Skip to main content
SpringerLink
Log in

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

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

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 \(n\).

References

  • Abrams, D.M., Strogatz, S.H.: Chimera states in a ring of nonlocally coupled oscillators. Int. J. Bifurc. Chaos Appl. Sci. Eng. 16(1), 21–37 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Absil, P.-A., Kurdyka, K.: On the stable equilibrium points of gradient systems. Syst. Control Lett. 55(7), 573–577 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Alon, N., Spencer, J.H.: The Probabilistic Method, Wiley-Interscience Series in Discrete Mathematics and Optimization, 3rd edn. Wiley, Hoboken (2008). With an appendix on the life and work of Paul Erdős

  • Arnold, V.I., Afrajmovich, V.S., Ilyashenko, Yu.S., Shilnikov, L.P.: Bifurcation Theory and Catastrophe Theory. Springer, Berlin (1999). Translated from the 1986 Russian original by N.D. Kazarinoff, Reprint of the 1994 English edition from the series Encyclopaedia of Mathematical Sciences [ıt Dynamical systems. V, Encyclopaedia Math. Sci., 5, Springer, Berlin, 1994; MR1287421 (95c:58058)]

  • Biggs, N.: Algebraic Graph Theory, 2nd edn. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  • Billingsley, P.: Probability and Measure. Willey, London (1995)

    MATH  Google Scholar 

  • Borgs, C., Chayes, J., Lovász, L., Sós, V., Vesztergombi, K.: Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing. Adv. Math. 219(6), 1801–1851 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  • Bressloff, P.C.: Spatiotemporal dynamics of continuum neural fields. J. Phys. A 45(3), 033001, 109 (2012)

    Article  MathSciNet  Google Scholar 

  • Bronski, J.C., De Ville, L., Park, M.J.: Fully synchronous solutions and the synchronization phase transition for the finite-N Kuramoto model. Chaos 22, 033133 (2012)

    Article  MathSciNet  Google Scholar 

  • Chung, F.R.K.: Spectral Graph Theory. CBMS Regional Conference Series in Mathematics, vol. 92. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1997)

  • Chung, F., Radcliffe, M.: On the spectra of general random graphs. Electron. J. Combin. 18(1), 215–229 (2011)

    MathSciNet  Google Scholar 

  • Chung, F.R.K., Graham, R.L., Wilson, R.M.: Quasirandom graphs. Proc. Natl. Acad. Sci. U.S.A. 85(4), 969–970 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  • Dorfler, F., Bullo, F.: Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators. SICON 50(3), 1616–1642 (2012)

    Article  MathSciNet  Google Scholar 

  • Girnyk, T., Hasler, M., Maistrenko, Y.: Multistability of twisted states in non-locally coupled Kuramoto-type models. Chaos 22, 013114 (2012)

    Article  MathSciNet  Google Scholar 

  • Hartman, P.: Ordinary Differential Equations. Classics in Applied Mathematics, vol. 38. Society for Industrial and Applied Mathematics. SIAM, Philadelphia (2002). Corrected reprint of the second (1982) edition [Birkhäuser, Boston, MA; MR0658490 (83e:34002)], With a foreword by Peter Bates

  • Hirsch, M.W.: Differential Topology, Graduate Texts in Mathematics, vol. 33. Springer, New York (1994). Corrected reprint of the 1976 original

  • Hoppensteadt, F.C., Izhikevich, E.M.: Weakly Connected Neural Networks. Springer, Berlin (1997)

    Book  Google Scholar 

  • Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)

    MATH  Google Scholar 

  • Krebs, M., Shaheen, A.: Expander Families and Caley Graphs: A Beginner’s Guide. Oxford University Press, Oxford (2011)

    Google Scholar 

  • Krivelevich, M., Sudakov, B.: Pseudo-random Graphs, More Sets, Graphs and Numbers, Bolyai Society of Mathematical Studies, vol. 15, pp. 199–262. Springer, Berlin (2006)

  • Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Springer, Berlin (1984a)

    Book  MATH  Google Scholar 

  • Kuramoto, Y.: Cooperative dynamics of oscillator community. Progr. Theor. Phys. Suppl. 79, 223–240 (1984b)

  • Kuramoto, Y., Battogtokh, D.: Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlinear Phenom. Complex Syst. 5, 380–385 (2002)

    Google Scholar 

  • Lovász, L.: Large Networks and Graph Limits. AMS, Providence (2012)

    MATH  Google Scholar 

  • Lovász, L., Szegedy, B.: Limits of dense graph sequences. J. Combin. Theory Ser. B 96(6), 933–957 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Magnus, W., Karrass, A., Solitar, D.: Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations. Wiley, New York (1966)

  • Malkin, I.G.: Metody Lyapunova i Puankare v teorii nelineĭnyh kolebaniĭ. OGIZ, Moscow (1949)

  • Medvedev, G.S.: Stochastic stability of continuous time consensus protocols. SIAM J. Control Optim. 50(4), 1859–1885 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  • Medvedev, G.S.: The nonlinear heat equation on dense graphs and graph limits. SIAM J. Math. Anal. 46(4), 2743–2766 (2014a)

  • Medvedev, G.S.: The nonlinear heat equation on W-random graphs. Arch. Ration. Mech. Anal. 212(3), 781–803 (2014b)

  • Medvedev, G.S.: Small-world networks of Kuramoto oscillators. Phys. D 266, 13–22 (2014c)

  • Medvedev, G.S., Zhuravytska, S.: The geometry of spontaneous spiking in neuronal networks. J. Nonlinear Sci. 22, 689–725 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  • Mirollo, R.E., Strogatz, S.H.: The spectrum of the locked state for the Kuramoto model of coupled oscillators. Phys. D 205(1–4), 249–266 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Omelchenko, O.E.: Coherence–incoherence patterns in a ring of non-locally coupled phase oscillators. Nonlinearity 26(9), 2469 (2013)

    Article  MathSciNet  Google Scholar 

  • Omelchenko, O.E., Wolfrum, M., Laing, C.R.: Partially coherent twisted states in arrays of coupled phase oscillators. Chaos Interdiscip. J. Nonlinear Sci. 24, 023102 (2014)

  • Terras, A.: Fourier Analysis on Finite Groups and Applications, London Mathematical Society Student Texts, vol. 43. Cambridge University Press, Cambridge (1999)

  • Thomason, A.: Pseudorandom graphs, Random Graphs ’85 (Poznań, 1985), North-Holland Mathematics Studies, vol. 144, pp 307–331. North-Holland, Amsterdam (1987)

  • Watanabe, S., Strogatz, S.H.: Constants of motion for superconducting Josephson arrays. Phys. D 74(34), 197–253 (1994)

    Article  MATH  Google Scholar 

  • Wiley, D.A., Strogatz, S.H., Girvan, M.: The size of the sync basin. Chaos 16(1), 015103, 8 (2006)

    Article  MathSciNet  Google Scholar 

  • Wolfrum, M., Omel’chenko, O.E., Yanchuk, S., Maistrenko, Y.: Spectral properties of chimera states. Chaos 21, 013112 (2011)

    Article  MathSciNet  Google Scholar 

  • Xie, J., Knobloch, E., Kao, H.-C.: Multi-cluster and traveling chimera states in nonlocal phase-coupled oscillators, preprint (2014)

Download references

Acknowledgments

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

Author information

Authors and Affiliations

  1. Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, PA, 19104, USA

    Georgi S. Medvedev & Xuezhi Tang

Authors
  1. Georgi S. Medvedev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xuezhi Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Georgi S. Medvedev.

Additional information

Communicated by Paul Newton.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00332-015-9252-y

Keywords

Mathematics Subject Classification

Search

Navigation