Local stability analysis of differential equations with state-dependent delay

Eugen  Stumpf

doi:10.3934/dcds.2016.36.3445

Article Contents

2016, Volume 36, Issue 6: 3445-3461. Doi: 10.3934/dcds.2016.36.3445

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Local stability analysis of differential equations with state-dependent delay

Eugen Stumpf^1,

1.
Department of Mathematics, University of Hamburg, Bundesstrasse 55, 20146 Hamburg

Received: January 31, 2015

Revised: August 31, 2015

Published: May 2017

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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.

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Mathematics Subject Classification: Primary: 34K19, 34K20; Secondary: 34K25.

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