Journal of Nonlinear Science

, Volume 25, Issue 6, pp 1169–1208 | Cite as

Stability of Twisted States in the Kuramoto Model on Cayley and Random Graphs

  • Georgi S. MedvedevEmail author
  • Xuezhi Tang


The Kuramoto model of coupled phase oscillators on complete, Paley, and Erdős–Rényi (ER) graphs is analyzed in this work. As quasirandom graphs, the complete, Paley, and ER graphs share many structural properties. For instance, they exhibit the same asymptotics of the edge distributions, homomorphism densities, graph spectra, and have constant graph limits. Nonetheless, we show that the asymptotic behavior of solutions in the Kuramoto model on these graphs can be qualitatively different. Specifically, we identify twisted states, steady-state solutions of the Kuramoto model on complete and Paley graphs, which are stable for one family of graphs but not for the other. On the other hand, we show that the solutions of the initial value problems for the Kuramoto model on complete and random graphs remain close on finite time intervals, provided they start from close initial conditions and the graphs are sufficiently large. Therefore, the results of this paper elucidate the relation between the network structure and dynamics in coupled nonlinear dynamical systems. Furthermore, we present new results on synchronization and stability of twisted states for the Kuramoto model on Cayley and random graphs.


Kuramoto model Twisted state Synchronization   Quasirandom graph Cayley graph Paley graph 

Mathematics Subject Classification

34C15 45J05 45L05 05C90 



This work was supported in part by the NSF Grants DMS 1109367 and DMS 1412066 (to GSM).


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsDrexel UniversityPhiladelphiaUSA

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