Skip to main content
Log in

Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators

  • Articles
  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

A Lyapunov function for the phase-locked state of the Kuramoto model of non-linearly coupled oscillators is presented. It is also valid for finite-range interactions and allows the introduction of thermodynamic formalism such as ground states and universality classes. For the Kuramoto model, a minimum of the Lyapunov function corresponds to a ground state of a system with frustration: the interaction between the oscillators,XY spins, is ferromagnetic, whereas the random frequencies induce random fields which try to break the ferromagnetic order, i.e., global phase locking. The ensuing arguments imply asymptotic stability of the phase-locked state (up to degeneracy) and hold for any probability distribution of the frequencies. Special attention is given to discrete distribution functions. We argue that in this case a perfect locking on each of the sublattices which correspond to the frequencies results, but that a partial locking of some but not all sublattices is not to be expected. The order parameter of the phase-locked state is shown to have a strictly positive lower bound (r ⩾ 1/2), so that a continuous transition to a nonlocked state with vanishing order parameter is to be excluded.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. B. van der Pol,London, Edinburgh, Dublin Phil. Mag. J. Sci. 3:65 (1927).

    Google Scholar 

  2. Y. Kuramoto, inInternational Symposium on Mathematical Problems in Theoretical Physics, H. Araki, ed. (Springer, Berlin, 1975), p. 420.

    Google Scholar 

  3. Y. Kuramoto,Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, 1984), pp. 68–77.

    Google Scholar 

  4. Y. Kuramoto and I. Nishikawa,J. Stat. Phys. 49:569 (1987).

    Google Scholar 

  5. Y. Kuramoto,Prog. Theor. Phys. Suppl. 79:223 (1984).

    Google Scholar 

  6. G. B. Ermentrout,J. Math. Biol. 22:1 (1985).

    Google Scholar 

  7. G. B. Ermentrout,Physica D 41:219 (1990).

    Google Scholar 

  8. G. B. Ermentrout and N. Kopell,SIAM J. Appl. Math. 50:125(1990), and references therein.

    Google Scholar 

  9. S. H. Strogatz and R. E. Mirollo,J. Phys. A: Math. Gen. 21:L699 (1988).

    Google Scholar 

  10. P. C. Matthews and S. H. Strogatz,Phys. Rev. Lett. 65:701 (1990).

    Google Scholar 

  11. R. E. Mirollo and S. H. Strogatz,SIAM J. Appl. Math. 50:108, 1645 (1990).

    Google Scholar 

  12. S. H. Strogatz and R. E. Mirollo,J. Stat. Phys. 63:613 (1991).

    Google Scholar 

  13. S. H. Strogatz, R. E. Mirollo, and P. C. Matthews,Phys. Rev. Lett. 68:2730 (1992).

    Google Scholar 

  14. J. Lamperti,Probability (Benjamin, New York, 1966), Sections 7 and 8.

    Google Scholar 

  15. M. W. Hirsch and S. Smale,Differential Equations, Dynamical Systems, and Linear Algebra (Academic Press, New York, 1974), Chapter 9.

    Google Scholar 

  16. J. L. van Hemmen, D. Grensing, A. Huber, and R. Kühn,Z. Phys. 65:53 (1986);J. Stat. Phys. 50:231 (1988); J. L. van Hemmen and R. Kühn,Phys. Rev. Lett. 57:913 (1986).

    Google Scholar 

  17. W. Rudin,Real and Complex Analysis (McGraw-Hill, New York, 1974), p. 63.

    Google Scholar 

  18. A. M. Ostrowski,Solutions of Equations in Euclidean and Banach Spaces (Academic Press, New York, 1973), Appendixes A and K.

    Google Scholar 

  19. J. Stoer and R. Bulirsch,Introduction to Numerical Analysis (Springer, Berlin, 1980).

    Google Scholar 

  20. G. Iooss and D. D. Joseph,Elementary Stability and Bifurcation Theory (Springer, Berlin, 1980).

    Google Scholar 

  21. R. Bellman,Introduction to Matrix Analysis, 2nd ed. (McGraw-Hill, New York, 1970), Section 12.5.

    Google Scholar 

  22. E. Niebur, D. M. Kammen, C. Koch, D. Ruderman, and H. G. Schuster, inNeural Information Processing Systems 3, R. P. Lippman, J. E. Moody, and D. S. Touretzky, eds. (Morgan Kaufmann, San Mateo, California, 1991), pp. 123–129.

    Google Scholar 

  23. H. Sakaguchi, S. Shinomoto, and Y. Kuramoto,Prog. Theor. Phys. 77:1005 (1987).

    Google Scholar 

  24. H. Sakaguchi, S. Shinomoto, and Y. Kuramoto,Prog. Theor. Phys. 79:1069 (1988).

    Google Scholar 

  25. M. Aizenman and J. Wehr,Commun. Math. Phys. 130:498 (1990).

    Google Scholar 

  26. H. Sompolinsky, D. Golomb, and D. Kleinfeld,Proc. Natl. Acad. Sci. USA 87:7200 (1990);Phys. Rev. A 43:6990 (1991).

    Google Scholar 

  27. R. Eckhorn, R. Bauer, W. Jordan, M. Brosch, W. Kruse, M. Munk, and H. J. Reitboeck,Biol. Cybernet. 60:121 (1988).

    Google Scholar 

  28. C. M. Gray and W. Singer,Proc. Natl. Acad. Sci. USA 86:1698 (1989).

    Google Scholar 

  29. D. O. Hebb,The Organization of Behavior (Wiley, New York, 1949), p. 62.

    Google Scholar 

  30. W. Gerstner and J. L. van Hemmen,Network 3:139 (1992).

    Google Scholar 

  31. W. Gerstner, R. Ritz, and J. L. van Hemmen,Biol. Cybernet. 68:363 (1993), and to be published.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

van Hemmen, J.L., Wreszinski, W.F. Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators. J Stat Phys 72, 145–166 (1993). https://doi.org/10.1007/BF01048044

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01048044

Key words

Navigation