Local stability analysis of differential equations with state-dependent delay

TL;DR

This paper investigates the local stability of differential equations with state-dependent delays, extending classical stability principles and introducing a reduction method applicable when linearization eigenvalues lie on the imaginary axis.

Contribution

It introduces an analog of the Pliss reduction principle for differential equations with state-dependent delays, enhancing stability analysis methods.

Findings

Extended classical stability principles to state-dependent delay equations.

Derived a reduction principle for cases with eigenvalues on the imaginary axis.

Provided criteria for local stability without eigenvalues with positive real parts.

Abstract

In the present article, we discuss some aspects of the local stability analysis for a class of abstract functional differential equations. This is done under smoothness assumptions which are often satisfied in the presence of a state-dependent delay. Apart from recapitulating the two classical principles of linearized stability and instability, we deduce the analogon of the Pliss reduction principle for the class of differential equations under consideration. This reduction principle enables to determine the local stability properties of a solution in the situation where the linearization does not have any eigenvalues with positive real part but at least one eigenvalue on the imaginary axis.