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

Anatoli F. Ivanov, Bernhard Lani-Wayda

TL;DR

The paper addresses the existence and structure of periodic solutions and a two‑dimensional invariant manifold for $(N{+}1)$‑dimensional delay systems with cyclic monotone negative feedback. It combines Mallet‑Paret–Sell Poincaré–Bendixson theory with Walther’s invariant‑manifold approach via a discrete Lyapunov functional, producing a global graph description of a strong unstable manifold whose forward extension is bounded by a single periodic orbit. A key result is that the closure of this manifold has the unique periodic orbit as its boundary, with the orbit lying inside the slowly oscillating set and attracting all nonzero forward trajectories on the manifold; the projection yields a planar, Jordan‑curve structure. The framework is then specialized to unidirectional coupling and applied to cyclic gene regulatory networks, including the repressilator, providing attractor location and oscillation/stability border analyses through explicit characteristic‑function criteria. Overall, the work extends scalar delay‑system insights to high‑dimensional cyclic networks, offering rigorous tools to analyze periodic behaviour in biological feedback loops and related delay systems.

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 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.

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

TL;DR

The paper addresses the existence and structure of periodic solutions and a two‑dimensional invariant manifold for ‑dimensional delay systems with cyclic monotone negative feedback. It combines Mallet‑Paret–Sell Poincaré–Bendixson theory with Walther’s invariant‑manifold approach via a discrete Lyapunov functional, producing a global graph description of a strong unstable manifold whose forward extension is bounded by a single periodic orbit. A key result is that the closure of this manifold has the unique periodic orbit as its boundary, with the orbit lying inside the slowly oscillating set and attracting all nonzero forward trajectories on the manifold; the projection yields a planar, Jordan‑curve structure. The framework is then specialized to unidirectional coupling and applied to cyclic gene regulatory networks, including the repressilator, providing attractor location and oscillation/stability border analyses through explicit characteristic‑function criteria. Overall, the work extends scalar delay‑system insights to high‑dimensional cyclic networks, offering rigorous tools to analyze periodic behaviour in biological feedback loops and related delay systems.

Abstract

For a class of -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 . 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.
Paper Structure (16 sections, 22 theorems, 131 equations, 1 figure)

This paper contains 16 sections, 22 theorems, 131 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Cyclic monotone negative feedback systems, linearization and discrete Lyapunov functional
  3. Cyclic monotone feedback
  4. Associated linear systems
  5. State space and solutions of linear and nonlinear systems.
  6. Attractor of the semiflow
  7. Discrete Lyapunov functional and eigenspace structure
  8. Linearization and local dynamics
  9. A two-dimensional invariant manifold and periodic solutions
  10. The forward extension $W$ of $W^{\mathrm{uu}}(\bar{r})$.
  11. The main theorem
  12. Attractor location, stability border and oscillation border for unidirectional coupling
  13. Approximate location of the attractor and of the manifold $W$ from Theorem \ref{['PerSolMfd']}.
  14. Oscillation border and stability border
  15. Cyclic gene regulatory networks
  16. ...and 1 more sections

Key Result

Lemma 2.1

Assume $n \in \mathbb N, \; t_0 \in \mathbb R$ and $A, \tilde{A}, B, \tilde{B} \in C^0([t_0, t_0 + \tau], \mathbb R^{n \times n})$, and consider the solutions $x,y$ of with the same initial value $x(t_0 + \cdot)\raisebox{-1.2ex}{$| \raisebox{-0.3ex}{$[- \tau, 0]$}$} = y(t_0 + \cdot)\raisebox{-1.2ex}{$| \raisebox{-0.3ex}{$[- \tau, 0]$}$} = \varphi \in C^0([-\tau, 0], \mathbb R^n)$. There exist p

Figures (1)

  • Figure 1: The argument of classical Poincaré-Bendixson-type (the sets $R_j$ and $M$ indicated)

Theorems & Definitions (48)

  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Remark 2.3
  • Corollary 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 38 more