Roger D. Nussbaum

A Survey on Functional Differential Equations

A discussion of the whole field of functional differential equations (F.D.E.'s) is, of course, not feasible. This survey will focus on certain classes of examples of nonlinear F.D.E.'s which are important from both a theoretical and a practical viewpoint. Two classes of equations which will be discussed are

x'(t) = b f(x(t), x(t-1)) (1)


x'(t) = b f(x(t), x(t-r))
r = r(x(t)),

where b > 0 is a constant and r and f are given real-valued functions. An important example of (1) is provided by the classical, much studied Wright's equation:

x'(t) = -b x(t-1) (1 + x(t)) . (3)

A simple-looking but nontrivial example of (2) is

x'(t) = -b x(t) - k b x(t-r)
r = 1 + c x(t),

where k > 1 and c > 0. We will describe some of the known results concerning (1) and (2), e.g., existence and properties of ``slowly oscillating'' periodic solutions x(t), uniqueness and nonuniqueness of periodic solutions, chaotic behaviour, boundary layer phenomena as b approaches infinity, Morse decompositions, Poincaré-Bendixson theory and real analyticity of solutions which are defined for all t.

Roger D. Nussbaum

Jan 09 1998