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

Abstract

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.

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Notes

  1. For convenience, we consider complete graphs with odd number of vertices, so that they can be interpreted as Cayley graphs.

  2. One zero eigenvalue is always present due to the translational invariance of twisted states.

  3. Note, however, that any finite abelian group is isomorphic to a direct product of cyclic groups. This can be used to extend the linear stability analysis of this section to the Kuramoto model on Cayley graphs of abelian groups (cf. Terras 1999, Chapter 10).

  4. Whenever we refer to \(P_n\), we assume implicitly that \(n\) is a prime and \(n=1\pmod 4\).

  5. The cut norm of graphon \(W\in {\mathcal {W}}_0\) is defined by

    $$\begin{aligned} \Vert W\Vert _\square =\sup _{S,T\in {\mathcal {L}}_I} \left| \int _{S\times T} W(x,y) dxdy\right| \end{aligned}$$

    where \({\mathcal {L}}_I\) stands for the set of all Lebesgue measurable subsets of \(I\).

  6. Note, however, that the positive eigenvalues in the spectrum of the linearized problem in this case are all \(o(1)\). Thus, the instability is rather weak for large \(n\).

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Acknowledgments

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

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Correspondence to Georgi S. Medvedev.

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Communicated by Paul Newton.

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Medvedev, G.S., Tang, X. Stability of Twisted States in the Kuramoto Model on Cayley and Random Graphs. J Nonlinear Sci 25, 1169–1208 (2015). https://doi.org/10.1007/s00332-015-9252-y

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Keywords

  • Kuramoto model
  • Twisted state
  • Synchronization
  • Quasirandom graph
  • Cayley graph
  • Paley graph

Mathematics Subject Classification

  • 34C15
  • 45J05
  • 45L05
  • 05C90