Complete synchronisation of linear oscillator networks

Abstract

Oscillators with individual natural frequencies are not synchronisable by diffusive couplings, which are usually used for solving consensus and synchronisation problems. This article proves that, however, such oscillators can be synchronised by dynamical couplings if and only if the interconnection network includes local dynamical units with integrator dynamics. A necessary and sufficient synchronisation condition shows that oscillators with arbitrary positive parameters are synchronised by any coupling network that possesses a spanning tree and has sufficiently large coupling strength. Simulation results illustrate the results of this article.

Keywords:

• synchronisation
• oscillator network
• networked control system

Acknowledgements

The authors wishes to thank Dr Johannes Dastych (Ruhr-University Bochum) for his hint at the publication by Bilharz (1944).