Slowly oscillating periodic solutions of autonomous state-dependent delay equations

Abstract

more than a dozen years since the peak of activity on the question of the existence of periodic, slowly oscillating, solutions of autonomous delay differential equations. Following the early work of Jones [l], Wright [2] and Grafton [3], the work of Nussbaum [4, 51 is to be specially noted for providing several new fixed point results and a global bifurcation theorem which are particularly useful for proving the existence of periodic solutions. Other important works include those of Hadeler and Tomiuk [6], Kaplan and Yorke [7], Chow [8], Chow and Hale [9], Alt [lo, 111 and Walther [12, 131. (See Hale’s book [14] for an overview.) To the best of our knowledge, only Nussbaum [5] and Alt [ 1 l] considered the question of the existence of periodic solutions of autonomous state-dependent delay differential equations. Nussbaum considered a special equation, (0.3) below, in [5] but did not prove a general result. Alt [ 1 l] considered more complicated, integral threshold-type, state-dependent delays which have arisen in various models in epidemiology and structured population models. Alt obtained a theorem for a general class of state-dependent equations on which we comment below. The lack of results on periodic solutions for state-dependent equations is probably explained by the fact that it is not clear what kinds of state-dependent delays are interesting or natural and on a lack of compelling examples of such equations arising from plausible mathematical models. However, some recent papers of Belair and Mackey [15] and Belair [16] contain some interesting state-dependent delay equations arising in economics and population biology. State-dependent delays of threshold type arise naturally in structured population models (see [lo, 111). The motivation for this paper stems from consideration of the innocent-looking equation x’(t) = -kx(t - r(x(t))) (0.1) where r(x) is bell-shaped, e.g. r(x) = 01 emXZ + 1 - 01, 0 I 01 5 1. If k > 7r/2 and r(O) = 1, then the zero solution of (0.1) is unstable. A formal linearization would yield z’(t) = -kz(t - 1) which is known to be unstable for