Frequency synchronization and phase agreement in Kuramoto oscillator networks with delays

Authors:
Gerd S. Schmidt;Antonis Papachristodoulou;Ulrich MüNz;Frank AllgöWer
Affiliations:
Institute for Systems Theory and Automatic Control, University of Stuttgart, Pfaffenwaldring 9, 70550 Stuttgart, Germany;Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, United Kingdom;Institute for Systems Theory and Automatic Control, University of Stuttgart, Pfaffenwaldring 9, 70550 Stuttgart, Germany;Institute for Systems Theory and Automatic Control, University of Stuttgart, Pfaffenwaldring 9, 70550 Stuttgart, Germany
Venue:
Automatica (Journal of IFAC)
Year:
2012

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Abstract

We investigate frequency synchronization and phase agreement in networks of non-identical Kuramoto oscillators with delayed coupling. We provide delay-dependent lower bounds for the coupling gain so that a network with all-to-all coupling and constant heterogeneous delays achieves synchronization. The given bounds are determined explicitly by the system parameters. Furthermore, we show that frequency synchronization and phase agreement are possible in general directed networks of non-identical oscillators which contain a spanning tree if the delays satisfy a self-consistency condition. Networks with delays differ at this point notably from networks without delays, because phase agreement is not possible in undelayed non-identical Kuramoto oscillator networks. In addition to the sufficient conditions for synchronization, we investigate the connection between phase agreement and the existence of solutions of a necessary self-consistency condition.