THE STRONG UNSTABLE MANIFOLD AND PERIODIC SOLUTIONS IN DIFFERENTIAL DELAY EQUATIONS WITH CYCLIC MONOTONE NEGATIVE FEEDBACK

For a class of (N + 1)-dimensional systems of differential delay equations with a cyclic and monotone negative feedback structure, we construct a two-dimensional invariant manifold, on which phase curves spiral outward towards a bounding periodic orbit. For this to happen, we assume essentially only instability of the zero equilibrium. Methods of the Poincaré-Bendixson theory due to Mallet-Paret and Sell are combined with techniques used by Walther for the scalar case (N = 0). Statements on the attractor location and on parameter borders concerning stability and oscillation are included. The results apply to models for gene regulatory systems, e.g. the ‘repressilator’ system.