Asymptotic analysis of differential-delay equations and nonuniqueness of periodic solutions

We study the differential-delay equation x′(t) = ?αf(x(t–1)), where α is a positive parameter and f is an odd function which decays like x^?r at infinity. In particular, we consider the case r ? 2, and prove the existence of periodic solutions with special symmetries which are different from previously known periodic solutions of minimal period 4. For r = 2 we prove sharp asymptotic estimates for the minimal periods of these solutions. Our results disprove a conjecture of R. D. Nussbaum.