Discrete Lyapunov functional for cyclic systems of differential equations with time-variable or state-dependent delay

TL;DR

This work develops a time-variable delay generalization of the discrete Lyapunov functional $V$ for unidirectional cyclic delay differential equations, extending the Mallet-Paret–Sell sign-change framework to nonautonomous and state-dependent delay settings. By reducing the nonlinear dynamics to a linear, positive-coefficient system, the authors establish monotonicity of $V$, criteria for its decrease, regularity results placing solution segments in the continuous regime $\mathcal{R}_{-\tau(t)}$, and finiteness of $V$ on backwards-bounded trajectories. The theory applies to two broad classes of state-dependent delays: threshold-type delays and implicit delays defined by two values, with conditions ensuring well-posedness and the applicability of the $V$-based analysis. These results pave the way for Poincaré–Bendixson-type conclusions and Morse decompositions for the global attractors of cyclic SDDEs, with concrete implications for delay-phenomenology in applications. Overall, the paper provides a rigorous framework to study oscillatory structure, attractors, and the global dynamics of cyclic systems with time-varying or state-dependent delays using a sign-change based Lyapunov functional.

Abstract

We consider nonautonomous cyclic systems of delay differential equations with variable delay. Under suitable feedback assumptions, we define an (integer valued) Lyapunov functional related to the number of sign changes of the coordinate functions of solutions. We prove that this functional possesses properties analogous to those established by Mallet-Paret and Sell for the constant delay case and by Krisztin and Arino for the scalar case. We also apply the results to equations with state-dependent delays.

Figures (1)

• Figure 3.1: Example for the choice of $\theta_i$, $i=0,\dots,k$. Here $N=10$, $\tau(t)=4$, $k=6$, and $n=3$.

Theorems & Definitions (26)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Remark 3.1
• Remark 3.2
• Theorem 3.3
• proof
• Theorem 3.4
• proof