Control by time delayed feedback near a Hopf bifurcation point

• Main Authors: Sjoerd Verduyn Lunel, Babette de Wolff
• Format: Article
• Language: English
• Published: University of Szeged 2017-12-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: pyragas control; time delayed feedback control; hopf bifurcation; neutral equations
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=6042

In this paper we study the stabilization of rotating waves using time delayed feedback control. It is our aim to put some recent results in a broader context by discussing two different methods to determine the stability of the target periodic orbit in the controlled system: 1) by directly studying the Floquet multipliers and 2) by use of the Hopf bifurcation theorem. We also propose an extension of the Pyragas control scheme for which the controlled system becomes a functional differential equation of neutral type. Using the observation that we are able to determine the direction of bifurcation by a relatively simple calculation of the root tendency, we find stability conditions for the periodic orbit as a solution of the neutral type equation.