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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
Article

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.

Keywords

Kuramoto model Twisted state Synchronization   Quasirandom graph Cayley graph Paley graph 

Mathematics Subject Classification

34C15 45J05 45L05 05C90 

Notes

Acknowledgments

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

References

  1. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  2. Absil, P.-A., Kurdyka, K.: On the stable equilibrium points of gradient systems. Syst. Control Lett. 55(7), 573–577 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 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 ErdősGoogle Scholar
  4. 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)]Google Scholar
  5. Biggs, N.: Algebraic Graph Theory, 2nd edn. Cambridge University Press, Cambridge (1993)Google Scholar
  6. Billingsley, P.: Probability and Measure. Willey, London (1995)zbMATHGoogle Scholar
  7. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  8. Bressloff, P.C.: Spatiotemporal dynamics of continuum neural fields. J. Phys. A 45(3), 033001, 109 (2012)MathSciNetCrossRefGoogle Scholar
  9. 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)MathSciNetCrossRefGoogle Scholar
  10. 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)Google Scholar
  11. Chung, F., Radcliffe, M.: On the spectra of general random graphs. Electron. J. Combin. 18(1), 215–229 (2011)MathSciNetGoogle Scholar
  12. Chung, F.R.K., Graham, R.L., Wilson, R.M.: Quasirandom graphs. Proc. Natl. Acad. Sci. U.S.A. 85(4), 969–970 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  13. Dorfler, F., Bullo, F.: Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators. SICON 50(3), 1616–1642 (2012)MathSciNetCrossRefGoogle Scholar
  14. Girnyk, T., Hasler, M., Maistrenko, Y.: Multistability of twisted states in non-locally coupled Kuramoto-type models. Chaos 22, 013114 (2012)MathSciNetCrossRefGoogle Scholar
  15. 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 BatesGoogle Scholar
  16. Hirsch, M.W.: Differential Topology, Graduate Texts in Mathematics, vol. 33. Springer, New York (1994). Corrected reprint of the 1976 originalGoogle Scholar
  17. Hoppensteadt, F.C., Izhikevich, E.M.: Weakly Connected Neural Networks. Springer, Berlin (1997)CrossRefGoogle Scholar
  18. Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)zbMATHGoogle Scholar
  19. Krebs, M., Shaheen, A.: Expander Families and Caley Graphs: A Beginner’s Guide. Oxford University Press, Oxford (2011)Google Scholar
  20. 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)Google Scholar
  21. Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Springer, Berlin (1984a)zbMATHCrossRefGoogle Scholar
  22. Kuramoto, Y.: Cooperative dynamics of oscillator community. Progr. Theor. Phys. Suppl. 79, 223–240 (1984b)Google Scholar
  23. Kuramoto, Y., Battogtokh, D.: Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlinear Phenom. Complex Syst. 5, 380–385 (2002)Google Scholar
  24. Lovász, L.: Large Networks and Graph Limits. AMS, Providence (2012)zbMATHGoogle Scholar
  25. Lovász, L., Szegedy, B.: Limits of dense graph sequences. J. Combin. Theory Ser. B 96(6), 933–957 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  26. Magnus, W., Karrass, A., Solitar, D.: Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations. Wiley, New York (1966)Google Scholar
  27. Malkin, I.G.: Metody Lyapunova i Puankare v teorii nelineĭnyh kolebaniĭ. OGIZ, Moscow (1949)Google Scholar
  28. Medvedev, G.S.: Stochastic stability of continuous time consensus protocols. SIAM J. Control Optim. 50(4), 1859–1885 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  29. Medvedev, G.S.: The nonlinear heat equation on dense graphs and graph limits. SIAM J. Math. Anal. 46(4), 2743–2766 (2014a)Google Scholar
  30. Medvedev, G.S.: The nonlinear heat equation on W-random graphs. Arch. Ration. Mech. Anal. 212(3), 781–803 (2014b)Google Scholar
  31. Medvedev, G.S.: Small-world networks of Kuramoto oscillators. Phys. D 266, 13–22 (2014c)Google Scholar
  32. Medvedev, G.S., Zhuravytska, S.: The geometry of spontaneous spiking in neuronal networks. J. Nonlinear Sci. 22, 689–725 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  33. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  34. Omelchenko, O.E.: Coherence–incoherence patterns in a ring of non-locally coupled phase oscillators. Nonlinearity 26(9), 2469 (2013)MathSciNetCrossRefGoogle Scholar
  35. 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)Google Scholar
  36. Terras, A.: Fourier Analysis on Finite Groups and Applications, London Mathematical Society Student Texts, vol. 43. Cambridge University Press, Cambridge (1999)Google Scholar
  37. Thomason, A.: Pseudorandom graphs, Random Graphs ’85 (Poznań, 1985), North-Holland Mathematics Studies, vol. 144, pp 307–331. North-Holland, Amsterdam (1987)Google Scholar
  38. Watanabe, S., Strogatz, S.H.: Constants of motion for superconducting Josephson arrays. Phys. D 74(34), 197–253 (1994)zbMATHCrossRefGoogle Scholar
  39. Wiley, D.A., Strogatz, S.H., Girvan, M.: The size of the sync basin. Chaos 16(1), 015103, 8 (2006)MathSciNetCrossRefGoogle Scholar
  40. Wolfrum, M., Omel’chenko, O.E., Yanchuk, S., Maistrenko, Y.: Spectral properties of chimera states. Chaos 21, 013112 (2011)MathSciNetCrossRefGoogle Scholar
  41. Xie, J., Knobloch, E., Kao, H.-C.: Multi-cluster and traveling chimera states in nonlocal phase-coupled oscillators, preprint (2014)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsDrexel UniversityPhiladelphiaUSA

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