Time-delayed feedback control of breathing localized structures in a three-component reaction-diffusion system

Gurevich Svetlana V.

Abstract

Svetlana V. Gurevich⇑Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Strasse 9, 48149 Münster, Germany e-mail: gurevics{at}uni-muenster.deAbstract The dynamics of a single breathing localized structure in a three-component reaction{\textendash}diffusion system subjected to time-delayed feedback is investigated. It is shown that variation of the delay time and the feedback strength can lead either to stabilization of the breathing or to delay-induced periodic or quasi-periodic oscillations of the localized structure. A bifurcation analysis of the system in question is provided and an order parameter equation is derived that describes the dynamics of the localized structure in the vicinity of the Andronov{\textendash}Hopf bifurcation. With the aid of this equation, the boundaries of the stabilization domains as well as the dependence of the oscillation radius on delay parameters can be explicitly derived, providing a robust mechanism to control the behaviour of the breathing localized structure in a straightforward manner. reaction{\textendash}diffusion systemstime-delayed feedbackbifurcation analysislocalized structuresFootnotesOne contribution of 19 to a Theme Issue {\textquoteleft}Localized structures in dissipative media: from optics to plant ecology{\textquoteright}.{\textcopyright} 2014 The Author(s) Published by the Royal Society. All rights reserved.