Periodic solutions of a class of differential equations with delay
by
Anatoli F. Ivanov
Pennsylvania State University

The differential delay equation

× x   (t)=f(x(t), x(t-1))

(1)

is called symmetric if the function f is even in the first variable and odd in the second one. E.g.,

f(-x, y)=f(x, y)=-f(x, -y)   for all    (x, y) ∈ U,

where U is the entire plane or a part of it. A periodic solution x=p(t) of equation (1) is called symmetric if

p(t+w)=-p(t)   for all    t ∈ R   and some   w > 0.

Of special interest are the so-called special symmetric periodic solutions (SSPSs for brief), when w = 2. The SSPSs are slowly oscillating periodic solutions with period 4. We study the problems of existence, uniqueness, bifircation, stability/instability of SSPSs of equation (1). The analysis is based on a reduction to studies of certain corresponding properties of ordinary differential equations in the plane. We demonstrate our results and discuss open problems using the cubic equation

x'(t)=-ax(t-1)[1+bx2(t)+gx2(t-1)].

Some of the reported results are obtained jointly with P.Dormayer, B.Lani-Wayda, and H.-O.Walther.

Date received: March 31, 2007