The strong unstable manifold and periodic solutions in differential delay equations with cyclic monotone negative feedbck

TL;DR

This paper constructs a two-dimensional invariant manifold in certain delay differential equations with cyclic negative feedback, showing phase curves spiral outward to a periodic orbit, with implications for gene regulatory models.

Contribution

It introduces a method to analyze the unstable manifold and periodic solutions in high-dimensional delay systems with cyclic monotone feedback, extending scalar case techniques.

Findings

Existence of a two-dimensional invariant manifold.

Spiral outward phase curves towards a periodic orbit.

Application to gene regulatory network models.

Abstract

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 Poincar\'e-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.