential equations in which x'(t) is a specified function not only of x(t) but also of the prior history of the function. Rigorously speaking, in a sense we shall describe, these should be understood as "infinite dimensional dynamical systems". Deceptively simple-looking examples are provided by

1. ax'(t)=exp(x(t-1)) (Wright's equation)
2. ax'(t)= -x(t)-kx(t-r), r:=1+x(t).

In equations (1) and (2), a>0 is a constant and k>1, and one may ask, for example, about the existence and properties of periodic solutions x(t). Despite their apparent simplicity, a variety of sophisticated techniques are needed to study these examples. We shall try to give some flavor of the subject by concentrating on a few examples, mentioning some of the techniques which have been used in their study, describing some theorems and indicating some open questions.