# Generalised Lyapunov-Razumikhin techniques for scalar state-dependent delay differential equations

## TL;DR

This paper introduces generalized Lyapunov-Razumikhin methods for analyzing the stability of scalar delay differential equations, including state-dependent delays, providing tools to establish stability or bounds on dynamics.

## Contribution

The paper extends Lyapunov-Razumikhin techniques to scalar delay differential equations with state-dependent delays, enabling stability analysis and bounds derivation for complex delay systems.

## Key findings

- Identified parameter regions for stability of the sawtooth delay equation.
- Derived bounds on periodic orbits when steady-state is unstable.
- Applicable to both constant and variable delay problems.

## Abstract

We present generalised Lyapunov-Razumikhin techniques for establishing global asymptotic stability of steady-state solutions of scalar delay differential equations. When global asymptotic stability cannot be established, the technique can be used to derive bounds on the persistent dynamics. The method is applicable to constant and variable delay problems, and we illustrate the method by applying it to the state-dependent delay differential equation known as the sawtooth equation, to find parameter regions for which the steady-state solution is globally asymptotically stable. We also establish bounds on the periodic orbits that arise when the steady-state is unstable. This technique can be readily extended to apply to other scalar delay differential equations with negative feedback.