Table of Contents
ISRN Mathematical Analysis
Volume 2013 (2013), Article ID 816353, 9 pages
Research Article

Time-Delayed Interactions in Networks of Self-Adapting Hopf Oscillators

1Fakultät für Physik, Universität Bielefeld, 33501 Bielefeld, Germany
2STI/IMT/LPM, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland

Received 30 August 2012; Accepted 24 September 2012

Academic Editors: A. Carpio, J. V. Stokman, and C. Zhu

Copyright © 2013 Julio Rodriguez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A network of coupled limit cycle oscillators with delayed interactions is considered. The parameters characterizing the oscillator’s frequency and limit cycle are allowed to self-adapt. Adaptation is due to time-delayed state variables that mutually interact via a network. The self-adaptive mechanisms ultimately drive all coupled oscillators to a consensual cyclostationary state, where the values of the parameters are identical for all local systems. They are analytically expressible. The interplay between the spectral properties of the coupling matrix and the time delays determines the conditions for which convergence towards a consensual state takes place. Once reached, this consensual state subsists even if interactions are removed. In our class of models, the consensual values of the parameters depend neither on the delays nor on the network’s topologies.