J Dyn Diff Equat (2011) 23:727–790 DOI 10.1007/s10884-011-9225-2 Large-Amplitude Periodic Solutions for Differential Equations with Delayed Monotone Positive Feedback Tibor Krisztin · Gabriella Vas Received: 28 August 2010 / Published online: 1 September 2011 © Springer Science+Business Media, LLC 2011 Abstract The aim of this paper is to show that the structure of the global attractor for delayed monotone positive feedback can be more complicated than the union of spindle- like structures between consecutive stable equilibria with respect to the pointwise ordering. Large amplitude periodic orbits—in the sense that they are not between two consecutive stable equilibria—are constructed for nonlinearities close to a step function. For some non- linearities there are exactly two large amplitude periodic orbits. By describing the unstable sets of these periodic orbits, a complete picture is obtained about the global attractor outside the spindle-like structures. Keywords Delay differential equation · Positive feedback · Periodic orbit · Large amplitude · Discrete Lyapunov functional · Hyperbolicity · Return map · Heteroclinic orbit Mathematics Subject Classification (2000) 34K13 · 37D05 · 37L25 · 37L45 1 Introduction The delay differential equation ˙ x (t ) =-μx (t ) + f (x (t - 1)) (1.1) with μ ≥ 0 and smooth monotone nonlinearity f : R → R appears in several applications, see e.g. [6, 7, 11, 18, 31] and the references therein. T. Krisztin (B ) Bolyai Institute, University of Szeged, Szeged, Hungary e-mail: krisztin@math.u-szeged.hu G. Vas Analysis and Stochastic Research Group of the Hungarian Academy of Sciences, University of Szeged, Szeged, Hungary e-mail: vasg@math.u-szeged.hu 123