On differential equations with state-dependent delay: The principles of linearized stability and instability revisited

• Main Author: Eugen Stumpf
• Format: Article
• Language: English
• Published: University of Szeged 2016-09-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: linearized stability; linearized instability; state-dependent delay; stability analysis; functional differential equation
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=5301
• ISSN: 1417-3875; 1417-3875

This paper deals with a dynamical systems approach for studying nonlinear autonomous differential equations with bounded state-dependent delay. Starting with the semiflow generated by solutions of such an equation, we revisit the principles of linearized stability and instability enabling the local stability analysis of equilibria via linearization. In particular, we prove both principles in an elementary way by using only a quantitative version of continuous dependence of the semiflow on initial data together with basic properties of the discrete semi-dynamical system induced by iterations of some time-$t$-map.