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

TL;DR

This paper introduces a Lyapunov functional for cyclic delay differential systems with variable delays, extending previous results to nonautonomous and state-dependent delay cases, and demonstrating its properties and applications.

Contribution

It defines a new Lyapunov functional for nonautonomous cyclic delay systems with variable and state-dependent delays, generalizing existing scalar delay results.

Findings

The Lyapunov functional measures sign changes of solutions.

Properties similar to constant delay case are established.

Applications to state-dependent delay equations are demonstrated.

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.