Abstract
The nonlinear Kuramoto equations for n coupled oscillators are derived and studied. The oscillators are defined to be synchronized when they oscillate at the same frequency and their phases are all equal. A control-theoretic viewpoint reveals that synchronized states of Kuramoto oscillators are locally asymptotically stable if every oscillator is coupled to all others. The problem of synchronization in Kuramoto oscillators is closely related to rendezvous, consensus, and flocking problems in distributed control. These problems, with their elegant solution by graph theory, are discussed briefly.
This is a preview of subscription content, log in via an institution to check access.
Bibliography
Dörfler F, Bullo F (2011) On the critical coupling for Kuramoto oscillators. SIAM J Appl Dyn Syst 10(3):1070–1099
F. Dörfler and F. Bullo. Synchronization in Complex Networks of Phase Oscillators: A Survey. Automatica, 50(6), June 2014. To appear.
Feintuch A, Francis B (2012) Infinite chains of kinematic points. Automatica 48:901–908
Jadbabaie A, Lin J, Morse AS (2003) Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans Automatic Control 48(6):988–1001
Kuramoto Y (1975) Self-entrainment of a population of coupled nonlinear oscillators. In: Araki H (ed) Volume 39 of International symposium on mathematical problems in theoretical physics, Kyoto. Lecture Notes in Physics. Springer, p 420
Lin Z, Francis BA, Maggiore M (2007) State agreement for coupled nonlinear systems with time-varying interaction. SIAM J Control Optim 46:288–307
Moreau L (2005) Stability of multi-agent systems with time-dependent communication links. IEEE Trans Automatic Control 50:169–182
Pantaleone J. Webpage. http://salt.uaa.alaska.edu/jim/
Pantaleone J (2002) Synchronization of metronomes. Am J Phys 70(10):992–1000
Scardovi L, Sarlette A, Sepulchre R (2007) Synchronization and balancing on the N-torus. Syst Control Lett 56:335–341
Sepulchre R, Paley DA, Leonard NE (2007) Stabilization of planar collective motion: all-to-all communication. IEEE Trans Automatic Control 52(5):811–824
Sepulchre R, Paley DA, Leonard NE (2008) Stabilization of planar collective motion with limited communication. IEEE Trans Automatic Control 53(3):706–719
Strogatz SH (2000) From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143:1–20
Strogatz SH (2004) Sync: the emerging science of spontaneous order. Hyperion Books, New York
Strogatz SH, Stewart I (1993) Coupled oscillators and biological synchronization. Sci Am 269:102–109
Wieland P, Sepulchre R, Allgower F (2011) An internal model principle is necessary and sufficient for linear output synchronization. Automatica 47:1068–1074
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag London
About this entry
Cite this entry
Francis, B.A. (2014). Oscillator Synchronization. In: Baillieul, J., Samad, T. (eds) Encyclopedia of Systems and Control. Springer, London. https://doi.org/10.1007/978-1-4471-5102-9_216-1
Download citation
DOI: https://doi.org/10.1007/978-1-4471-5102-9_216-1
Received:
Accepted:
Published:
Publisher Name: Springer, London
Online ISBN: 978-1-4471-5102-9
eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering