Skip to main content

Chimeras Unfolded

Abstract

The instability of mixing in the Kuramoto model of coupled phase oscillators is the key to understanding a range of spatiotemporal patterns, which feature prominently in collective dynamics of systems ranging from neuronal networks, to coupled lasers, to power grids. In this paper, we describe a codimension–2 bifurcation of mixing whose unfolding, in addition to the classical scenario of the onset of synchronization, also explains the formation of clusters and chimeras. We use a combination of linear stability analysis and Penrose diagrams to identify and analyze a variety of spatiotemporal patterns including stationary and traveling coherent clusters and twisted states, as well as their combinations with regions of incoherent behavior called chimera states. Penrose diagrams are used to locate the bifurcation of mixing and to determine its type. The linear stability analysis, on the other hand, yields the velocity distribution of the pattern emerging from the bifurcation. Furthermore, we show that network topology can endow chimera states with nontrivial spatial organization. In particular, we present twisted chimera states, whose coherent regions are organized as stationary or traveling twisted states. The analytical results are illustrated with numerical bifurcation diagrams computed for the Kuramoto model with uni-, bi-, and trimodal frequency distributions and all-to-all and nonlocal nearest-neighbor connectivity.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Data Availability Statement

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Notes

  1. From this point on, we explicitly indicate the dependence of \(\mathcal {C},\) x,  and P on \(\mu \).

  2. The Penrose diagram similar to that in Fig. 3e was discussed in [11], but the connection to the AH bifurcation was not made.

  3. Twisted states are also commonly referred to as splay states (cf. [41]).

References

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

    MathSciNet  MATH  Article  Google Scholar 

  2. Abrams, D.M., Mirollo, R., Strogatz, S.H., Wiley, D.A.: Solvable model for chimera states of coupled oscillators. Phys. Rev. Lett. 101, 084103 (2008)

    ADS  Article  Google Scholar 

  3. Chiba, H.: A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model. Ergodic Theory Dynam. Syst. 35(3), 762–834 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  4. Chiba, H.: A spectral theory of linear operators on rigged Hilbert spaces under analyticity conditions. Adv. Math. 273, 324–379 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  5. Chiba, H.: A Hopf bifurcation in the Kuramoto-Daido model (2016). arXiv:1610.02834

  6. Chiba, H., Medvedev, G.S.: The mean field analysis of the Kuramoto model on graphs I. The mean field equation and transition point formulas. Discret. Contin. Dyn. Syst. 39(1), 131–155 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  7. Chiba, H., Medvedev, G.S.: The mean field analysis of the Kuramoto model on graphs II. Asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. Discret. Contin. Dyn. Syst. 39(7), 3897–3921 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  8. Chiba, H., Medvedev, G.S., Mizuhara, M.S.: Bifurcations in the Kuramoto model on graphs. Chaos 28(7), 073109, 10 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  9. Chiba, H., Medvedev, G.S., Mizuhara, M.S.: Instability of mixing in the Kuramoto model: from bifurcations to patterns (2020). arXiv:2009.00103

  10. Chiba, H., Medvedev, G.S., Mizuhara, M.S.: Bifurcations and patterns in the Kuramoto model with inertia (2021)

  11. Dietert, H.: Stability and bifurcation for the Kuramoto model. J. Math. Pures Appl. (9) 105(4), 451–489 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  12. Dietert, H., Fernandez, B.: The mathematics of asymptotic stability in the Kuramoto model. Proceedings A 474(2220), 20180467, 20 (2018)

    MathSciNet  MATH  Google Scholar 

  13. Dudkowski, D., Maistrenko, Y., Kapitaniak, T.: Occurrence and stability of chimera states in coupled externally excited oscillators. Chaos 26(11), 116306, 9 (2016)

    MathSciNet  Article  Google Scholar 

  14. Eidelman, Y., Milman, V., Tsolomitis, A.: Functional Analysis, Graduate Studies in Mathematics, vol. 66. American Mathematical Society, Providence, RI (2004)

    MATH  Google Scholar 

  15. Georgi, S.: Medvedev, The continuum limit of the Kuramoto model on sparse random graphs. Commun. Math. Sci. 17(4), 883–898 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  16. John David Crawford: Amplitude expansions for instabilities in populations of globally-coupled oscillators. J. Stat. Phys. 74(5–6), 1047–1084 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  17. Kaliuzhnyi-Verbovetskyi, D., Medvedev, G.S.: The mean field equation for the Kuramoto model on graph sequences with non-Lipschitz limit. SIAM J. Math. Anal. 50(3), 2441–2465 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  18. Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Springer Series in Synergetics, vol. 19. Springer, Berlin (1984)

    MATH  Book  Google Scholar 

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

    Google Scholar 

  20. Laing, R.C.: The dynamics of chimera states in heterogeneous Kuramoto networks. Physics D 238(16), 1569–1588 (2009)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  21. Lancellotti, C.: On the Vlasov limit for systems of nonlinearly coupled oscillators without noise. Transport Theory Stat. Phys. 34(7), 523–535 (2005)

    ADS  MathSciNet  MATH  Article  Google Scholar 

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

    MATH  Book  Google Scholar 

  23. Martens, E.A., Barreto, E., Strogatz, S.H., Ott, E., So, P., Antonsen, T.M.: Exact results for the Kuramoto model with a bimodal frequency distribution. Phys. Rev. E (3) 79(2), 026204, 11 (2009)

    ADS  MathSciNet  Article  Google Scholar 

  24. Medvedev, G.S., Mizuhara, M.S.: Stability of clusters in the second-order Kuramoto model on random graphs. J. Stat. Phys. 182(2), 30 (2021)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  25. Mirollo, R., Strogatz, S.H.: The spectrum of the partially locked state for the Kuramoto model. J. Nonlinear Sci. 17(4), 309–347 (2007)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  26. Nkomo, S., Tinsley, M.R., Showalter, K.: Chimera and chimera-like states in populations of nonlocally coupled homogeneous and heterogeneous chemical oscillators. Chaos 26(9), 094826, 10 (2016)

    MathSciNet  Article  Google Scholar 

  27. Omel’chenko, O.E.: Coherence-incoherence patterns in a ring of non-locally coupled phase oscillators. Nonlinearity 26(9), 2469 (2013)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  28. Omel’chenko, O.E.: The mathematics behind chimera states. Nonlinearity 31(5), R121–R164 (2018)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  29. Omel’chenko, O.E.: Traveling chimera states. J. Phys. A 52(10), 104001 (2019)

    ADS  MathSciNet  Article  Google Scholar 

  30. Omel’chenko, O.E., Maistrenko, Y.L., Tass, P.A.: Chimera states: the natural link between coherence and incoherence. Phys. Rev. Lett. 100, 044105 (2008)

    ADS  Article  Google Scholar 

  31. Omel’chenko, O.E., Wolfrum, M., Maistrenko, Y.L.: Chimera states as chaotic spatiotemporal patterns. Phys. Rev. E 81(6), 065201 (2010)

    ADS  MathSciNet  Article  Google Scholar 

  32. Omel’chenko, O.E., Wolfrum, M., Laing, C.R.: Partially coherent twisted states in arrays of coupled phase oscillators. Chaos 24(2), 023102 (2014)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  33. Ott, E., Antonsen, T.M.: Low dimensional behavior of large systems of globally coupled oscillators. Chaos 18(3), 037113, 6 (2008)

    MathSciNet  MATH  Google Scholar 

  34. Panaggio, J.M., Abrams, D.M.: Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators. Nonlinearity 28(3), R67–R87 (2015)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  35. Penrose, O.: Electrostatic instabilities of a uniform non-Maxwellian plasma. Phys. Fluids 3(2), 258–265 (1960)

    ADS  MATH  Article  Google Scholar 

  36. Rodrigues, F.A., Peron, T.K.D.M., Ji, P., Kurths, J.: The Kuramoto model in complex networks. Phys. Rep. 610, 1–98 (2016)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  37. Schmidt, L., Schönleber, K., Krischer, K., García-Morales, V.: Coexistence of synchrony and incoherence in oscillatory media under nonlinear global coupling. Chaos 24(1), 013102, 7 (2014)

    MathSciNet  Article  Google Scholar 

  38. Simon, B.: Basic Complex Analysis, A Comprehensive Course in Analysis, Part 2A. American Mathematical Society, Providence, RI (2015)

    MATH  Google Scholar 

  39. Strogatz, H.S.: From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physics D 143(1–4), 1–20 (2000)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  40. Strogatz, S.H., Mirollo, R.E.: Stability of incoherence in a population of coupled oscillators. J. Stat. Phys. 63(3–4), 613–635 (1991)

    ADS  MathSciNet  Article  Google Scholar 

  41. Xie, J., Knobloch, E., Kao, H.-C.: Multicluster and traveling chimera states in nonlocal phase-coupled oscillators. Phys. Rev. E 90, 022919 (2014)

    ADS  Article  Google Scholar 

Download references

Acknowledgements

The authors thank Oleh Omelchenko for illuminating discussions on chimera states. This work was supported in part by NSF grant DMS 2009233 (to GSM). Numerical simulations were completed using the high performance computing cluster (ELSA) at the School of Science, The College of New Jersey. Funding of ELSA is provided in part by NSF OAC-1828163. MSM was additionally supported by a Support of Scholarly Activities Grant at The College of New Jersey.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Georgi S. Medvedev.

Additional information

Communicated by Shin-ichi Sasa.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Medvedev, G.S., Mizuhara, M.S. Chimeras Unfolded. J Stat Phys 186, 46 (2022). https://doi.org/10.1007/s10955-022-02881-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10955-022-02881-y